![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
Computation, Information, and the Arrow of Time 3
that only erases its input and we make a movie of its execution and play
it backward. What would we see? We see a machine creating information
out of nothing, just the same way the toddler in the reversed movie was
pulling neat French fries out of his mouth. So also in this case, if we reverse
the arrow of time, destruction of information becomes creation and vice
versa. In previous papers the first author has investigated the relation
between learning and data compression ([2, 4]). Here we are interested in
the converse problem: How do data-sets from which we can learn something
emerge in the world? What processes grow information?
There is a class of deterministic processes that discard or destroy in-
formation. Examples are: simple erasure of bits, (lossy) data compression,
and learning. There is another class of processes that seems to create infor-
mation: coin flipping, growth, evolution. In general, stochastic processes
create information, exactly because we are uncertain of their future, and
deterministic processes discard information, precisely because the future
of the process is known. The basic paradigm of a stochastic information
generating process is coin flipping. If we flip a coin in such a way that the
probability of heads is equal to the probability of tails, and we note the
results as a binary string, then with high probability this string is random
and incompressible. The string will then have maximal Kolmogorov com-
plexity, i.e. a program that generates the string on a computer will be at
least as long as the string itself ([8]). On the other hand if we generate a
string by means of a simple deterministic program (say “For x = 1 to k
print("1")”) then the string is highly compressible and by definition has
a low Kolmogorov complexity which approximates log k for large enough k.
In the light of these observations one could formulate the following research
question: Given the fact that creation and destruction of information seem
to be symmetrical over the time axis, could one develop a time-invariant
description of computational processes for which creation of information
is the same process as destruction of information with the time arrow re-
versed? A more concise version of the same question is: Are destruction
and creation of information computationally symmetrical in time? The
main part of this paper is dedicated to a positive answer to this question.
Prima facie it seems that we compute to get new information. So if
we want to know what the exact value of 10! is, then the answer 3628800
really contains information for us. It tells us something we did not know.
We also have the intuition, that the harder it is to compute a function, the
more value (i.e. information) the answer contains. So 10! in a way contains](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-16-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
4 P. Adriaans & P. van Emde Boas
more information than 102
. Yet from a mathematical point of view 10! and
3628800 are just different descriptions of the same number. The situation
becomes even more intriguing if we turn our intention to the simulation of
processes on a computer that really seem to create new information like
the growth of a tree, game playing, or the execution of a genetic algorithm.
What is happening here if computation cannot generate information? What
is the exact relation between information generating processes that we find
in our world and our abstract models of computation?
In most curricula, theories about information and computation are
treated in isolation. That is probably the reason why the rather funda-
mental question studied in this paper up till now has received little at-
tention in computer science: What is the interaction between information
and computation? Samson Abramsky has posed this question in a recent
publication with some urgency (without offering a definitive answer): We
compute in order to gain information, but how is this possible logically or
thermodynamically? How can it be reconciled with the point of view of In-
formation Theory? How does information increase appear in the various
extant theories? ([1], p. 487). Below we will formulate a partial answer
to this question by means of an analysis of time invariant descriptions of
computational processes.
1.2. A Formal Framework: Meta-computational Space
In order to study the interplay between entropy, information, and compu-
tation we need to develop a formal framework. For this purpose we develop
the notion of meta-computational space in this section: formally, the space
of the graphs of all possible computations of all possible Turing machines.
The physical equivalent would be the space of all possible histories of all
possible universes.
C(x) will be the classical Kolmogorov complexity of a binary string x,
i.e. the length of the shortest program p that computes x on a reference
universal Turing machine U. Given the correspondence between natural
numbers and binary strings, M consists of an enumeration of all possible
self-delimiting programs for a preselected arbitrary universal Turing ma-
chine U. Let x be an arbitrary bit string. The shortest program that
produces x on U is x∗
= argminM∈M(U(M) = x) and the Kolmogorov
complexity of x is C(x) = |x∗
|. The conditional Kolmogorov complexity of
a string x given a string y is C(x|y), this can be interpreted as the length
of a program for x given input y. A string is defined to be random if](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-17-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
Computation, Information, and the Arrow of Time 5
C(x) ≥ |x|. I(x) is the classical integer complexity function that assigns to
each integer x another integer C(x) [8].
We will follow the standard textbook of Hopcroft, Motwani, and Ullman
for the basic definitions ([7]). A Turing machine (TM) is described by a
7-tuple
M = (Q, Σ, Γ, δ, q0, B, F).
Here, as usual, Q is the finite set of states, Σ is the finite set of input symbols
with Σ ⊂ Γ, where Γ is the complete set of tape symbols, δ is a transition
function such that δ(q, X) = (p, Y, D), if it is defined, where p ∈ Q is the
next state, X ∈ Γ is the symbol read in the cell being scanned, Y ∈ Γ is
the symbol written in the cell being scanned, D ∈ {L, R} is the direction
of the move, either left or right, q0 ∈ Q is the start state, B ∈ Γ − Σ is the
blank default symbol on the tape, and F ⊂ Q is the set of accepting states.
A move of a TM is determined by the current content of the cell that is
scanned and the state the machine is in. It consists of three parts:
(1) Change state;
(2) Write a tape symbol in the current cell;
(3) Move the read-write head to the tape cell on the left or right.
A non-deterministic Turing machine (NTM) is equal to a deterministic
TM with the exception that the range of the transition function consists of
sets of triples:
δ(q, X) = {(p1, Y1, D1), (p2, Y2, D2), ..., (pk, Yk, Dk)}.
A TM is a reversible Turing machine (RTM) if the transition function
δ(q, X) = (p, Y, D) is one-to-one, with the additional constraint that the
movement D of the read-write head is uniquely determined by the target
state p.
Definition 1.1. An Instantaneous Description (ID) of a TM during its
execution is a string X1X2...Xi−1qXiXi+1...Xn in which q is the state
of the TM, the tape head is scanning the i-th head from the left,
X1X2...Xn is the portion of the tape between the leftmost and the rightmost
blank. Given an Instantaneous Description X1X2...Xi−1qXiXi+1...Xn it
will be useful to define an Extensional Instantaneous Description (EID)
X1X2...Xi−1XiXi+1...Xn, that only looks at the content of the tape and
ignores the internal state of the machine and an Intensional Instantaneous
Description (IID) qXiD, that only looks at the content of the current cell](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-18-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
Computation, Information, and the Arrow of Time 15
In computations that use counters, Program Complexity and Program
Counter Complexity are mixed up during the execution. In fact one can
characterize various types of computations by means of their “information
signature”. Informally, at extremes of the spectrum, one could distinguish:
• Pure Information Discarding Processes: in such processes the program
counter does not play any role. They reach an accepting state by means
of systematically reducing the input. Summation of a set of numbers,
or erasing of a string are examples.
• Pure Counting Processes: For x=1 to i write(1): The condi-
tional complexity of the tape configuration grows from 0 to log i and
then diminishes to 0 again.
• Pure Search Processes: In such processes the input is not reduced but
is kept available during the whole process. The information in the
program counter is used to explore the search space. Standard decision
procedures for NP-hard programs, where the checking function is tested
on an enumeration of all possible solutions, are an example.
A deeper analysis of various information signatures of computational pro-
cesses and their consequences for complexity theory is a subject of future
work.
1.5. Discussion
We can draw some conclusions and formulate some observations on the
basis of the analysis given above.
1) Erasing and creating information are indeed, as suggested in the
introduction, from a time invariant computational point of view the same
processes: The quasi-reversible machine that is associated with a simple de-
terministic machine that erases information is a non-deterministic machine
writing arbitrary bit-strings on the tape. This symmetry also implies that
creation of information in positive time involves destruction of information
in negative time.
2) The class of quasi-reversible machines indeed describes the class of
data-sets from which we can learn something in the following way: If L is
the language accepted by M then M−1
generates L. M−1
is an informer
for L in the sense of [6], every sentence in L will be non-deterministically
produced by M−1
in the limit. QRT M is the class of all informers for
type-0 languages.](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-28-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
16 P. Adriaans P. van Emde Boas
3) These insights suggests that we can describe stochastic processes in
the real world as deterministic processes in negative time: e.g. throwing a
dice in positive time is erasing information about its “future” in negative
time, the evolution of species in positive time could be described as the
“deterministic” computation of their ancestor in negative time. A necessary
condition for the description of such growth processes as computational
processes is that the number of bits that can be produced per time unit is
restricted. A stochastic interpretation of a QRTM can easily be developed
by assigning a set of probabilities to each split in the δ relation. The
resulting stochastic-QRTM is a sufficient statistic for the data-sets that are
generated.
4) The characterization of the class NP in terms of quasi-reversible com-
puting seems to be more moderate than the classical description in terms
of full non-deterministic computing. The full power of non-deterministic
computing is never realized in a system with only one time direction.
5) Processes like game playing and genetic algorithms seem to be meta-
computational processes in which non-deterministic processes (throwing a
dice, adding mutations) seem to be intertwined with deterministic phases
(making moves, checking the fitness function).
6) Time-symmetry has consequences for some philosophical positions.
The idea that the evolution of our universe can be described as a determin-
istic computational process has been proposed by several authors (Zuse,
Bostrom, Schmidthuber, Wolfram [10], Lloyd [9], etc.). Nowadays it is re-
ferred to as pancomputationalism [5]. If deterministic computation is an
information discarding process then it implies that the amount of informa-
tion in the universe rapidly decreases. This contradicts the second law of
thermodynamics. On the other hand, if the universe evolves in a quasi-
reversible way, selecting possible configurations according to some quasi-
reversible computational model, it computes the Big Bang in negative time.
The exact implications of these observations can only be explained by means
of the notion of facticity [3], but that is another discussion. The concept of
quasi-reversible computing seems to be relevant for these discussions [2].
1.6. Conclusion
Computing is moving through meta-computational space. For a fixed Tur-
ing machine Mi such movement is confined to one local infinite graph
IDMi , ⊢Mi . If Mi is deterministic then M−1
i is non-deterministic.
If M is information discarding then M−1
“creates” information. The two](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-29-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
Computation, Information, and the Arrow of Time 17
fundamental complexities involved in a deterministic computation are Pro-
gram Complexity and Program Counter Complexity. Programs can be
classified in terms of their “information signature” with pure counting pro-
grams and pure information discarding programs as two ends of the spec-
trum. The class NP is simply the class of polynomial deterministic time
calculations in negative time. Thinking in terms of meta-computational
space allows us to conceptualize computation as movement in a certain
space and is thus a source of new intuitions to study computation. Specif-
ically a deeper analysis of various information signatures of computational
(and other) processes is a promising subject for further study.
References
[1] S. Abramsky. Information, Processes and Games. In eds. P. W. Adriaans
and J. F. A. K. van Benthem, Handbook of the Philosophy of Information,
In Handbooks of the Philosophy of Science, series edited by D. M. Gabbay,
P. Thagard and J. Woods, pp. 483–550. Elsevier, (2008).
[2] P. W. Adriaans and J. F. A. K. van Benthem, eds., Handbook of the Phi-
losophy of Information. In Handbooks of the Philosophy of Science, series
edited by D. M. Gabbay, P. Thagard and J. Woods. Elsevier, (2008).
[3] P. W. Adriaans, Between order and chaos: The quest for meaningful infor-
mation, Theor. Comp. Sys. 45(4), (2009).
[4] P. W. Adriaans and P. Vitányi, Approximation of the two-part MDL code,
IEEE Transactions on Information Theory. 55(1), 444–457, (2009).
[5] L. Floridi. Trends in the philosophy of information. In eds. P. W. Adriaans
and J. F. A. K. van Benthem, Handbook of the Philosophy of Information,
In Handbooks of the Philosophy of Science, series edited by D. M. Gabbay,
P. Thagard and J. Woods, pp. 113–132. Elsevier, (2008).
[6] E. M. Gold, Language identification in the limit, Information and Control.
10(5), 447–474, (1967).
[7] J. E. Hopcroft, R. Motwani, and J. D. Ullman, Introduction to Automata
Theory, Languages, and Computation. Addison-Wesley, (2001), second edi-
tion.
[8] M. Li and P. Vitányi, An Introduction to Kolmogorov Complexity and its
Applications. Springer-Verlag, (2008), third edition.
[9] S. Lloyd, Ultimate physical limits to computation, Nature. 406, 1047–1054,
(2000).
[10] S. Wolfram, A New Kind of Science. Wolfram Media Inc., (2002).](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-30-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
Chapter 2
The Isomorphism Conjecture for NP
Manindra Agrawal ∗
Indian Institute of Technology
Kanpur, India
E-mail: manindra@iitk.ac.in
In this chapter, we survey the arguments and known results for and
against the Isomorphism Conjecture.
Contents
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Formulation and Early Investigations . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 A Counter Conjecture and Relativizations . . . . . . . . . . . . . . . . . . . . 26
2.5 The Conjectures for Other Classes . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 The Conjectures for Other Reducibilities . . . . . . . . . . . . . . . . . . . . . 30
2.6.1 Restricting the input head movement . . . . . . . . . . . . . . . . . . . 31
2.6.2 Reducing space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6.3 Reducing depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.7 A New Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.8 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.1. Introduction
The Isomorphism Conjecture for the class NP states that all polynomial-
time many-one complete sets for NP are polynomial-time isomorphic to each
other. It was made by Berman and Hartmanis [21]a
, inspired in part by
a corresponding result in computability theory for computably enumerable
sets [50], and in part by the observation that all the existing NP-complete
∗N Rama Rao Professor, Indian Institute of Technology, Kanpur. Research supported
by J C Bose Fellowship FLW/DST/CS/20060225.
aThe conjecture is also referred as Berman–Hartmanis Conjecture after the proposers.
19](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-32-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
20 M. Agrawal
sets known at the time were indeed polynomial-time isomorphic to each
other. This conjecture has attracted a lot of attention because it predicts
a very strong structure of the class of NP-complete sets, one of the funda-
mental classes in complexity theory.
After an initial period in which it was believed to be true, Joseph and
Young [40] raised serious doubts against the conjecture based on the notion
of one-way functions. This was followed by investigation of the conjecture
in relativized worlds [27, 33, 46] which, on the whole, also suggested that
the conjecture may be false. However, disproving the conjecture using one-
way functions, or proving it, remained very hard (either implies DP 6= NP).
Hence research progressed in three distinct directions from here.
The first direction was to investigate the conjecture for complete degrees
of classes bigger than NP. Partial results were obtained for classes EXP and
NEXP [20, 29].
The second direction was to investigate the conjecture for degrees other
than complete degrees. For degrees within the 2-truth-table-complete degree
of EXP, both possible answers to the conjecture were found [41, 43, 44].
The third direction was to investigate the conjecture for reducibilities
weaker than polynomial-time. For several such reducibilities it was found
that the isomorphism conjecture, or something close to it, is true [1, 2, 8, 16].
These results, especially from the third direction, suggest that the Iso-
morphism Conjecture for the class NP may be true contrary to the evidence
from the relativized world. A recent work [13] shows that if all one-way
functions satisfy a certain property then a non-uniform version of the con-
jecture is true.
An excellent survey of the conjecture and results related to the first two
directions is in [45].
2.2. Definitions
In this section, we define most of the notions that we will need.
We fix the alphabet to Σ = {0, 1}. Σ∗
denotes the set of all finite strings
over Σ and Σn
denotes the set of strings of size n. We start with defining
the types of functions we use.
Definition 2.1. Let r be a resource bound on Turing machines. Function
f, f : Σ∗
7→ Σ∗
, is r-computable if there exists a Turing machine (TM, in
short) M working within resource bound of r that computes f. We also
refer to f as an r-function.](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-33-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
The Isomorphism Conjecture for NP 21
Function f is size-increasing if for every x, |f(x)| |x|. f is honest if
there exists a polynomial p(·) such that for every x, p(|f(x)|) |x|.
For function f, f−1
denotes a function satisfying the property that
for all x, f(f−1
(f(x))) = f(x). We say f is r-invertible if some f−1
is
r-computable.
For function f, its range is denoted as: range(f) = {y | (∃x) f(x) = y}.
We will be primarily interested in the resource bound of polynomial-
time, and abbreviate it as p. We now define several notions of reducibilities.
Definition 2.2. Let r be a resource bound. Set A r-reduces to set B if
there exists an r-function f such that for every x, x ∈ A iff f(x) ∈ B. We
also write this as A ≤r
m B via f. Function f is called an r-reduction of A
to B.
Similarly, A ≤r
1 B (A ≤r
1,si B; A ≤r
1,si,i B) if there exists a 1-1 (1-1 and
size-increasing; 1-1, size-increasing and r-invertible) r-function f such that
A ≤r
m B via f.
A ≡r
m B if A ≤r
m B and B ≤r
m A. An r-degree is an equivalence class
induced by the relation ≡r
m.
Definition 2.3. A is r-isomorphic to B if A ≤r
m B via f where f is a 1-1,
onto, r-invertible r-function.
The definitions of complexity classes DP, NP, PH, EXP, NEXP etc. can
be found in [52]. We define the notion of completeness we are primarily
interested in.
Definition 2.4. Set A is r-complete for NP if A ∈ NP and for every B ∈
NP, B ≤r
m A. For r = p, set A is called NP-complete in short. The class of
r-complete sets is also called the complete r-degree of NP.
Similarly one defines complete sets for other classes.
The Satisfiability problem (SAT) is one of the earliest known NP-
complete problems [25]. SAT is the set of all satisfiable propositional
Boolean formulas.
We now define one-way functions. These are p-functions that are not
p-invertible on most of the strings. One-way functions are one of the fun-
damental objects in cryptography.
Without loss of generality (see [30]), we can assume that one-way func-
tions are honest functions f for which the input length determines the
output length, i.e., there is a length function ℓ such that |f(x)| = ℓ(|x|) for
all x ∈ Σ∗
.](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-34-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
22 M. Agrawal
Definition 2.5. Function f is a s(n)-secure one-way function if (1) f
is a p-computable, honest function and (2) the following holds for every
polynomial-time randomized Turing machine M and for all sufficiently large
n:
Pr
x∈U Σn
[ f(M(f(x))) = f(x) ]
1
s(n)
.
In the above, the probability is also over random choices of M, and x ∈U Σn
mean that x is uniformly and randomly chosen from strings of size n.
We impose the property of honesty in the above definition since a func-
tion that shrinks length by more than a polynomial is trivially one-way.
It is widely believed that 2nǫ
-secure one-way functions exist for some
ǫ 0. We give one example. Start by defining a modification of the
multiplication function:
Mult(x, y) =
1z if x and y are both prime numbers
and z is the product x ∗ y
0xy otherwise.
In the above definition, (·, ·) is a pairing function. In this paper, we
assume the following definition of (·, ·): (x, y) = xyℓ where |ℓ| = ⌈log |xy|⌉
and ℓ equals |x| written in binary. With this definition, |(x, y)| = |x|+|y|+
⌈log |xy|⌉. This definition is easily extended for m-tuples for any m.
Mult is a p-function since testing primality of numbers is in DP [11].
Computing the inverse of Mult is equivalent to factorization, for which no
efficient algorithm is known. However, Mult is easily invertible on most of
the inputs, e.g., when any of x and y is not prime. The density estimate
for prime numbers implies that Mult is p-invertible on at least 1 − 1
nO(1)
fraction of inputs. It is believed that Mult is (1 − 1
nO(1) )-secure, and it
remains so even if one lets the TM M work for time 2nδ
for some small
δ 0. From this assumption, one can show that arbitrary concatenation
of Mult:
MMult(x1, y1, x2, y2, . . . , xm, ym) =
Mult(x1, y1) · Mult(x2, y2) · · · Mult(xm, ym)
is a 2nǫ
-secure one-way function [30](p. 43).
One-way functions that are 2nǫ
-secure are not p-invertible almost any-
where. The weakest form of one-way functions are worst-case one-way
functions:](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-35-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
The Isomorphism Conjecture for NP 23
Definition 2.6. Function f is a worst-case one-way function if (1) f is a
p-computable, honest function, and (2) f is not p-invertible.
2.3. Formulation and Early Investigations
The conjecture was formulated by Berman and Hartmanis [21] in 1977.
Part of their motivation for the conjecture was a corresponding result in
computability theory for computably enumerable sets [50]:
Theorem 2.1. (Myhill) All complete sets for the class of computably enu-
merable sets are isomorphic to each other under computable isomorphisms.
The non-trivial part in the proof of this theorem is to show that complete
sets for the class of computably enumerable sets reduce to each other via 1-1
reductions. It is then easy to construct isomorphisms between the complete
sets. In many ways, the class NP is the resource bounded analog of the
computably enumerable class, and polynomial-time functions the analog of
computable functions. Hence it is natural to ask if the resource bounded
analog of the above theorem holds.
Berman and Hartmanis noted that the requirement for p-isomorphisms
is stronger. Sets reducing to each other via 1-1 p-reductions does not guar-
antee p-isomorphisms as p-functions do not have sufficient time to perform
exponential searches. Instead, one needs p-reductions that are 1-1, size-
increasing, and p-invertible:
Theorem 2.2. (Berman–Hartmanis) If A ≤p
1,si,i B and B ≤p
1,si,i A
then A is p-isomorphic to B.
They defined the paddability property which ensures the required kind
of reductions.
Definition 2.7. Set A is paddable if there exists a p-computable padding
function p, p : Σ∗
× Σ∗
7→ Σ∗
, such that:
• Function p is 1-1, size-increasing, and p-invertible,
• For every x, y ∈ Σ∗
, p(x, y) ∈ A iff x ∈ A.
Theorem 2.3. (Berman–Hartmanis) If B ≤p
m A and A is paddable,
then B ≤p
1,si,i A.
Proof. Suppose B ≤p
m A via f. Define function g as: g(x) = p(f(x), x).
Then, x ∈ B iff f(x) ∈ A iff g(x) = p(f(x), x) ∈ A. By its definition and](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-36-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
24 M. Agrawal
the fact that p is 1-1, size-increasing, and p-invertible, it follows that g is
also 1-1, size-increasing, and p-invertible.
Berman and Hartmanis next showed that the known complete sets for
NP at the time were all paddable and hence p-isomorphic to each other.
For example, the following is a padding function for SAT:
pSAT (x, y1y2 · · · ym) = x ∧
m
^
i=1
zi
m
^
i=1
ci
where ci = zm+i if bit yi = 1 and ci = z̄i if yi = 0 and the Boolean variables
z1, z2, . . ., z2m do not occur in the formula x.
This observation led them to the following conjecture:
Isomorphism Conjecture. All NP-complete sets are p-isomorphic to
each other.
The conjecture immediately implies DP 6= NP:
Proposition 2.1. If the Isomorphism Conjecture is true then DP 6= NP.
Proof. If DP = NP then all sets in DP are NP-complete. However, DP
has both finite and infinite sets and there cannot exist an isomorphism
between a finite and an infinite set. Hence the Isomorphism Conjecture is
false.
This suggests that proving the conjecture is hard because the problem
of separating DP from NP has resisted all efforts so far. A natural question,
therefore, is: Can one prove the conjecture assuming a reasonable hypoth-
esis such as DP 6= NP? We address this question later in the paper. In
their paper, Berman and Hartmanis also asked a weaker question: Does
DP 6= NP imply that no sparse set can be NP-complete?
Definition 2.8. Set A is sparse if there exist constants k, n0 0 such that
for every n n0, the number of strings in A of length ≤ n is at most nk
.
This was answered in the affirmative by Mahaney [49]:
Theorem 2.4. (Mahaney) If DP 6= NP then no sparse set is NP-
complete.
Proof Sketch. We give a proof based on an idea of [9, 19, 51]. Suppose
there is a sparse set S such that SAT ≤p
m S via f. Let F be a Boolean
formula on n variables. Start with the set T = {F} and do the following:](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-37-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
The Isomorphism Conjecture for NP 25
Replace each formula F̂ ∈ T by F̂0 and F̂1 where F̂0 and F̂1 are obtained
by setting the first variable of F̂ to 0 and 1 respectively. Let T =
{F1, F2, . . . , Ft}. If t exceeds a certain threshold t0, then let Gj = F1 ∨
Fj and zj = f(Gj ) for 1 ≤ j ≤ t. If all zj’s are distinct then drop
F1 from T . Otherwise, zi = zj for some i 6= j. Drop Fi from T and
repeat until |T | ≤ t0. If T has only formulas with no variables, then
output Satisfiable if T contains a True formula else output Unsatisfiable.
Otherwise, go to the beginning of the algorithm and repeat.
The invariant maintained during the entire algorithm is that F is sat-
isfiable iff T contains a satisfiable formula. It is true in the beginning, and
remains true in each iteration after replacing every formula F̂ ∈ T with F̂0
and F̂1. The threshold t0 must be such that t0 is a upper bound on the
number of strings in the set S of size maxj |f(Gj)|. This is a polynomial in
|F| since |Gj| ≤ 2|F|, f is a p-function, and S is sparse. If T has more than
t0 formulas at any stage then the algorithm drops a formula from T . This
formula is F1 when all zj’s are distinct. This means there are more than t0
zj’s all of size bounded by maxj |f(Gj)|. Not all of these can be in S due
to the choice of t0 and hence F1 6∈ SAT. If zi = zj then Fi is dropped. If Fi
is satisfiable then so is Gi. And since zi = zj and f is a reduction of SAT
to S, Gj is also satisfiable; hence either F1 or Fj is satisfiable. Therefore
dropping Fi from T maintains the invariant.
The above argument shows that the size of T does not exceed a poly-
nomial in |F| at any stage. Since the number of iterations of the algorithm
is bounded by n ≤ |F|, the overall time complexity of the algorithm is
polynomial. Hence SAT ∈ DP and therefore, DP = NP.
The “searching-with-pruning” technique used in the above proof has
been used profitably in many results subsequently. The Isomorphism Con-
jecture, in fact, implies a much stronger density result: All NP-complete
sets are dense.
Definition 2.9. Set A is dense if there exist constants ǫ, n0 0 such that
for every n n0, the number of strings in A of length ≤ n is at least 2nǫ
.
Buhrman and Hitchcock [22] proved that, under a plausible hypothesis,
every NP-complete set is dense infinitely often:
Theorem 2.5. (Buhrman–Hitchcock) If PH is infinite then for any
NP-complete set A, there exists ǫ 0 such that for infinitely many n, the
number of strings in A of length ≤ n is at least 2nǫ
.](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-38-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
26 M. Agrawal
Later, we show that a stronger density theorem holds if 2nǫ
-secure one-
way functions exist.
2.4. A Counter Conjecture and Relativizations
After Mahaney’s result, there was not much progress on the conjecture
although researchers believed it to be true. However, this changed in 1984
when Joseph and Young [40] argued that the conjecture is false. Their
argument was as follows (paraphrased by Selman [53]). Let f be any 1-1,
size-increasing, 2nǫ
-secure one-way function. Consider the set A = f(SAT).
Set A is clearly NP-complete. If it is p-isomorphic to SAT, there must exist
a 1-1, honest p-reduction of SAT to A which is also p-invertible. However,
the set A is, in a sense, a “coded” version of SAT such that on most of the
strings of A, it is hard to “decode” it (because f is not p-invertible on most
of the strings). Thus, there is unlikely to be a 1-1, honest p-reduction of
SAT to A which is also p-invertible, and so A is unlikely to be p-isomorphic
to SAT. This led them to make a counter conjecture:
Encrypted Complete Set Conjecture. There exists a 1-1, size-
increasing, one-way function f such that SAT and f(SAT) are not p-
isomorphic to each other.
It is useful to observe here that this conjecture is false in computable
setting: The inverse of any 1-1, size-increasing, computable function is also
computable. The restriction to polynomial-time computability is what gives
rise to the possible existence of one-way functions.
It is also useful to observe that this conjecture too implies DP 6= NP:
Proposition 2.2. If the Encrypted Complete Set Conjecture is true then
DP 6= NP.
Proof. If DP = NP then every 1-1, size-increasing p-function is also
p-invertible. Hence for every such function, SAT and f(SAT) are p-
isomorphic.
The Encrypted Complete Set conjecture fails if one-way functions do
not exist. Can it be shown to follow from the existence of strong one-
way functions, such as 2nǫ
-secure one-way functions? This is not clear.
(In fact, later we argue the opposite.) Therefore, to investigate the two
conjectures further, the focus moved to relativized worlds. Building on a
result of Kurtz [42], Hartmanis and Hemachandra [33] showed that there](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-39-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
The Isomorphism Conjecture for NP 27
is an oracle relative to which DP = UP and the Isomorphism Conjecture is
false. This shows that both the conjectures fail in a relativized world since
DP = UP implies that no one-way functions exist.
Kurtz, Mahaney, and Royer [46] defined the notion of scrambling func-
tions:
Definition 2.10. Function f is scrambling function if f is 1-1, size-
increasing, p-computable, and there is no dense polynomial-time subset
in range(f).
Kurtz et al. observed that,
Proposition 2.3. If scrambling functions exist then the Encrypted Com-
plete Set Conjecture is true.
Proof. Let f be a scrambling function, and consider A = f(SAT). Set
A is NP-complete. Suppose it is p-isomorphic to SAT and let p be the
isomorphism between SAT and A. Since SAT has a dense polynomial-time
subset, say D, p(D) is a dense polynomial time subset of A. This contradicts
the scrambling property of f.
Kurtz et al., [46], then showed that,
Theorem 2.6. (Kurtz, Mahaney, Royer) Relative to a random oracle,
scrambling functions exist.
Proof Sketch. Let O be an oracle. Define function f as:
f(x) = O(x)O(x1)O(x11) · · · O(x12|x|
)
where O(z) = 1 if z ∈ O, 0 otherwise. For a random choice of O, f
is 1-1 with probability 1. So, f is a 1-1, size-increasing, pO
-computable
function. Suppose a polynomial-time TM M with oracle O accepts a subset
of range(f). In order to distinguish a string in range of f from those outside,
M needs to check the answer of oracle O on several unique strings. And
since M can query only polynomially many strings from O, M can accept
only a sparse subset of range(f).
Therefore, relative to a random oracle, the Encrypted Complete Set
Conjecture is true and the Isomorphism Conjecture is false. The question
of existence of an oracle relative to which the Isomorphism Conjecture is
true was resolved by Fenner, Fortnow, and Kurtz [27]:
Theorem 2.7. (Fenner, Fortnow, Kurtz) There exists an oracle rela-
tive to which Isomorphism Conjecture is true.](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-40-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
28 M. Agrawal
Thus, there are relativizations in which each of the three possible an-
swers to the two conjectures is true. However, the balance of evidence
provided by relativizations is towards the Encrypted Complete Set Conjec-
ture since properties relative to a random oracle are believed to be true in
unrelativized world too.b
2.5. The Conjectures for Other Classes
In search of more evidence for the two conjectures, researchers translated
them to classes bigger than NP. The hope was that diagonalization argu-
ments that do not work within NP can be used for these classes to prove
stronger results about the structure of complete sets. This hope was real-
ized, but not completely. In this section, we list the major results obtained
for classes EXP and NEXP which were the two main classes considered.
Berman [20] showed that,
Theorem 2.8. (Berman) Let A be a p-complete set for EXP. Then for
every B ∈ EXP, B ≤p
1,si A.
Proof Sketch. Let M1, M2, . . . be an enumeration of all polynomial-time
TMs such that Mi halts, on input x, within time |x||i|
+ |i| steps. Let
B ∈ EXP and define B̂ to be the set accepted by the following algorithm:
Input (i, x). Let Mi(i, x) = y. If |y| ≤ |x|, accept iff y 6∈ A. If there
exists a z, z x (in lexicographic order), such that Mi(i, z) = y, then
accept iff z 6∈ B. Otherwise, accept iff x ∈ B.
The set B̂ is clearly in EXP. Let B̂ ≤p
m A via f. Let the TM Mj compute
f. Define function g as: g(x) = f(j, x). It is easy to argue that f is 1-1
and size-increasing on inputs of the form (j, ⋆) using the definition of B̂
and the fact that f is a reduction. It follows that g is a 1-1, size-increasing
p-reduction of B to A.
Remark 2.1. A case can be made that the correct translation of the iso-
morphism result of [50] to the polynomial-time realm is to show that the
complete sets are also complete under 1-1, size-increasing reductions. As
observed earlier, the non-trivial part of the result in the setting of com-
putability is to show the above implication. Inverting computable reduc-
tions is trivial. This translation will also avoid the conflict with Encrypted
Complete Set Conjecture as it does not require p-invertibility. In fact, as
bThere are notable counterexamples of this though. The most prominent one is the
result IP = PSPACE [48, 54] which is false relative to a random oracle [24].](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-41-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
The Isomorphism Conjecture for NP 29
will be shown later, one-way functions help in proving an analog of the
above theorem for the class NP! However, the present formulation has a
nice symmetry to it (both the isomorphism and its inverse require the same
amount of resources) and hence is the preferred one.
For the class NEXP, Ganesan and Homer [29] showed that,
Theorem 2.9. (Ganesan–Homer) Let A be a p-complete set for NEXP.
Then for every B ∈ NEXP, B ≤p
1 A.
The proof of this uses ideas similar to the previous proof for EXP. The
result obtained is not as strong since enforcing the size-increasing property
of the reduction requires accepting the complement of a NEXP set which
cannot be done in NEXP unless NEXP is closed under complement, a very
unlikely possibility. Later, the author [5] proved the size-increasing property
for reductions to complete sets for NEXP under a plausible hypothesis.
While the two conjectures could not be settled for the complete p-degree
of EXP (and NEXP), answers have been found for p-degrees close to the
complete p-degree of EXP. The first such result was shown by Ko, Long,
and Du [41]. We need to define the notion of truth-table reductions to state
this result.
Definition 2.11. Set A k-truth-table reduces to set B if there exists a p-
function f, f : Σ∗
7→ Σ∗
× Σ∗
× · · · × Σ∗
| {z }
k
×Σ2k
such that for every x ∈ Σ∗
,
if f(x) = (y1, y2, . . . , yk, T ) then x ∈ A iff T (B(y1)B(y2) · · · B(yk)) = 1
where B(yi) = 1 iff yi ∈ B and T (s), |s| = k, is the sth bit of string T .
Set B is k-truth-table complete for EXP if B ∈ EXP and for every A ∈
EXP, A k-truth-table reduces to B.
The notion of truth-table reductions generalizes p-reductions. For both
EXP and NEXP, it is known that complete sets under 1-truth-table reduc-
tions are also p-complete [23, 38], and not all complete sets under 2-truth-
table reductions are p-complete [55]. Therefore, the class of 2-truth-table
complete sets for EXP is the smallest class properly containing the complete
p-degree of EXP.
Ko, Long, and Du [41] related the structure of certain p-degrees to the
existence of worst-case one-way functions:
Theorem 2.10. (Ko–Long–Du) If there exist worst-case one-way func-
tions then there is a p-degree in EXP such that the sets in the degree are not](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-42-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
30 M. Agrawal
all p-isomorphic to each other. Further, sets in this degree are 2-truth-table
complete for EXP.
Kurtz, Mahaney, and Royer [43] found a p-degree for which the sets are
unconditionally not all p-isomorphic to each other:
Theorem 2.11. (Kurtz–Mahaney–Royer) There exists a p-degree in
EXP such that the sets in the degree are not all p-isomorphic to each other.
Further, sets in this degree are 2-truth-table complete for EXP.
Soon afterwards, Kurtz, Mahaney, and Royer [44] found another p-
degree with the opposite structure:
Theorem 2.12. (Kurtz–Mahaney–Royer) There exists a p-degree in
EXP such that the sets in the degree are all p-isomorphic to each other.
Further, this degree is located inside the 2-truth-table complete degree of
EXP.
The set of results above on the structure of complete (or nearly com-
plete) p-degree of EXP and NEXP do not favor any of the two conjectures.
However, they do suggest that the third possibility, viz., both the conjec-
tures being false, is unlikely.
2.6. The Conjectures for Other Reducibilities
Another direction from which to approach the two conjectures is to weaken
the power of reductions instead of the class NP, the hope being that for
reductions substantially weaker than polynomial-time, one can prove un-
conditional results. For several weak reductions, this was proven correct
and in this section we summarize the major results in this direction.
The two conjectures for r-reductions can be formulated as:
r-Isomorphism Conjecture. All r-complete sets for NP are r-
isomorphic to each other.
r-Encrypted Complete Set Conjecture. There is a 1-1, size-
increasing, r-function f such that SAT and f(SAT) are not r-
isomorphic to each other.
Weakening p-reductions to logspace-reductions (functions computable by
TMs with read-only input tape and work tape space bounded by O(log n),
n is the input size) does not yield unconditional results as any such result](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-43-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
The Isomorphism Conjecture for NP 31
will separate NP from L, another long-standing open problem. So we need
to weaken it further. There are three major ways of doing this.
2.6.1. Restricting the input head movement
Allowing the input head movement in only one direction leads to the notion
of 1-L-functions.
Definition 2.12. A 1-L-function is computed by deterministic TMs with
read-only input tape, the workspace bounded by O(log n) where n is the
input length, and the input head restricted to move in one direction only
(left-to-right by convention). In other words, the TM is allowed only one
scan of its input. To ensure the space bound, the first O(log n) cells on the
work tape are marked at the beginning of the computation.
These functions were defined by Hartmanis, Immerman, and Ma-
haney [34] to study the complete sets for the class L. They also ob-
served that the “natural” NP-complete sets are also complete under 1-L-
reductions. Structure of complete sets under 1-L-reductions attracted a lot
of attention, and the first result was obtained by Allender [14]:
Theorem 2.13. (Allender) For the classes PSPACE and EXP, complete
sets under 1-L-reductions are p-isomorphic to each other.
While this shows a strong structure of complete sets of some classes
under 1-L-reductions, it does not answer the 1-L-Isomorphism Conjecture.
After a number of extensions and improvements [10, 29, 37], the author [1]
showed that,
Theorem 2.14. (Agrawal) Let A be a 1-L-complete set for NP. Then for
every B ∈ NP, B ≤1−L
1,si,i A.
Proof Sketch. We first show that A is also complete under forgetful 1-
L-reductions. Forgetful 1-L-reductions are computed by TMs that, imme-
diately after reading a bit of the input, forget its value. This property is
formalized by defining configurations: A configuration of a 1-L TM is a
tuple hq, j, wi where q is a state of the TM, j its input head position, and
w the contents of its worktape including the position of the worktape head.
A forgetful TM, after reading a bit of the input and before reading the
next bit, reaches a configuration which is independent of the value of the
bit that is read.](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-44-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
32 M. Agrawal
Let B ∈ NP, and define B̂ to be the set accepted by the following
algorithm:
Input x. Let x = y10b
1k
. Reject if b is odd or |y| 6= tb for some integer
t. Otherwise, let y = y1y2 · · · yt with |yi| = b. Let vi = 1 if yi = uu for
some u, |u| = b
2
; vi = 0 otherwise. Accept iff v1v2 · · · vt ∈ B.
The set B̂ is a “coded” version of set B and reduces to B via a p-
reduction. Hence, B̂ ∈ NP. Let f be a 1-L-reduction of B̂ to A computed
by TM M. Consider the workings of M on inputs of size n. Since M
has O(log n) space, the number of configurations of M will be bounded
by a polynomial, say q(·), in n. Let b = k⌈log n⌉ such that 2b/2
q(n).
Let C0 be the initial configuration of M. By the Pigeon Hole Principle, it
follows that there exist two distinct strings u1 and u′
1, |u1| = |u′
1| = b
2 , such
that M reaches the same configuration, after reading either of u1 and u′
1.
Let C1 be the configuration reached from this configuration after reading
u1. Repeat the same argument starting from C1 to obtain strings u2, u′
2,
and configuration C2. Continuing this way, we get triples (ui, u′
i, Ci) for
1 ≤ i ≤ t = ⌊n−b−1
b ⌋. Let k = n − b − 1 − bt. It follows that the TM M
will go through the configurations C0, C1, . . ., Ct on any input of the form
y1y2 . . . yt10b
1k
with yi ∈ {uiui, u′
iui}. Also, that the pair (ui, u′
i) can be
computed in logspace without reading the input.
Define a reduction g of B to B̂ as follows: On input v, |v| = t, compute
b such that 2b/2
q(b + 1 + bt), and consider M on inputs of size b + 1 + bt.
For each i, 1 ≤ i ≤ t, compute the pair (ui, u′
i) and output uiui if the ith
bit of v is 1, output uiu′
i otherwise. It is easy to argue that the composition
of f and g is a forgetful 1-L-reduction of B to A.
Define another set B′
as accepted by the following algorithm:
Input x. Reject if |x| is odd. Otherwise, let x = x1x2 · · · xns1s2 · · · sn.
Accept if exactly one of s1, s2, . . ., sn, say sj , is zero and xj = 1. Accept
if all of s1, s2, . . ., sn are one and x1x2 · · · xn ∈ B. Reject in all other
cases.
Set B′
∈ NP. As argued above, there exists a forgetful 1-L-reduction of
B′
to A, say h. Define a reduction g′
of B to B′
as: g′
(v) = v1|v|
. It is easy
to argue that h ◦ g′
is a size-increasing, 1-L-invertible, 1-L-reduction of B
to A and h ◦ g′
is 1-1 on strings of size n for all n. Modifying this to get a
reduction that is 1-1 everywhere is straightforward.
The above result strongly suggests that the 1-L-Isomorphism Conjecture
is true. However, the author [1] showed that,](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-45-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
The Isomorphism Conjecture for NP 33
Theorem 2.15. (Agrawal) 1-L-complete sets for NP are all 2-L-
isomorphic to each other but not 1-L-isomorphic.
The 2-L-isomorphism above is computed by logspace TMs that are al-
lowed two left-to-right scans of their input. Thus, the 1-L-Isomorphism
Conjecture fails and a little more work shows that the 1-L-Encrypted Com-
plete Set Conjecture is true! However, the failure of the Isomorphism Con-
jecture here is for a very different reason: it is because 1-L-reductions are
not powerful enough to carry out the isomorphism construction as in The-
orem 2.2. For a slightly more powerful reducibility, 1-NL-reductions, this
is not the case.
Definition 2.13. A 1-NL-function is computed by TMs satisfying the re-
quirements of definition 2.12, but allowed to be non-deterministic. The
non-deterministic TM must output the same string on all paths on which
it does not abort the computation.
For 1-NL-reductions, the author [1] showed, using proof ideas similar to
the above one, that,
Theorem 2.16. (Agrawal) 1-NL-complete sets for NP are all 1-NL-
isomorphic to each other.
The author [1] also showed similar results for c-L-reductions for constant
c (functions that are allowed at most c left-to-right scans of the input).
2.6.2. Reducing space
The second way of restricting logspace reductions is by allowing the TMs
only sublogarithmic space, i.e., allowing the TM space o(log n) on input of
size n; we call such reductions sublog-reductions. Under sublog-reductions,
NP has no complete sets, and the reason is simple: Every sublog-reduction
can be computed by deterministic TMs in time O(n2
) and hence if there
is a complete set for NP under sublog-reductions, then NTIME(nk+1
) =
NTIME(nk
) for some k 0, which is impossible [26]. On the other hand,
each of the classes NTIME(nk
), k ≥ 1, has complete sets under sublog-
reductions.
The most restricted form for sublog-reductions is 2-DFA-reductions:
Definition 2.14. A 2-DFA-function is computed by a TM with read-only
input tape and no work tape.](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-46-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
34 M. Agrawal
2-DFA functions do not require any space for their computation, and
therefore are very weak. Interestingly, the author [4] showed that sublog-
reductions do not add any additional power for complete sets:
Theorem 2.17. (Agrawal) For any k ≥ 1, sublog-complete sets for
NTIME(nk
) are also 2-DFA-complete.
For 2-DFA-reductions, the author and Venkatesh [12] proved that,
Theorem 2.18. (Agrawal-Venkatesh) Let A be a 2-DFA-complete set
for NTIME(nk
) for some k ≥ 1. Then, for every B ∈ NTIME(nk
), B ≤2DFA
1,si
A via a reduction that is mu-DFA-invertible.
muDFA-functions are computed by TMs with no space and multiple
heads, each moving in a single direction only. The proof of this is also
via forgetful TMs. The reductions in the theorem above are not 2-DFA-
invertible, and in fact, it was shown in [12] that,
Theorem 2.19. (Agrawal-Venkatesh) Let f(x) = xx. Function f is a
2-DFA-function and for any k ≥ 1, there is a 2-DFA-complete set A for
NTIME(nk
) such that A 6≤2DFA
1,si,i f(A).
The above theorem implies that 2-DFA-Encrypted Complete Set Con-
jecture is true.
2.6.3. Reducing depth
Logspace reductions can be computed by (unbounded fan-in) circuits of
logarithmic depth.c
Therefore, another type of restricted reducibility is
obtained by further reducing the depth of the circuit family computing the
reduction. Before proceeding further, let us define the basic notions of a
circuit model.
Definition 2.15. A circuit family is a set {Cn : n ∈ N} where each Cn is an
acyclic circuit with n Boolean inputs x1, . . . , xn (as well as the constants 0
and 1 allowed as inputs) and some number of output gates y1, . . . , yr. {Cn}
has size s(n) if each circuit Cn has at most s(n) gates; it has depth d(n) if
the length of the longest path from input to output in Cn is at most d(n).
A circuit family has a notion of uniformity associated with it:
cFor a detailed discussion on the circuit model of computation, see [52].](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-47-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
The Isomorphism Conjecture for NP 35
Definition 2.16. A family C = {Cn} is uniform if the function n 7→ Cn
is easy to compute in some sense. This can also be defined using the
complexity of the connection set of the family:
conn(C) = {(n, i, j, Ti, Tj) | the output of gate i of type Ti
is input to gate j of type Tj in Cn}.
Here, gate type Ti can be Input, Output, or some Boolean operator.
Family C is Dlogtime-uniform [18] if conn(C) is accepted by a linear-time
TM. It is p-uniform [15] if conn(C) is accepted by a exponential-time TM
(equivalently, by a TM running in time bounded by a polynomial in the
circuit size). If we assume nothing about the complexity of conn(C), then
we say that the family is non-uniform.
An important restriction of logspace functions is to functions computed
by constant depth circuits.
Definition 2.17. Function f is a u-uniform AC0
-function if there is a u-
uniform circuit family {Cn} of size nO(1)
and depth O(1) consisting of
unbounded fan-in AND and OR and NOT gates such that for each input
x of length n, the output of Cn on input x is f(x).
Note that with this definition, an AC0
-function cannot map strings of
equal size to strings of different sizes. To allow this freedom, we adopt the
following convention: Each Cn will have nk
+k log(n) output bits (for some
k). The last k log n output bits will be viewed as a binary number r, and
the output produced by the circuit will be the binary string contained in
the first r output bits.
It is worth noting that, with this definition, the class of Dlogtime-
uniform AC0
-functions admits many alternative characterizations, includ-
ing expressibility in first-order logic with {+, ×, ≤} [18, 47], the logspace-
rudimentary reductions [17, 39], logarithmic-time alternating Turing ma-
chines with O(1) alternations [18] etc. Moreover, almost all known NP-
complete sets are also complete under Dlogtime-uniform AC0
-reductions
(an exception is provided by [7]). We will refer to Dlogtime-uniform AC0
-
functions also as first-order-functions.
AC0
-reducibility is important for our purposes too, since the complete
sets under the reductions of the previous two subsections are also complete
under AC0
-reductions (with uniformity being Dlogtime- or p-uniform). This
follows from the fact that these sets are also complete under some appro-
priate notion of forgetful reductions. Therefore, the class of AC0
-complete
sets for NP is larger than all of the previous classes of this section.](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-48-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
36 M. Agrawal
The first result for depth-restricted functions was proved by Allender,
Balcázar, and Immerman [16]:
Theorem 2.20. (Allender–Balcázar–Immerman) Complete sets for
NP under first-order projections are first-order-isomorphic to each other.
First-order projections are computed by a very restricted kind of
Dlogtime-uniform AC0
family in which no circuit has AND and OR gates.
This result was generalized by the author and Allender [6] to NC0
-functions,
which are functions computed by AC0
family in which the fan-in of every
gate of every circuit is at most two.
Theorem 2.21. (Agrawal–Allender) Let A be a non-uniform NC0
-
complete set for NP. Then for any B ∈ NP, B non-uniform NC0
-reduces
to A via a reduction that is 1-1, size-increasing, and non-uniform AC0
-
invertible. Further, all non-uniform NC0
-complete sets for NP are non-
uniform AC0
-isomorphic to each other where these isomorphisms can be
computed and inverted by depth three non-uniform AC0
circuits.
Proof Sketch. The proof we describe below is the one given in [3]. Let
B ∈ NP, and define B̂ to be the set accepted by the following algorithm:
On input y, let y = 1k
0z. If k does not divide |z|, then reject. Otherwise,
break z into blocks of k consecutive bits each. Let these be u1u2u3 . . . up.
Accept if there is an i, 1 ≤ i ≤ p, such that ui = 1k
. Otherwise, reject
if there is an i, 1 ≤ i ≤ p, such that ui = 0k
. Otherwise, for each i,
1 ≤ i ≤ p, label ui as null if the number of ones in it is 2 modulo 3; as
zero if the number of ones in it is 0 modulo 3; and as one otherwise. Let
vi = ǫ if ui is null, 0 if ui is zero, and 1 otherwise. Let x = v1v2 · · · vp,
and accept iff x ∈ B.
Clearly, B̂ ∈ NP. Let {Cn} be the NC0
circuit family computing a reduction
of B̂ to A. Fix size n and consider circuit Ck+1+n for k = 4⌈log n⌉. Let C
be the circuit that results from setting the first input k + 1 bits of Ck+1+n
to 1k
0. Randomly set each of the n input bits of C in the following way:
With probability 1
2 , leave it unset; with probability 1
4 each, set it to 0 and
1 respectively. The probability that any block of k bits is completely set
is at most 1
n4 . Similarly, the probability that there is a block that has at
most three unset bits is at most 1
n , and therefore, with high probability,
every block has at least four unset bits.
Say that an output bit is good if, after the random assignment to the
input bits described above is completed, the value of the output bit depends
on exactly one unset input bit. Consider an output bit. Since C is an NC0](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-49-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
The Isomorphism Conjecture for NP 37
circuit, the value of this bit depends on at most a constant, say c, number
of input bits. Therefore, the probability that this bit is good after the
assignment is at least 1
2 · 1
4c−1 . Therefore, the expected number of good
output bits is at least m
4c , where m is the number of output bits of C
whose value depends on some input bit. Using the definition of set B̂, it
can be argued that Ω(n) output bits depend on some input bit, and hence
Ω(n) output bits are expected to be good after the assignment. Fix any
assignment that does this, as well as leaves at least four unset bits in each
block. Now set some more input bits so that each block that is completely
set is null, each block that has exactly two unset bits has number of ones
equal to 0 modulo 3, and there are no blocks with one, three, or more unset
bits. Further, for at least one unset input bit in a block, there is a good
output bit that depends on the bit, and there are Ω( n
log n ) unset input bits.
It is easy to see that all these conditions can be met.
Now define a reduction of B to B̂ as: On input x, |x| = p, consider
Ck+1+n such that the number of unset input bits in Ck+1+n after doing the
above process is at least p. Now map the ith bit of x to the unset bit in a
block that influences a good output bit and set the other unset input bit in
the block to zero. This reduction can be computed by an NC0
circuit (in
fact, the circuit does not need any AND or OR gate).
Define a reduction of B to A given by the composition of the above two
reductions. This reduction is a superprojection: it is computed by circuit
family {Dp} with each Dp being an NC0
circuit such that for every input
bit to Dp, there is an output bit that depends exactly on this input bit. A
superprojection has the input written in certain bit positions of the output.
Therefore, it is 1-1 and size-increasing. Inverting the function is also easy:
Given string y, identify the locations where the input is written, and check
if the circuit Dp (p = number of locations) on this input outputs y. This
checking can be done by a depth two AC0
circuit.
This gives a 1-1, size-increasing, AC0
-invertible, NC0
-reduction of B
to A. The circuit family is non-uniform because it is not clear how to
deterministically compute the settings of the input bits. Exploiting the
fact that the input is present in the output of the reductions, an AC0
-
isomorphism, computed by depth three circuits, can be constructed between
two complete sets following [21] (see [8] for details).
Soon after, the author, Allender, and Rudich [8] extended it to all
AC0
-functions, proving the Isomorphism Conjecture for non-uniform AC0
-
functions.](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-50-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
38 M. Agrawal
Theorem 2.22. (Agrawal–Allender–Rudich) Non-uniform AC0
-com-
plete sets for NP are non-uniform AC0
-isomorphic to each other. Fur-
ther, these isomorphisms can be computed and inverted by depth three non-
uniform AC0
circuits.
Proof Sketch. The proof shows that complete sets for NP under AC0
-
reductions are also complete under NC0
-reductions and invokes the above
theorem for the rest. Let A be a complete set for NP under AC0
-reductions.
Let B ∈ NP. Define set B̂ exactly as in the previous proof. Fix an AC0
-
reduction of B̂ to A given by family {Cn}. Fix size n, and consider Ck+1+n
for k = n1−ǫ
for a suitable ǫ 0 to be fixed later. Let D be the circuit that
results from setting the first k + 1 input bits of Ck+1+n to 1k
0.
Set each input bit of D to 0 and 1 with probability 1
2 − 1
2n1−2ǫ each and
leave it unset with probability 1
n1−2ǫ . By the Switching Lemma of Furst,
Saxe, and Sipser [28], the circuit D will reduce, with high probability, to an
NC0
circuit on the unset input bits for a suitable choice of ǫ 0. In each
block of k bits, the expected number of unset bits will be nǫ
, and therefore,
with high probability, each block has at least three unset bits. Fix any
settings satisfying both of the above.
Now define a reduction of B to B̂ that, on input x, |x| = p, identifies n
for which the circuit D has at least p blocks, and then maps ith bit of input
x to an unset bit of the ith block of the input to D, setting the remaining
bits of the block so that the sum of ones in the block is 0 modulo 3. Unset
bits in all remaining blocks are set so that the sum of ones in the block
equals 2 modulo 3.
The composition of the reduction of B to B̂ and B̂ to A is an NC0
-
reduction of B to A. Again, it is non-uniform due to the problem of finding
the right settings of the input bits.
The focus then turned towards removing the non-uniformity in the
above two reductions. In the proof of Theorem 2.21 given in [6], the uni-
formity condition is p-uniform. In [7], the uniformity of 2.22 was improved
to p-uniform by giving a polynomial-time algorithm that computes the cor-
rect settings of input bits. Both the conditions were further improved to
logspace-uniform in [3] by constructing a more efficient derandomization
of the random assignments. And finally, in [2], the author obtained very
efficient derandomizations to prove that,
Theorem 2.23. (Agrawal) First-order-complete sets for NP are first-
order-isomorphic.](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-51-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
The Isomorphism Conjecture for NP 39
The isomorphisms in the theorem above are no longer computable by
depth three circuits; instead, their depth is a function of the depth of the
circuits computing reductions between the two complete sets.
2.6.4. Discussion
At first glance, the results for the weak reducibilities above seem to provide
equal support to both the conjectures: The Isomorphism Conjecture is
true for 1-NL and AC0
-reductions for any reasonable notion of uniformity,
while the Encrypted Complete Set Conjecture is true for 1-L and 2-DFA
reductions. However, on a closer look a pattern begins to emerge. First of
all, we list a common feature of all the results above:
Corollary 2.1. For r ∈ {1-L, 1-NL, 2-DFA, NC0
, AC0
}, r-complete sets
for NP are also complete under 1-1, size-increasing, r-reductions.
The differences arise in the resources required to invert the reductions
and to construct the isomorphism. Some of the classes of reductions that
we consider are so weak, that for a given function f in the class, there is no
function in the class that can check, on input x and y, whether f(x) = y.
For example, suppose f is an NC0
-function and one needs to construct
a circuit that, on input x and y, outputs 1 if y = f(x), and outputs 0
otherwise. Given x and y, an NC0
circuit can compute f(x), and can check
if the bits of f(x) are equal to the corresponding bits of y; however, it cannot
output 1 if f(x) = y, since this requires taking an AND of |y| bits. Similarly,
some of the reductions are too weak to construct the isomorphism between
two sets given two 1-1, size-increasing, and invertible reductions between
them. Theorems 2.14 and 2.15 show this for 1-L-reductions, and the same
can be shown for NC0
-reductions too. Observe that p-reductions do not
suffer from either of these two drawbacks. Hence we cannot read too much
into the failure of the Isomorphism Conjecture for r-reductions. We now
formulate another conjecture that seems better suited to getting around
the above drawbacks of some of the weak reducibilities. This conjecture
was made in [1].
Consider a 1-1, size-increasing r-function f for a resource bound r. Con-
sider the problem of accepting the set range(f). A TM accepting this set
will typically need to guess an x and then verify whether f(x) = y. It
is, therefore, a non-deterministic TM with resource bound at least r. Let
rrange
≥ r be the resource bound required by this TM. For a circuit accept-
ing range(f), the non-determinism is provided as additional “guess bits”](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-52-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
40 M. Agrawal
and its output is 1 if the circuit evaluates to 1 on some settings of the guess
bits. We can similarly define rrange
to be the resource bound required by
such a non-deterministic circuit to accept range(f).
r-Complete Degree Conjecture. r-Complete sets for NP are also com-
plete under 1-1, size-increasing, r-reductions that are rrange
-invertible.
Notice that the invertibility condition in the conjecture does not allow
non-determinism. For p-reductions,
Proposition 2.4. The p-Complete Degree Conjecture is equivalent to the
Isomorphism Conjecture.
Proof. Follows from the observation that prange
= p as range of a p-
function can be accepted in non-deterministic polynomial-time, and from
Theorem 2.2.
Moreover, for the weaker reducibilities that we have considered, one can
show that,
Theorem 2.24. For r ∈ {1-L, 1-NL, 2-DFA, NC0
, AC0
}, the r-Complete
Degree Conjecture is true.
Proof. It is an easy observation that for r ∈ {1-L, 1-NL, AC0
}, rrange
=
r. The conjecture follows from Theorems 2.14, 2.16, and 2.23.
Accepting range of a 2-DFA-function requires verifying the output of
2-DFA TM on each of its constant number of passes on the input. The
minimum resources required for this are to have multiple heads stationed
at the beginning of the output of each pass, guess the input bit-by-bit, and
verify the outputs on this bit for each pass simultaneously. Thus, the TM
is a non-deterministic TM with no space and multiple heads, each moving
in one direction only. So Theorem 2.18 proves the conjecture.
Accepting range of an NC0
-function requires a non-deterministic AC0
circuit. Therefore, Theorems 2.21 and 2.23 prove the conjecture for r =
NC0
.
In addition to the reducibilities in the above theorem, the r-Complete
Degree Conjecture was proven for some more reducibilities in [1].
These results provide evidence that r-Complete Degree Conjecture is
true for all reasonable resource bounds; in fact, there is no known example
of a reasonable reducibility for which the conjecture is false.](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-53-320.jpg)
![January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability
The Isomorphism Conjecture for NP 41
The results above also raise doubts about the intuition behind the En-
crypted Complete Set Conjecture as we shall argue now. Consider AC0
-
reductions. There exist functions computable by depth d, Dlogtime-uniform
AC0
circuits that cannot be inverted on most of the strings by depth three,
non-uniform AC0
circuits [35]. However, by Theorem 2.22, AC0
-complete
sets are also complete under AC0
-reductions that are invertible by depth
two, non-uniform AC0
circuits and the isomorphisms between all such sets
are computable and invertible by depth three, non-uniform AC0
circuits.
So, for every 1-1, size-increasing, AC0
-function, it is possible to efficiently
find a dense subset on which the function is invertible by depth two AC0
circuits.
Therefore, the results for weak reducibilities provide evidence that the
Isomorphism Conjecture is true.
2.7. A New Conjecture
In this section, we revert to the conjectures in their original form. The
investigations for weak reducibilities provide some clues about the struc-
ture of NP-complete sets. They strongly suggest that all NP-complete sets
should also be complete under 1-1, size-increasing p-reductions. Proving
this, of course, is hard as it implies DP 6= NP (Proposition 2.1). Can we
prove this under a reasonable assumption? This question was addressed
and partially answered by the author in [5], and subsequently improved by
the author and Watanabe [13]:
Theorem 2.25. (Agrawal–Watanabe) If there exists a 1-1, 2nǫ
-secure
one-way function for some ǫ 0, then all NP-complete sets are also com-
plete under 1-1, and size-increasing, P/poly-reductions.
In the above theorem, P/poly-functions are those computed by
polynomial-size, non-uniform circuit families.
Proof Sketch. Let A be an NP-complete set and let B ∈ NP. Let f0 be a
1-1, 2nǫ
-secure one-way function. Recall that we have assumed that |f0(y)|
is determined by |y| for all y. Håstad et al., [36], showed how to construct a
pseudorandom generator using any one-way function. Pseudorandom gen-
erators are size-increasing functions whose output cannot be distinguished
from random strings by polynomial-time probabilistic TMs. Let G be the
pseudorandom generator constructed from f0. Without loss of generality,
we can assume that |G(y)| = 2|y| + 1 for all y. We also modify f0 to f
as: f(y, r) = f0(y)rb where |r| = |y| and b = y · r, the inner product of](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-54-320.jpg)
![Two Opposite
Views of
Liberty
the world and a fuller grasp of the essentials of liberty had, meanwhile,
developed in the Englishman's mind[1] a love for free religious thought and
practice to which his belief in schools and his affection for literature and the
press added strength and character. The Dutchman was nomadic in life,
pastoral in pursuit, lazy and sluggish in disposition. The Englishman was at
times restless in seeking wealth or pleasure, but upon the whole he liked to
settle down in a permanent home and with surroundings which he could
make his own in ever-increasing comfort and usefulness. He drew the line at
no single occupation and made, as the case might be, a good farmer, or
artisan, or labourer, or merchant. And he was usually of active mind as well
as body.
[1] I use the word Englishman here in a general sense, and inclusive of
the Scotchman or Irishman.
The Dutchman in South Africa wanted liberty to do as
he liked and to live as he chose, but he did not wish to
accord that liberty to inferior races, or to attempt the
training of them in its use and application. The Englishman,
on the other hand, loved liberty in a broad way, and wanted nothing better
than to see it applied to others as freely and fully as to himself. The one race
looked upon the negro as only fitted to be a human chattel and as not being
even a possible subject for improvement, education or elevation. The other,
in all parts of the world as well as in the Dark Continent, believed in the
humanity of the coloured man, whether black, or red, or brown, and looked
upon him as fitted for civilization, for Christianity and for freedom. He
considered him as material for good government and for fair play. Both
views, however, have been carried to an extreme in South Africa and upon
either side evil resulted. The Boer treated the native from the standpoint of
an intolerant and ignorant slave-owner. The Colonial Office tried to treat him
solely from the standpoint of the sympathizing and often prejudiced
missionary. Hence, in part, the Great Trek; hence some of the Kaffir raids
and consequent sufferings of the early settlers; hence an addition to the
growing racial antagonism.](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-70-320.jpg)
![Varied
Opportunities
for Settlers
Statistics and
Finances of
South Africa
and temperate growth. Within the million and a half square
miles of South African territory were room and verge for a
vastly greater white population than has yet touched its
shores; while every racial peculiarity or pursuit could find a
place in its towns and farms and mines and upon its rolling veldt. To the
lover of quiet village life and retirement nothing could be more pleasant than
parts of Natal and Cape Colony, and of the two Republics. To the keen
business man, eager for gain and intent upon quick returns, the rapid and
wealth-producing progress of the great mining towns gave all that could be
desired. To the adventurous spirit, willing to suffer hardships and endure
labor in its severest form for a possibly glittering return, the diamond and
gold fields offered untold opportunities. To the hunter and tourist and
traveller the myriad wild animals of the interior gave a pleasure only second
to that felt by the Kaffir and the Boer when hunting the lion to his lair or the
elephant in its native jungle. To the man fond of country life the vast plains,
stretching in varied degrees of value and elevation from Cape Town to the
Zambesi, afforded room for pastoral occupation and the raising of cattle and
sheep upon a veritable thousand hills. To the seeker after new industries,
ostrich farming, mohair, the feather industry and diamond mining have from
time to time proved the greatest attraction. To the farmer or planter parts of
the region were eminently fitted for the raising of wheat and other cereals,
and the cultivation of tobacco, cotton, sugar and rice. To the restless and
wandering Boer, South Africa seems to have given for a time everything that
his spirit desired—isolation, land, wild animals to hunt, independence of
control, freedom from the trammels of education and taxation and
civilization. To the quieter Dutchman of Cape Colony has been given every
element of British liberty and privilege of British equality; as well as land in
plenty, and for thirty years, at least, the pledge of internal peace.
According, also, to the latest figures[1] the material
progress and recent position of all these countries has been
good. Cape Colony, in 1897-98, had a revenue of
$36,940,000, an expenditure of $34,250,000 and an
indebtedness of $136,400,000; a tonnage of British vessels, entered and
cleared, amounting to 12,137,000, together with 2,835 miles of railway and
6,609 miles of telegraph; exports of $108,300,000, and imports of
$90,000,000; and 132,000 scholars in its schools. Natal and Zululand,
combined, had a revenue of $11,065,000, an expenditure of $8,120,000 and](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-72-320.jpg)
![an indebtedness of $38,720,000; a tonnage of British vessels, entering and
clearing, of 2,132,000, together with 487 miles of railway and 960 of
telegraph; exports of $8,100,000 and imports of $30,000,000; and 19,222
scholars in its schools. The exports of Basutoland, under purely native
control, had grown to $650,000 and its imports to half a million. The length
of railway in the Bechuanaland Protectorate was 586 miles and in Rhodesia
1,086 miles; while the telegraph lines of the former region covered 1,856
miles. The South African Republic, or Transvaal, had a revenue of
$22,400,000, an expenditure of $21,970,000 and an indebtedness of
$13,350,000; announced imports of $107,575,000 and no declared exports;
railways of 774 miles in total length and telegraph lines of 2,000 miles; and
scholars numbering 11,552. The Orange Free State had a revenue of
$2,010,000, an expenditure of $1,905,000 and an indebtedness of $200,000;
imports of $6,155,000—chiefly from Cape Colony—and exports of
$8,970,000, which were divided principally between Cape Colony and the
Transvaal; 366 miles of railway, 1,762 miles of telegraph and 7,390 scholars
in its schools. The following table[1] gives an easily comprehended view of
South Africa as divided amongst its Kaffir, Dutch and English communities
in respect to mode of government and measure of British responsibility:
[1] British Empire Series. Vol. II. Kegan Paul, Trench, Trübner 4 Co.,
Limited. London, 1809.
[1] South Africa. By W. Basil Worsfold, M.A. London, 1895.
MODE OF
GOVERNMENT.
{ Cape Colony } Responsible
Government
Three British Colonies { Natal }
{
{ Bechuanaland } Crown Colony.
{ South African } Full internal
freedom
{ Republic } within terms of](https://image.slidesharecdn.com/1182215-250519004139-7bba6dcb/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-S-Barry-Cooper-73-320.jpg)
Computability In Context Computation And Logic In The Real World S Barry Cooper
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- 6. Computability in Context Computation and Logic in the Real World P577 tp.indd 1 1/14/11 10:07 AM
- 7. This page intentionally left blank This page intentionally left blank
- 8. Imperial College Press ICP editors S Barry Cooper University of Leeds, UK Andrea Sorbi Università degli Studi di Siena, Italy Computability in Context Computation and Logic in the Real World P577 tp.indd 2 1/14/11 10:07 AM
- 9. Library of Congress Cataloging-in-Publication Data Computability in context : computation and logic in the real world / edited by S. Barry Cooper & Andrea Sorbi. p. cm. Includes bibliographical references. ISBN-13: 978-1-84816-245-7 (hardcover : alk. paper) ISBN-10: 1-84816-245-6 (hardcover : alk. paper) 1. Computable functions. 2. Computational intelligence. 3. Set theory. 4. Mathematics--Philosophy. I. Cooper, S. B. (S. Barry) II. Sorbi, Andrea, 1956– QA9.59.C655 2011 511.3'52--dc22 2010039227 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Printed in Singapore. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. Copyright © 2011 by Imperial College Press EH - Computability in context.pmd 12/13/2010, 3:16 PM 1
- 10. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability Preface Computability has played a crucial role in mathematics and computer sci- ence – leading to the discovery, understanding and classification of decid- able/undecidable problems, paving the way to the modern computer era and affecting deeply our view of the world. Recent new paradigms of com- putation, based on biological and physical models, address in a radically new way questions of efficiency and even challenge assumptions about the so-called Turing barrier. This book addresses various aspects of the ways computability and the- oretical computer science enable scientists and philosophers to deal with mathematical and real world issues, ranging through problems related to logic, mathematics, physical processes, real computation and learning the- ory. At the same time it focuses on different ways in which computability emerges from the real world, and how this affects our way of thinking about everyday computational issues. But the title Computability in Context has been carefully chosen. The contributions to be found here are not strictly speaking ‘applied computability’. The literature directly addressing everyday computational questions has grown hugely since the days of Turing and the computer pioneers. The Computability in Europe conference series and association is built on the recognition of the complementary role that mathematics and fundamental science plays in progressing practical work; and, at the same time, of the vital importance of a sense of context of basic research. This book positions itself at the interface between applied and fundamental re- search, prioritising mathematical approaches to computational barriers. For us, the conference Computability in Europe 2007: Computation and Logic in the Real World was a hugely exciting – and taxing – experience. It brought together a remarkable assembly of speakers, and a level of par- ticipation around issues of computability that would surely have astounded Turing and those other early pioneers of ‘computing with understanding’. All of the contributions here come from invited plenary speakers or Pro- v
- 11. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability vi Preface gramme Committee members of CiE 2007. Many of these articles are likely to become key contributions to the literature of computability and its real- world significance. The authors are all world leaders in their fields, all much in demand as speakers and writers. As editors, we very much appreciate their work. Barry Cooper and Andrea Sorbi
- 12. January 24, 2011 16:5 World Scientific Review Volume - 9in x 6in computability Contents Preface v 1. Computation, Information, and the Arrow of Time 1 P. Adriaans & P. van Emde Boas 2. The Isomorphism Conjecture for NP 19 M. Agrawal 3. The Ershov Hierarchy 49 M. M. Arslanov 4. Complexity and Approximation in Reoptimization 101 G. Ausiello, V. Bonifaci, & B. Escoffier 5. Definability in the Real Universe 131 S. B. Cooper 6. HF-Computability 169 Y. L. Ershov, V. G. Puzarenko, & A. I. Stukachev 7. The Mathematics of Computing between Logic and Physics 243 G. Longo & T. Paul vii
- 13. January 24, 2011 16:5 World Scientific Review Volume - 9in x 6in computability viii Contents 8. Liquid State Machines: Motivation, Theory, and Applications 275 W. Maass 9. Experiments on an Internal Approach to Typed Algo- rithms in Analysis 297 D. Normann 10. Recursive Functions: An Archeological Look 329 P. Odifreddi 11. Reverse Mathematics and Well-ordering Principles 351 M. Rathjen & A. Weiermann 12. Discrete Transfinite Computation Models 371 P. D. Welch
- 14. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability Chapter 1 Computation, Information, and the Arrow of Time Pieter Adriaans & Peter van Emde Boas Adriaans ADZA Beheer B.V., and FNWI, University of Amsterdam, 1098 XG Amsterdam, The Netherlands E-mail: pieter@pieter-adriaans.com Bronstee.com B.V., Heemstede, and ILLC, FNWI, University of Amsterdam 1090 GE Amsterdam, The Netherlands E-mail: peter@bronstee.com In this chapter we investigate the relation between information and com- putation under time symmetry. We show that there is a class of non- deterministic automata, the quasi-reversible automata (QRTM), that is the class of classical deterministic Turing machines operating in negative time, and that computes all the languages in NP. The class QRTM is isomorphic to the class of standard deterministic Turing machines TM, in the sense that for every M ∈ TM there is a M−1 in QRTM such that each computation on M is mirrored by a computation on M−1 with the arrow of time reversed. This suggests that non-deterministic computing might be more aptly described as deterministic computing in negative time. If Mi is deterministic then M−1 i is non deterministic. If M is information discarding then M−1 “creates” information. The two fundamental complexities involved in a deterministic computation are Program Complexity and Program Counter Complexity. Programs can be classified in terms of their “information signature” with pure counting programs and pure information discarding programs as two ends of the spectrum. The chapter provides a formal basis for a further analysis of such diverse domains as learning, creative processes, growth, and the study of the interaction between computational processes and thermodynamics. 1
- 15. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 2 P. Adriaans & P. van Emde Boas Contents 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 A Formal Framework: Meta-computational Space . . . . . . . . . . . . . . . . 4 1.3 Time Symmetries in Meta-computational Space . . . . . . . . . . . . . . . . . 7 1.4 The Interplay of Computation and Information . . . . . . . . . . . . . . . . . 11 1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.1. Introduction The motivation behind this research is expressed in a childhood memory of one of the authors: “When I was a toddler my father was an enthusiastic 8-mm movie amateur. The events captured in these movies belong to my most vivid memories. One of the things that fascinated me utterly was the fact that you could reverse the time. In my favorite movie I was eating a plate of French fries. When played forward one saw the fries vanish in my mouth one by one, but when played backward a miracle happened. Like a magician pulling a rabbit out of a hat I was pulling undamaged fries out of my mouth. The destruction of fries in positive time was associated with the creation of fries in negative time.” This is a nice example of the kind of models we have been discussing when we were working on the research for this paper. It deals with com- putation and the growth and destruction of information. Deterministic computation seems to be incapable of creating new information. In fact most recursive functions are non-reversible. They discard information. If one makes a calculation like a + b = c then the input contains roughly (log a + log b) bits of information whereas the answer contains log(a + b) bits which is in general much less. Somewhere in the process of transform- ing the input to the output we have lost bits. The amount of information we have lost is exactly the information needed to separate c in to a and b. There are many ways to select two numbers a and b that add up to c. So there are many inputs that could create the output. The information about the exact history of the computation is discarded by the algorithm. This leaves us with an interesting question: If there is so much information in the world and computation does not generate information, then where does the information come from? Things get more fascinating if we consider the Turing machine version of the French fries example above. Suppose we make a Turing machine
- 16. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability Computation, Information, and the Arrow of Time 3 that only erases its input and we make a movie of its execution and play it backward. What would we see? We see a machine creating information out of nothing, just the same way the toddler in the reversed movie was pulling neat French fries out of his mouth. So also in this case, if we reverse the arrow of time, destruction of information becomes creation and vice versa. In previous papers the first author has investigated the relation between learning and data compression ([2, 4]). Here we are interested in the converse problem: How do data-sets from which we can learn something emerge in the world? What processes grow information? There is a class of deterministic processes that discard or destroy in- formation. Examples are: simple erasure of bits, (lossy) data compression, and learning. There is another class of processes that seems to create infor- mation: coin flipping, growth, evolution. In general, stochastic processes create information, exactly because we are uncertain of their future, and deterministic processes discard information, precisely because the future of the process is known. The basic paradigm of a stochastic information generating process is coin flipping. If we flip a coin in such a way that the probability of heads is equal to the probability of tails, and we note the results as a binary string, then with high probability this string is random and incompressible. The string will then have maximal Kolmogorov com- plexity, i.e. a program that generates the string on a computer will be at least as long as the string itself ([8]). On the other hand if we generate a string by means of a simple deterministic program (say “For x = 1 to k print("1")”) then the string is highly compressible and by definition has a low Kolmogorov complexity which approximates log k for large enough k. In the light of these observations one could formulate the following research question: Given the fact that creation and destruction of information seem to be symmetrical over the time axis, could one develop a time-invariant description of computational processes for which creation of information is the same process as destruction of information with the time arrow re- versed? A more concise version of the same question is: Are destruction and creation of information computationally symmetrical in time? The main part of this paper is dedicated to a positive answer to this question. Prima facie it seems that we compute to get new information. So if we want to know what the exact value of 10! is, then the answer 3628800 really contains information for us. It tells us something we did not know. We also have the intuition, that the harder it is to compute a function, the more value (i.e. information) the answer contains. So 10! in a way contains
- 17. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 4 P. Adriaans & P. van Emde Boas more information than 102 . Yet from a mathematical point of view 10! and 3628800 are just different descriptions of the same number. The situation becomes even more intriguing if we turn our intention to the simulation of processes on a computer that really seem to create new information like the growth of a tree, game playing, or the execution of a genetic algorithm. What is happening here if computation cannot generate information? What is the exact relation between information generating processes that we find in our world and our abstract models of computation? In most curricula, theories about information and computation are treated in isolation. That is probably the reason why the rather funda- mental question studied in this paper up till now has received little at- tention in computer science: What is the interaction between information and computation? Samson Abramsky has posed this question in a recent publication with some urgency (without offering a definitive answer): We compute in order to gain information, but how is this possible logically or thermodynamically? How can it be reconciled with the point of view of In- formation Theory? How does information increase appear in the various extant theories? ([1], p. 487). Below we will formulate a partial answer to this question by means of an analysis of time invariant descriptions of computational processes. 1.2. A Formal Framework: Meta-computational Space In order to study the interplay between entropy, information, and compu- tation we need to develop a formal framework. For this purpose we develop the notion of meta-computational space in this section: formally, the space of the graphs of all possible computations of all possible Turing machines. The physical equivalent would be the space of all possible histories of all possible universes. C(x) will be the classical Kolmogorov complexity of a binary string x, i.e. the length of the shortest program p that computes x on a reference universal Turing machine U. Given the correspondence between natural numbers and binary strings, M consists of an enumeration of all possible self-delimiting programs for a preselected arbitrary universal Turing ma- chine U. Let x be an arbitrary bit string. The shortest program that produces x on U is x∗ = argminM∈M(U(M) = x) and the Kolmogorov complexity of x is C(x) = |x∗ |. The conditional Kolmogorov complexity of a string x given a string y is C(x|y), this can be interpreted as the length of a program for x given input y. A string is defined to be random if
- 18. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability Computation, Information, and the Arrow of Time 5 C(x) ≥ |x|. I(x) is the classical integer complexity function that assigns to each integer x another integer C(x) [8]. We will follow the standard textbook of Hopcroft, Motwani, and Ullman for the basic definitions ([7]). A Turing machine (TM) is described by a 7-tuple M = (Q, Σ, Γ, δ, q0, B, F). Here, as usual, Q is the finite set of states, Σ is the finite set of input symbols with Σ ⊂ Γ, where Γ is the complete set of tape symbols, δ is a transition function such that δ(q, X) = (p, Y, D), if it is defined, where p ∈ Q is the next state, X ∈ Γ is the symbol read in the cell being scanned, Y ∈ Γ is the symbol written in the cell being scanned, D ∈ {L, R} is the direction of the move, either left or right, q0 ∈ Q is the start state, B ∈ Γ − Σ is the blank default symbol on the tape, and F ⊂ Q is the set of accepting states. A move of a TM is determined by the current content of the cell that is scanned and the state the machine is in. It consists of three parts: (1) Change state; (2) Write a tape symbol in the current cell; (3) Move the read-write head to the tape cell on the left or right. A non-deterministic Turing machine (NTM) is equal to a deterministic TM with the exception that the range of the transition function consists of sets of triples: δ(q, X) = {(p1, Y1, D1), (p2, Y2, D2), ..., (pk, Yk, Dk)}. A TM is a reversible Turing machine (RTM) if the transition function δ(q, X) = (p, Y, D) is one-to-one, with the additional constraint that the movement D of the read-write head is uniquely determined by the target state p. Definition 1.1. An Instantaneous Description (ID) of a TM during its execution is a string X1X2...Xi−1qXiXi+1...Xn in which q is the state of the TM, the tape head is scanning the i-th head from the left, X1X2...Xn is the portion of the tape between the leftmost and the rightmost blank. Given an Instantaneous Description X1X2...Xi−1qXiXi+1...Xn it will be useful to define an Extensional Instantaneous Description (EID) X1X2...Xi−1XiXi+1...Xn, that only looks at the content of the tape and ignores the internal state of the machine and an Intensional Instantaneous Description (IID) qXiD, that only looks at the content of the current cell
- 19. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 6 P. Adriaans & P. van Emde Boas of the tape, the internal state of the machine, and the direction D in which the head will move. We make the jump from an object- to a meta-level of descriptions of computations by means of considering the set of all possible transitions between instantaneous descriptions. Definition 1.2. Let < IDM , ⊢M > be the configuration graph of all possible transformations of a machine M, i.e. IDM is the countable set of all possible instantaneous descriptions and for IDi,j ∈ IDM : IDi ⊢M IDj if and only if TM can reach IDj in one move from IDi. IDm is reachable from IDi iff there exists a sequence of transformations from one to the other: (IDi ⊢∗ M IDm) ⇔ IDi ⊢M IDj ⊢M IDk...IDl ⊢M IDm. The intensional description of such a transformation will be: (IIDi ⊢∗ M IIDm). The extensional description will be: (EIDi ⊢∗ M EIDm). Note that two machines can perform computations that are extensionally isomorphic without intensional isomorphism and vice versa. We refer here to transformations rather than computations since, in most cases, only a subpart of the configuration graph represents valid computations that begin with a start state and end in an accepting state. Note that the class of all possible instantaneous descriptions for a certain machine contains for each possible tape configuration, at each possible position of the head on the tape, an instance for each possible internal state. Most of these configurations will only be the result, or lead to, fragments of computations. On the other hand, all valid computations that begin with a start state and either continue forever or end in an accepting state, will be represented in the configuration graph. Note that there is a strict relation between the structure of the transi- tion function δ and the configuration graph: For a deterministic machine the configuration graph has only one outgoing edge for each configuration, for a non-deterministic machine the configuration graph can have multiple outgoing edges per ID, for a reversible machine the graph consists only of a number of linear paths without bifurcations either way. Lemma 1.1. Let M be a Turing machine. We have C(< IDM , ⊢M >) < C(M) + O(1).
- 20. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability Computation, Information, and the Arrow of Time 7 Proof. Given M the graph < IDM , ⊢M > can be constructed by the fol- lowing algorithm: Create IDM by enumerating the language of all possible IDs, at each step of this process run M for one step on all IDs created so far and add appropriate edges to ⊢M when M transforms IDi in IDj. The finite object M and the infinite object IDM , ⊢M identify the same structure. We use here two variants of the Kolmogorov complexity: The complexity of the finite object M is defined by the smallest program that computes the object on a universal Turing machine and then halts; the complexity of IDM , ⊢M is given by the shortest program that creates the object in an infinite run. Definition 1.3. Given an enumeration of Turing machines the meta- computational space is defined as the disjunct sum of all configuration graphs IDMi , ⊢Mi for i ∈ N. The meta-computational space is a very rich object in which we can study a number of fundamental questions concerning the interaction be- tween information and computation. We can also restrict ourselves to the study of either extensional or intensional descriptions of computations and this will prove useful, e.g. when we want to study the class of all compu- tational histories that have descriptions with isomorphic pre- or suffixes. For the moment we want to concentrate on time symmetries in meta- computational space. 1.3. Time Symmetries in Meta-computational Space In this paragraph we study the fact that some well-known classes of compu- tational processes can be interpreted as each others’ symmetrical images in time, i.e. processes in one class can be described as processes in the other class with the time arrow reversed, or, to say it differently, as processes tak- ing place in negative time. We can reverse the time arrow for all possible computations of a certain machine by means of reversing all the edges in the computational graph. This motivates the following notation: Definition 1.4. (IDi ⊢ IDj) ⇔ (IDj ⊣ IDi) (IDi ⊢∗ IDk) ⇔ (IDk ⊣∗ IDi).
- 21. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 8 P. Adriaans P. van Emde Boas The analysis of valid computations of T M can now be lifted to the study of reachability in the configuration graph. The introduction of such a meta-computational model allows us to study a much more general class of computations in which the arrow of time can be reversed. We will introduce the following shorthand notation that allows us to say that M−1 is the same machine as M with the arrow of time reversed: M = IDM , ⊢M ⇔ M−1 = IDM , ⊣M . Intuitively the class of languages that is “computed” in negative time by a certain Turing machine is the class of accepting tape configurations that can be reached from a start state. We have to stress however, that moving back in time in the configuration graph describes a process that is fundamentally different from the standard notion of “computation” as we know it. We give some differences: • The standard definition of a Turing machine knows only one starting state and possibly several accepting states. Computing in negative time will trace back from several accepting states to one start state. • The interpretation of the δ-function or relation is different. In positive time we use the δ-function to decide which action to take given a certain state-symbol combination. In negative time this situation is reversed: We use the δ-function to decide which state-symbol-move combination could have led to a certain action. • At the start of a computation there could be a lot of rubbish on the tape that is simply not used during the computation. All computations starting with arbitrary rubbish are in the configuration graph. We want to exclude these from our definitions and stick to some minimal definition of the input of a computation in negative time. In order to overcome these difficulties the following lemma will be useful: Lemma 1.2. (Minimal Input-Output Reconstruction) If an inten- sional description of a fragment of a (deterministic or non-deterministic) computation of a machine M: (IIDi ⊢∗ M IIDm) can be interpreted as the trace of a valid computation then there exist a minimal input configuration IDi and a minimal output configuration IDm for which M will reach IDm starting at IDi. Otherwise the minimal input and output configuration are undefined. Proof. The proof first gives a construction for the minimal output in a positive sweep and then the minimal input in a negative sweep.
- 22. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability Computation, Information, and the Arrow of Time 9 Positive sweep: Note that (IIDi ⊢∗ M IIDm) consists of a sequence of descriptions: qiXiDi ⊢ qi+1Xi+1Di+1 ⊢ ... ⊢ qmXmDm. Reconstruct a computation in the following way: Start with an infinite tape for which all of the symbols are unknown. Position the read-write head at an arbitrary cell and perform the following interpretation operation: Interpret this as the state-symbol-move configuration qiXiDi. Now we know the contents of the cell Xi, the state qi, and the direction D of the move of the read-write head. The action will consist of writing a symbol in the current cell and moving the read-write head left or right. Perform this action. The content of one cell is now fixed. Now there are two possibilities: (1) We have the read-write head in a new cell with unknown content. From the intensional description we know that the state-symbol combination is qi+1Xi+1Di+1, so we can repeat the interpretation operation for the new cell. (2) We have visited this cell before in our reconstruction and it already contains a symbol. From the intensional description we know that the state-symbol combination should be qi+1Xi+1Di+1. If this is inconsis- tent with the content of the current cell, the reconstruction stops and the minimal output is undefined. If not, we can repeat the interpreta- tion operation for the new cell. Repeat this operation till the intensional description is exhausted. Cells on the tape that still have unknown content have not been visited by the computational process: We may consider them to contain blanks. We now have the minimal output configuration on the tape IDm. Negative sweep: start with the minimal output configuration IDm. We know the current location of the read-write head and the content of the cell. From the intensional description (IIDi ⊢∗ M IIDm) we know which state- symbol combination qmXmDm has led to IDm: from this we can construct IDm−1. Repeat this process till the intensional description is exhausted and we read IDi, which is the minimal input configuration. Lemma 1.2 gives us a way to tame the richness of the configuration graphs: We can restrict ourselves to the study of computational processes that are intensionally equivalent, specifically intensionally equivalent pro- cesses that start with a starting state and end in an accepting state. This facilitates the following definition:
- 23. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 10 P. Adriaans P. van Emde Boas Definition 1.5. If (IIDi ⊢∗ M IIDm) is an intensional description of a computation then INPUT(IIDi ⊢∗ M IIDm) = x gives the minimal input x and OUTPUT(IIDi ⊢∗ M IIDm) = y gives the minimal output y. With some abuse of notation we will also apply these functions to histories of full IDs. Definition 1.6. Given a Turing machine M the language recognized by its counterpart M−1 in negative time is the set of minimal output configura- tions associated with intensional descriptions of computations on M that begin in a start state and end in an accepting state. Definition 1.7. The class P−1 is the class of languages that are recognized by an M−1 i with i ∈ N in time polynomial to the length of minimal input configuration. Note that, after a time reversal operation, the graph of a deterministic machine is transformed into a specific non-deterministic graph with the characteristic that each ID has only one incoming edge. We will refer to such a model of computation as quasi-reversible. The essence of this analysis is that, given a specific machine M, we can study its behavior under reversal of the arrow of time. We can use the symmetry between deterministic and quasi-reversible computing in proofs. Whatever we prove about the execution of a program on M also holds for M−1 with the time reversed and vice versa. Let QRT M be the class of quasi-reversible non-deterministic machines that are the mirror image in time of the class of deterministic machines T M, and QRP be the class of languages that can be recognized by QRT M in polynomial time. The lemma below is at first sight quite surprising. The class of languages that we can recognize non-deterministically in polynomial time is the same class as the class of polynomial quasi-reversible languages: Lemma 1.3. The class LQRP of languages recognized by a QRT M in poly- nomial time is NP. Proof. 1) LQRP ⊆ NP: The class of languages recognized by quasi- reversible machines is a subclass of the class of languages recognized by
- 24. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability Computation, Information, and the Arrow of Time 11 non-deterministic machines. This is trivial since there is a non-deterministic machine that produces any {0, 1}≤k in time k. 2) NP ⊆ LQRP : The class NP is defined in a standard way in terms of a checking relation R ⊆ Σ∗ × Σ∗ 1 for some finite alphabets Σ∗ and Σ∗ 1. We associate with each such relation R a language LR over Σ∗ ∪Σ∗ 1 ∪# defined by LR = {w#y|R(w, y)} where the symbol # is not in Σ. We say that R is polynomial-time iff LR ∈ P. Now we define the class NP of languages by the condition that a language L over Σ is in NP iff there is k ∈ N and a polynomial-time checking relation R such that for all w ∈ Σ∗ , w ∈ L ⇔ ∃y(|y| |w|k R(w, y)) where |w| and |y| denote the lengths of w and y, respectively. Suppose that M implements a polynomial-time checking relation for R. Adapt M to form M′ that takes R(w, y) as input and erases y from the tape after checking the relation, the transformation of M to M−1 is polynomial. The corresponding QRTM M′−1 will start with guessing a value for y non- deterministically and will finish in a configuration for which the relation R(w, y) holds in polynomial time since |y| |w|k and the checking relation R is polynomial. We can formulate the following result: Theorem 1.1. NP = P−1 Proof. Immediate consequence of Lemma 1.3 and Definition 1.7. NP is the class of languages that can be recognized by deterministic Turing machines in negative time. This shows that quasi-reversible com- puting is in a way a more natural model of non-deterministic computing than the classical full-blown non-deterministic model. The additional power is unnecessary. 1.4. The Interplay of Computation and Information We now look at the interplay between information and computation. The tool we use will be the study of the changes in C(IDt), i.e. changes in the Kolmogorov complexity of instantaneous descriptions over time. We make some observations:
- 25. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 12 P. Adriaans P. van Emde Boas • If IDi ⊢M IDj then the information distance between the instanta- neous descriptions IDi and IDj is log k + 1 at most where k is the number of internal states of M. • If EIDi ⊢M EIDj then the information distance between the exten- sional descriptions EIDi and EIDj is 1 bit at most. • If IIDi ⊢M IIDj then the information distance between the intensional descriptions IIDi and IIDj is log k + 2 at most where k is the number of internal states of M. • Let x be the minimal input of a computational fragment (IIDi ⊢∗ M IIDm) and let y be the minimal output. We have C(x|IIDi ⊢∗ M IIDm) = C(y|IIDi ⊢∗ M IIDm) = O(1). This is an immediate consequence of Lemma 1.2. We can now identify some interesting typical machines: • No machine can produce information faster than 1 bit per computa- tional step. There is indeed a non-deterministic machine that reaches this “speed”: the non-deterministic “coin-flip” automaton that writes random bits. For such an automaton we have with high probability C(IDt) ≈ t. In negative time this machine is the maximal eraser. It erases information with the maximum “speed” of 1 bit per computa- tional step. • A unary counting machine produces information with a maximum speed of log t. Note that C(t) = I(t), i.e. the complexity at time t is equal to the value of the integer complexity function. The function I(x) has indefinite “dips” in complexity, i.e. at those places where it approaches a highly compressible number. When t approaches such a dip the information produced by a unary counting machine will drop as the machine continues to write bits. The counter part of the unary counter in negative time is the unary eraser. It erases information with the maximal speed of log t, although at times it will create information by erasing bits. • The slowest information producer for its size is the busy-beaver func- tion. When it is finished it will have written an enormous number of bits with a conditional complexity of O(1). Its counterpart in nega- tive time will be a busy-glutton automaton that “eats” an enormous number of bits of an exact size.
- 26. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability Computation, Information, and the Arrow of Time 13 These insights allow us to draw a picture that tells us how information and computation are intertwined in a deterministic process. Figure 1.1. Schematic representation of the various types of complexity estimates in- volved in a deterministic computation. The complexity of the history of a computation is related to the com- plexity of the input given the output. There are two forms of complexity involved in a deterministic computation: • Program Complexity: This is the complexity of the input and its sub- sequent configurations during the process. It cannot grow during the computation. Most computations reduce program complexity. • Program Counter Complexity: This is the descriptive complexity of the program counter during the execution of the process. It is 0 at the beginning, grows to log a in the middle, and reduces to 0 again at the end of the computation.
- 27. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 14 P. Adriaans P. van Emde Boas The relationship between these forms of complexity is given by the following theorem: Theorem 1.2. (Information exchange in Deterministic Comput- ing) Suppose M is a deterministic machine and IDi ⊢M IDa is a fragment of an accepting computation, where IDm contains an accepting state. For every i ≤ k ≤ a we have: (1) Determinism: C(IDi+k+1 ⊢M IDa|M, IDi+k) = O(1), i.e. at any moment of time if we have the present configuration and the definition of M then the future of the computation is known. (2) Program Counter Complexity from the start: C(IDt|ID0, M) (log k) + O(1), this constraint is known during the computation. (3) Program Counter Complexity from the end: C(IDt|ID0, M) (log a− k) + O(1), this constraint is not known during the computation. (4) Program complexity: C((IIDi+k ⊢∗ M IIDa)|M) = C(INPUT(IIDi+k ⊢∗ M IIDa)|M) + O(1). Proof. (1) Trivial, since M is deterministic. (2) Any state IDk at time k can be identified by information of size log k if the initial configuration and M are known. (3) Any state IDk at time k can be identified by information of size log(a− k) if the total description of the accepting computational process and M are known. (4) By the fact that the computation is deterministic it can be recon- structed from the minimal input, given M. By Lemma 1.2, given M, the minimal input can be reconstructed from (IIDi ⊢∗ M IIDa). This gives the equality modulo O(1). We cannot prove such a nice equality for the minimal output. Note that even if the following inequality holds: C((IIDi ⊢∗ M IIDa)|M) ≥ C((IIDi+k ⊢∗ M IIDa)|M) + O(1) this does not imply that: C(OUTPUT(IIDi ⊢∗ M IIDa)|M) ≥ C(OUTPUT(IIDi+k ⊢∗ M IIDa)|M)+O(1). As a counterexample, observe that a program that erases a random string has a string of blanks as minimal output. A longer string still can have a lower Kolmogorov complexity.
- 28. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability Computation, Information, and the Arrow of Time 15 In computations that use counters, Program Complexity and Program Counter Complexity are mixed up during the execution. In fact one can characterize various types of computations by means of their “information signature”. Informally, at extremes of the spectrum, one could distinguish: • Pure Information Discarding Processes: in such processes the program counter does not play any role. They reach an accepting state by means of systematically reducing the input. Summation of a set of numbers, or erasing of a string are examples. • Pure Counting Processes: For x=1 to i write(1): The condi- tional complexity of the tape configuration grows from 0 to log i and then diminishes to 0 again. • Pure Search Processes: In such processes the input is not reduced but is kept available during the whole process. The information in the program counter is used to explore the search space. Standard decision procedures for NP-hard programs, where the checking function is tested on an enumeration of all possible solutions, are an example. A deeper analysis of various information signatures of computational pro- cesses and their consequences for complexity theory is a subject of future work. 1.5. Discussion We can draw some conclusions and formulate some observations on the basis of the analysis given above. 1) Erasing and creating information are indeed, as suggested in the introduction, from a time invariant computational point of view the same processes: The quasi-reversible machine that is associated with a simple de- terministic machine that erases information is a non-deterministic machine writing arbitrary bit-strings on the tape. This symmetry also implies that creation of information in positive time involves destruction of information in negative time. 2) The class of quasi-reversible machines indeed describes the class of data-sets from which we can learn something in the following way: If L is the language accepted by M then M−1 generates L. M−1 is an informer for L in the sense of [6], every sentence in L will be non-deterministically produced by M−1 in the limit. QRT M is the class of all informers for type-0 languages.
- 29. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 16 P. Adriaans P. van Emde Boas 3) These insights suggests that we can describe stochastic processes in the real world as deterministic processes in negative time: e.g. throwing a dice in positive time is erasing information about its “future” in negative time, the evolution of species in positive time could be described as the “deterministic” computation of their ancestor in negative time. A necessary condition for the description of such growth processes as computational processes is that the number of bits that can be produced per time unit is restricted. A stochastic interpretation of a QRTM can easily be developed by assigning a set of probabilities to each split in the δ relation. The resulting stochastic-QRTM is a sufficient statistic for the data-sets that are generated. 4) The characterization of the class NP in terms of quasi-reversible com- puting seems to be more moderate than the classical description in terms of full non-deterministic computing. The full power of non-deterministic computing is never realized in a system with only one time direction. 5) Processes like game playing and genetic algorithms seem to be meta- computational processes in which non-deterministic processes (throwing a dice, adding mutations) seem to be intertwined with deterministic phases (making moves, checking the fitness function). 6) Time-symmetry has consequences for some philosophical positions. The idea that the evolution of our universe can be described as a determin- istic computational process has been proposed by several authors (Zuse, Bostrom, Schmidthuber, Wolfram [10], Lloyd [9], etc.). Nowadays it is re- ferred to as pancomputationalism [5]. If deterministic computation is an information discarding process then it implies that the amount of informa- tion in the universe rapidly decreases. This contradicts the second law of thermodynamics. On the other hand, if the universe evolves in a quasi- reversible way, selecting possible configurations according to some quasi- reversible computational model, it computes the Big Bang in negative time. The exact implications of these observations can only be explained by means of the notion of facticity [3], but that is another discussion. The concept of quasi-reversible computing seems to be relevant for these discussions [2]. 1.6. Conclusion Computing is moving through meta-computational space. For a fixed Tur- ing machine Mi such movement is confined to one local infinite graph IDMi , ⊢Mi . If Mi is deterministic then M−1 i is non-deterministic. If M is information discarding then M−1 “creates” information. The two
- 30. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability Computation, Information, and the Arrow of Time 17 fundamental complexities involved in a deterministic computation are Pro- gram Complexity and Program Counter Complexity. Programs can be classified in terms of their “information signature” with pure counting pro- grams and pure information discarding programs as two ends of the spec- trum. The class NP is simply the class of polynomial deterministic time calculations in negative time. Thinking in terms of meta-computational space allows us to conceptualize computation as movement in a certain space and is thus a source of new intuitions to study computation. Specif- ically a deeper analysis of various information signatures of computational (and other) processes is a promising subject for further study. References [1] S. Abramsky. Information, Processes and Games. In eds. P. W. Adriaans and J. F. A. K. van Benthem, Handbook of the Philosophy of Information, In Handbooks of the Philosophy of Science, series edited by D. M. Gabbay, P. Thagard and J. Woods, pp. 483–550. Elsevier, (2008). [2] P. W. Adriaans and J. F. A. K. van Benthem, eds., Handbook of the Phi- losophy of Information. In Handbooks of the Philosophy of Science, series edited by D. M. Gabbay, P. Thagard and J. Woods. Elsevier, (2008). [3] P. W. Adriaans, Between order and chaos: The quest for meaningful infor- mation, Theor. Comp. Sys. 45(4), (2009). [4] P. W. Adriaans and P. Vitányi, Approximation of the two-part MDL code, IEEE Transactions on Information Theory. 55(1), 444–457, (2009). [5] L. Floridi. Trends in the philosophy of information. In eds. P. W. Adriaans and J. F. A. K. van Benthem, Handbook of the Philosophy of Information, In Handbooks of the Philosophy of Science, series edited by D. M. Gabbay, P. Thagard and J. Woods, pp. 113–132. Elsevier, (2008). [6] E. M. Gold, Language identification in the limit, Information and Control. 10(5), 447–474, (1967). [7] J. E. Hopcroft, R. Motwani, and J. D. Ullman, Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, (2001), second edi- tion. [8] M. Li and P. Vitányi, An Introduction to Kolmogorov Complexity and its Applications. Springer-Verlag, (2008), third edition. [9] S. Lloyd, Ultimate physical limits to computation, Nature. 406, 1047–1054, (2000). [10] S. Wolfram, A New Kind of Science. Wolfram Media Inc., (2002).
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- 32. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability Chapter 2 The Isomorphism Conjecture for NP Manindra Agrawal ∗ Indian Institute of Technology Kanpur, India E-mail: manindra@iitk.ac.in In this chapter, we survey the arguments and known results for and against the Isomorphism Conjecture. Contents 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Formulation and Early Investigations . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 A Counter Conjecture and Relativizations . . . . . . . . . . . . . . . . . . . . 26 2.5 The Conjectures for Other Classes . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6 The Conjectures for Other Reducibilities . . . . . . . . . . . . . . . . . . . . . 30 2.6.1 Restricting the input head movement . . . . . . . . . . . . . . . . . . . 31 2.6.2 Reducing space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.6.3 Reducing depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.7 A New Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.8 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.1. Introduction The Isomorphism Conjecture for the class NP states that all polynomial- time many-one complete sets for NP are polynomial-time isomorphic to each other. It was made by Berman and Hartmanis [21]a , inspired in part by a corresponding result in computability theory for computably enumerable sets [50], and in part by the observation that all the existing NP-complete ∗N Rama Rao Professor, Indian Institute of Technology, Kanpur. Research supported by J C Bose Fellowship FLW/DST/CS/20060225. aThe conjecture is also referred as Berman–Hartmanis Conjecture after the proposers. 19
- 33. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 20 M. Agrawal sets known at the time were indeed polynomial-time isomorphic to each other. This conjecture has attracted a lot of attention because it predicts a very strong structure of the class of NP-complete sets, one of the funda- mental classes in complexity theory. After an initial period in which it was believed to be true, Joseph and Young [40] raised serious doubts against the conjecture based on the notion of one-way functions. This was followed by investigation of the conjecture in relativized worlds [27, 33, 46] which, on the whole, also suggested that the conjecture may be false. However, disproving the conjecture using one- way functions, or proving it, remained very hard (either implies DP 6= NP). Hence research progressed in three distinct directions from here. The first direction was to investigate the conjecture for complete degrees of classes bigger than NP. Partial results were obtained for classes EXP and NEXP [20, 29]. The second direction was to investigate the conjecture for degrees other than complete degrees. For degrees within the 2-truth-table-complete degree of EXP, both possible answers to the conjecture were found [41, 43, 44]. The third direction was to investigate the conjecture for reducibilities weaker than polynomial-time. For several such reducibilities it was found that the isomorphism conjecture, or something close to it, is true [1, 2, 8, 16]. These results, especially from the third direction, suggest that the Iso- morphism Conjecture for the class NP may be true contrary to the evidence from the relativized world. A recent work [13] shows that if all one-way functions satisfy a certain property then a non-uniform version of the con- jecture is true. An excellent survey of the conjecture and results related to the first two directions is in [45]. 2.2. Definitions In this section, we define most of the notions that we will need. We fix the alphabet to Σ = {0, 1}. Σ∗ denotes the set of all finite strings over Σ and Σn denotes the set of strings of size n. We start with defining the types of functions we use. Definition 2.1. Let r be a resource bound on Turing machines. Function f, f : Σ∗ 7→ Σ∗ , is r-computable if there exists a Turing machine (TM, in short) M working within resource bound of r that computes f. We also refer to f as an r-function.
- 34. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability The Isomorphism Conjecture for NP 21 Function f is size-increasing if for every x, |f(x)| |x|. f is honest if there exists a polynomial p(·) such that for every x, p(|f(x)|) |x|. For function f, f−1 denotes a function satisfying the property that for all x, f(f−1 (f(x))) = f(x). We say f is r-invertible if some f−1 is r-computable. For function f, its range is denoted as: range(f) = {y | (∃x) f(x) = y}. We will be primarily interested in the resource bound of polynomial- time, and abbreviate it as p. We now define several notions of reducibilities. Definition 2.2. Let r be a resource bound. Set A r-reduces to set B if there exists an r-function f such that for every x, x ∈ A iff f(x) ∈ B. We also write this as A ≤r m B via f. Function f is called an r-reduction of A to B. Similarly, A ≤r 1 B (A ≤r 1,si B; A ≤r 1,si,i B) if there exists a 1-1 (1-1 and size-increasing; 1-1, size-increasing and r-invertible) r-function f such that A ≤r m B via f. A ≡r m B if A ≤r m B and B ≤r m A. An r-degree is an equivalence class induced by the relation ≡r m. Definition 2.3. A is r-isomorphic to B if A ≤r m B via f where f is a 1-1, onto, r-invertible r-function. The definitions of complexity classes DP, NP, PH, EXP, NEXP etc. can be found in [52]. We define the notion of completeness we are primarily interested in. Definition 2.4. Set A is r-complete for NP if A ∈ NP and for every B ∈ NP, B ≤r m A. For r = p, set A is called NP-complete in short. The class of r-complete sets is also called the complete r-degree of NP. Similarly one defines complete sets for other classes. The Satisfiability problem (SAT) is one of the earliest known NP- complete problems [25]. SAT is the set of all satisfiable propositional Boolean formulas. We now define one-way functions. These are p-functions that are not p-invertible on most of the strings. One-way functions are one of the fun- damental objects in cryptography. Without loss of generality (see [30]), we can assume that one-way func- tions are honest functions f for which the input length determines the output length, i.e., there is a length function ℓ such that |f(x)| = ℓ(|x|) for all x ∈ Σ∗ .
- 35. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 22 M. Agrawal Definition 2.5. Function f is a s(n)-secure one-way function if (1) f is a p-computable, honest function and (2) the following holds for every polynomial-time randomized Turing machine M and for all sufficiently large n: Pr x∈U Σn [ f(M(f(x))) = f(x) ] 1 s(n) . In the above, the probability is also over random choices of M, and x ∈U Σn mean that x is uniformly and randomly chosen from strings of size n. We impose the property of honesty in the above definition since a func- tion that shrinks length by more than a polynomial is trivially one-way. It is widely believed that 2nǫ -secure one-way functions exist for some ǫ 0. We give one example. Start by defining a modification of the multiplication function: Mult(x, y) = 1z if x and y are both prime numbers and z is the product x ∗ y 0xy otherwise. In the above definition, (·, ·) is a pairing function. In this paper, we assume the following definition of (·, ·): (x, y) = xyℓ where |ℓ| = ⌈log |xy|⌉ and ℓ equals |x| written in binary. With this definition, |(x, y)| = |x|+|y|+ ⌈log |xy|⌉. This definition is easily extended for m-tuples for any m. Mult is a p-function since testing primality of numbers is in DP [11]. Computing the inverse of Mult is equivalent to factorization, for which no efficient algorithm is known. However, Mult is easily invertible on most of the inputs, e.g., when any of x and y is not prime. The density estimate for prime numbers implies that Mult is p-invertible on at least 1 − 1 nO(1) fraction of inputs. It is believed that Mult is (1 − 1 nO(1) )-secure, and it remains so even if one lets the TM M work for time 2nδ for some small δ 0. From this assumption, one can show that arbitrary concatenation of Mult: MMult(x1, y1, x2, y2, . . . , xm, ym) = Mult(x1, y1) · Mult(x2, y2) · · · Mult(xm, ym) is a 2nǫ -secure one-way function [30](p. 43). One-way functions that are 2nǫ -secure are not p-invertible almost any- where. The weakest form of one-way functions are worst-case one-way functions:
- 36. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability The Isomorphism Conjecture for NP 23 Definition 2.6. Function f is a worst-case one-way function if (1) f is a p-computable, honest function, and (2) f is not p-invertible. 2.3. Formulation and Early Investigations The conjecture was formulated by Berman and Hartmanis [21] in 1977. Part of their motivation for the conjecture was a corresponding result in computability theory for computably enumerable sets [50]: Theorem 2.1. (Myhill) All complete sets for the class of computably enu- merable sets are isomorphic to each other under computable isomorphisms. The non-trivial part in the proof of this theorem is to show that complete sets for the class of computably enumerable sets reduce to each other via 1-1 reductions. It is then easy to construct isomorphisms between the complete sets. In many ways, the class NP is the resource bounded analog of the computably enumerable class, and polynomial-time functions the analog of computable functions. Hence it is natural to ask if the resource bounded analog of the above theorem holds. Berman and Hartmanis noted that the requirement for p-isomorphisms is stronger. Sets reducing to each other via 1-1 p-reductions does not guar- antee p-isomorphisms as p-functions do not have sufficient time to perform exponential searches. Instead, one needs p-reductions that are 1-1, size- increasing, and p-invertible: Theorem 2.2. (Berman–Hartmanis) If A ≤p 1,si,i B and B ≤p 1,si,i A then A is p-isomorphic to B. They defined the paddability property which ensures the required kind of reductions. Definition 2.7. Set A is paddable if there exists a p-computable padding function p, p : Σ∗ × Σ∗ 7→ Σ∗ , such that: • Function p is 1-1, size-increasing, and p-invertible, • For every x, y ∈ Σ∗ , p(x, y) ∈ A iff x ∈ A. Theorem 2.3. (Berman–Hartmanis) If B ≤p m A and A is paddable, then B ≤p 1,si,i A. Proof. Suppose B ≤p m A via f. Define function g as: g(x) = p(f(x), x). Then, x ∈ B iff f(x) ∈ A iff g(x) = p(f(x), x) ∈ A. By its definition and
- 37. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 24 M. Agrawal the fact that p is 1-1, size-increasing, and p-invertible, it follows that g is also 1-1, size-increasing, and p-invertible. Berman and Hartmanis next showed that the known complete sets for NP at the time were all paddable and hence p-isomorphic to each other. For example, the following is a padding function for SAT: pSAT (x, y1y2 · · · ym) = x ∧ m ^ i=1 zi m ^ i=1 ci where ci = zm+i if bit yi = 1 and ci = z̄i if yi = 0 and the Boolean variables z1, z2, . . ., z2m do not occur in the formula x. This observation led them to the following conjecture: Isomorphism Conjecture. All NP-complete sets are p-isomorphic to each other. The conjecture immediately implies DP 6= NP: Proposition 2.1. If the Isomorphism Conjecture is true then DP 6= NP. Proof. If DP = NP then all sets in DP are NP-complete. However, DP has both finite and infinite sets and there cannot exist an isomorphism between a finite and an infinite set. Hence the Isomorphism Conjecture is false. This suggests that proving the conjecture is hard because the problem of separating DP from NP has resisted all efforts so far. A natural question, therefore, is: Can one prove the conjecture assuming a reasonable hypoth- esis such as DP 6= NP? We address this question later in the paper. In their paper, Berman and Hartmanis also asked a weaker question: Does DP 6= NP imply that no sparse set can be NP-complete? Definition 2.8. Set A is sparse if there exist constants k, n0 0 such that for every n n0, the number of strings in A of length ≤ n is at most nk . This was answered in the affirmative by Mahaney [49]: Theorem 2.4. (Mahaney) If DP 6= NP then no sparse set is NP- complete. Proof Sketch. We give a proof based on an idea of [9, 19, 51]. Suppose there is a sparse set S such that SAT ≤p m S via f. Let F be a Boolean formula on n variables. Start with the set T = {F} and do the following:
- 38. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability The Isomorphism Conjecture for NP 25 Replace each formula F̂ ∈ T by F̂0 and F̂1 where F̂0 and F̂1 are obtained by setting the first variable of F̂ to 0 and 1 respectively. Let T = {F1, F2, . . . , Ft}. If t exceeds a certain threshold t0, then let Gj = F1 ∨ Fj and zj = f(Gj ) for 1 ≤ j ≤ t. If all zj’s are distinct then drop F1 from T . Otherwise, zi = zj for some i 6= j. Drop Fi from T and repeat until |T | ≤ t0. If T has only formulas with no variables, then output Satisfiable if T contains a True formula else output Unsatisfiable. Otherwise, go to the beginning of the algorithm and repeat. The invariant maintained during the entire algorithm is that F is sat- isfiable iff T contains a satisfiable formula. It is true in the beginning, and remains true in each iteration after replacing every formula F̂ ∈ T with F̂0 and F̂1. The threshold t0 must be such that t0 is a upper bound on the number of strings in the set S of size maxj |f(Gj)|. This is a polynomial in |F| since |Gj| ≤ 2|F|, f is a p-function, and S is sparse. If T has more than t0 formulas at any stage then the algorithm drops a formula from T . This formula is F1 when all zj’s are distinct. This means there are more than t0 zj’s all of size bounded by maxj |f(Gj)|. Not all of these can be in S due to the choice of t0 and hence F1 6∈ SAT. If zi = zj then Fi is dropped. If Fi is satisfiable then so is Gi. And since zi = zj and f is a reduction of SAT to S, Gj is also satisfiable; hence either F1 or Fj is satisfiable. Therefore dropping Fi from T maintains the invariant. The above argument shows that the size of T does not exceed a poly- nomial in |F| at any stage. Since the number of iterations of the algorithm is bounded by n ≤ |F|, the overall time complexity of the algorithm is polynomial. Hence SAT ∈ DP and therefore, DP = NP. The “searching-with-pruning” technique used in the above proof has been used profitably in many results subsequently. The Isomorphism Con- jecture, in fact, implies a much stronger density result: All NP-complete sets are dense. Definition 2.9. Set A is dense if there exist constants ǫ, n0 0 such that for every n n0, the number of strings in A of length ≤ n is at least 2nǫ . Buhrman and Hitchcock [22] proved that, under a plausible hypothesis, every NP-complete set is dense infinitely often: Theorem 2.5. (Buhrman–Hitchcock) If PH is infinite then for any NP-complete set A, there exists ǫ 0 such that for infinitely many n, the number of strings in A of length ≤ n is at least 2nǫ .
- 39. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 26 M. Agrawal Later, we show that a stronger density theorem holds if 2nǫ -secure one- way functions exist. 2.4. A Counter Conjecture and Relativizations After Mahaney’s result, there was not much progress on the conjecture although researchers believed it to be true. However, this changed in 1984 when Joseph and Young [40] argued that the conjecture is false. Their argument was as follows (paraphrased by Selman [53]). Let f be any 1-1, size-increasing, 2nǫ -secure one-way function. Consider the set A = f(SAT). Set A is clearly NP-complete. If it is p-isomorphic to SAT, there must exist a 1-1, honest p-reduction of SAT to A which is also p-invertible. However, the set A is, in a sense, a “coded” version of SAT such that on most of the strings of A, it is hard to “decode” it (because f is not p-invertible on most of the strings). Thus, there is unlikely to be a 1-1, honest p-reduction of SAT to A which is also p-invertible, and so A is unlikely to be p-isomorphic to SAT. This led them to make a counter conjecture: Encrypted Complete Set Conjecture. There exists a 1-1, size- increasing, one-way function f such that SAT and f(SAT) are not p- isomorphic to each other. It is useful to observe here that this conjecture is false in computable setting: The inverse of any 1-1, size-increasing, computable function is also computable. The restriction to polynomial-time computability is what gives rise to the possible existence of one-way functions. It is also useful to observe that this conjecture too implies DP 6= NP: Proposition 2.2. If the Encrypted Complete Set Conjecture is true then DP 6= NP. Proof. If DP = NP then every 1-1, size-increasing p-function is also p-invertible. Hence for every such function, SAT and f(SAT) are p- isomorphic. The Encrypted Complete Set conjecture fails if one-way functions do not exist. Can it be shown to follow from the existence of strong one- way functions, such as 2nǫ -secure one-way functions? This is not clear. (In fact, later we argue the opposite.) Therefore, to investigate the two conjectures further, the focus moved to relativized worlds. Building on a result of Kurtz [42], Hartmanis and Hemachandra [33] showed that there
- 40. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability The Isomorphism Conjecture for NP 27 is an oracle relative to which DP = UP and the Isomorphism Conjecture is false. This shows that both the conjectures fail in a relativized world since DP = UP implies that no one-way functions exist. Kurtz, Mahaney, and Royer [46] defined the notion of scrambling func- tions: Definition 2.10. Function f is scrambling function if f is 1-1, size- increasing, p-computable, and there is no dense polynomial-time subset in range(f). Kurtz et al. observed that, Proposition 2.3. If scrambling functions exist then the Encrypted Com- plete Set Conjecture is true. Proof. Let f be a scrambling function, and consider A = f(SAT). Set A is NP-complete. Suppose it is p-isomorphic to SAT and let p be the isomorphism between SAT and A. Since SAT has a dense polynomial-time subset, say D, p(D) is a dense polynomial time subset of A. This contradicts the scrambling property of f. Kurtz et al., [46], then showed that, Theorem 2.6. (Kurtz, Mahaney, Royer) Relative to a random oracle, scrambling functions exist. Proof Sketch. Let O be an oracle. Define function f as: f(x) = O(x)O(x1)O(x11) · · · O(x12|x| ) where O(z) = 1 if z ∈ O, 0 otherwise. For a random choice of O, f is 1-1 with probability 1. So, f is a 1-1, size-increasing, pO -computable function. Suppose a polynomial-time TM M with oracle O accepts a subset of range(f). In order to distinguish a string in range of f from those outside, M needs to check the answer of oracle O on several unique strings. And since M can query only polynomially many strings from O, M can accept only a sparse subset of range(f). Therefore, relative to a random oracle, the Encrypted Complete Set Conjecture is true and the Isomorphism Conjecture is false. The question of existence of an oracle relative to which the Isomorphism Conjecture is true was resolved by Fenner, Fortnow, and Kurtz [27]: Theorem 2.7. (Fenner, Fortnow, Kurtz) There exists an oracle rela- tive to which Isomorphism Conjecture is true.
- 41. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 28 M. Agrawal Thus, there are relativizations in which each of the three possible an- swers to the two conjectures is true. However, the balance of evidence provided by relativizations is towards the Encrypted Complete Set Conjec- ture since properties relative to a random oracle are believed to be true in unrelativized world too.b 2.5. The Conjectures for Other Classes In search of more evidence for the two conjectures, researchers translated them to classes bigger than NP. The hope was that diagonalization argu- ments that do not work within NP can be used for these classes to prove stronger results about the structure of complete sets. This hope was real- ized, but not completely. In this section, we list the major results obtained for classes EXP and NEXP which were the two main classes considered. Berman [20] showed that, Theorem 2.8. (Berman) Let A be a p-complete set for EXP. Then for every B ∈ EXP, B ≤p 1,si A. Proof Sketch. Let M1, M2, . . . be an enumeration of all polynomial-time TMs such that Mi halts, on input x, within time |x||i| + |i| steps. Let B ∈ EXP and define B̂ to be the set accepted by the following algorithm: Input (i, x). Let Mi(i, x) = y. If |y| ≤ |x|, accept iff y 6∈ A. If there exists a z, z x (in lexicographic order), such that Mi(i, z) = y, then accept iff z 6∈ B. Otherwise, accept iff x ∈ B. The set B̂ is clearly in EXP. Let B̂ ≤p m A via f. Let the TM Mj compute f. Define function g as: g(x) = f(j, x). It is easy to argue that f is 1-1 and size-increasing on inputs of the form (j, ⋆) using the definition of B̂ and the fact that f is a reduction. It follows that g is a 1-1, size-increasing p-reduction of B to A. Remark 2.1. A case can be made that the correct translation of the iso- morphism result of [50] to the polynomial-time realm is to show that the complete sets are also complete under 1-1, size-increasing reductions. As observed earlier, the non-trivial part of the result in the setting of com- putability is to show the above implication. Inverting computable reduc- tions is trivial. This translation will also avoid the conflict with Encrypted Complete Set Conjecture as it does not require p-invertibility. In fact, as bThere are notable counterexamples of this though. The most prominent one is the result IP = PSPACE [48, 54] which is false relative to a random oracle [24].
- 42. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability The Isomorphism Conjecture for NP 29 will be shown later, one-way functions help in proving an analog of the above theorem for the class NP! However, the present formulation has a nice symmetry to it (both the isomorphism and its inverse require the same amount of resources) and hence is the preferred one. For the class NEXP, Ganesan and Homer [29] showed that, Theorem 2.9. (Ganesan–Homer) Let A be a p-complete set for NEXP. Then for every B ∈ NEXP, B ≤p 1 A. The proof of this uses ideas similar to the previous proof for EXP. The result obtained is not as strong since enforcing the size-increasing property of the reduction requires accepting the complement of a NEXP set which cannot be done in NEXP unless NEXP is closed under complement, a very unlikely possibility. Later, the author [5] proved the size-increasing property for reductions to complete sets for NEXP under a plausible hypothesis. While the two conjectures could not be settled for the complete p-degree of EXP (and NEXP), answers have been found for p-degrees close to the complete p-degree of EXP. The first such result was shown by Ko, Long, and Du [41]. We need to define the notion of truth-table reductions to state this result. Definition 2.11. Set A k-truth-table reduces to set B if there exists a p- function f, f : Σ∗ 7→ Σ∗ × Σ∗ × · · · × Σ∗ | {z } k ×Σ2k such that for every x ∈ Σ∗ , if f(x) = (y1, y2, . . . , yk, T ) then x ∈ A iff T (B(y1)B(y2) · · · B(yk)) = 1 where B(yi) = 1 iff yi ∈ B and T (s), |s| = k, is the sth bit of string T . Set B is k-truth-table complete for EXP if B ∈ EXP and for every A ∈ EXP, A k-truth-table reduces to B. The notion of truth-table reductions generalizes p-reductions. For both EXP and NEXP, it is known that complete sets under 1-truth-table reduc- tions are also p-complete [23, 38], and not all complete sets under 2-truth- table reductions are p-complete [55]. Therefore, the class of 2-truth-table complete sets for EXP is the smallest class properly containing the complete p-degree of EXP. Ko, Long, and Du [41] related the structure of certain p-degrees to the existence of worst-case one-way functions: Theorem 2.10. (Ko–Long–Du) If there exist worst-case one-way func- tions then there is a p-degree in EXP such that the sets in the degree are not
- 43. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 30 M. Agrawal all p-isomorphic to each other. Further, sets in this degree are 2-truth-table complete for EXP. Kurtz, Mahaney, and Royer [43] found a p-degree for which the sets are unconditionally not all p-isomorphic to each other: Theorem 2.11. (Kurtz–Mahaney–Royer) There exists a p-degree in EXP such that the sets in the degree are not all p-isomorphic to each other. Further, sets in this degree are 2-truth-table complete for EXP. Soon afterwards, Kurtz, Mahaney, and Royer [44] found another p- degree with the opposite structure: Theorem 2.12. (Kurtz–Mahaney–Royer) There exists a p-degree in EXP such that the sets in the degree are all p-isomorphic to each other. Further, this degree is located inside the 2-truth-table complete degree of EXP. The set of results above on the structure of complete (or nearly com- plete) p-degree of EXP and NEXP do not favor any of the two conjectures. However, they do suggest that the third possibility, viz., both the conjec- tures being false, is unlikely. 2.6. The Conjectures for Other Reducibilities Another direction from which to approach the two conjectures is to weaken the power of reductions instead of the class NP, the hope being that for reductions substantially weaker than polynomial-time, one can prove un- conditional results. For several weak reductions, this was proven correct and in this section we summarize the major results in this direction. The two conjectures for r-reductions can be formulated as: r-Isomorphism Conjecture. All r-complete sets for NP are r- isomorphic to each other. r-Encrypted Complete Set Conjecture. There is a 1-1, size- increasing, r-function f such that SAT and f(SAT) are not r- isomorphic to each other. Weakening p-reductions to logspace-reductions (functions computable by TMs with read-only input tape and work tape space bounded by O(log n), n is the input size) does not yield unconditional results as any such result
- 44. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability The Isomorphism Conjecture for NP 31 will separate NP from L, another long-standing open problem. So we need to weaken it further. There are three major ways of doing this. 2.6.1. Restricting the input head movement Allowing the input head movement in only one direction leads to the notion of 1-L-functions. Definition 2.12. A 1-L-function is computed by deterministic TMs with read-only input tape, the workspace bounded by O(log n) where n is the input length, and the input head restricted to move in one direction only (left-to-right by convention). In other words, the TM is allowed only one scan of its input. To ensure the space bound, the first O(log n) cells on the work tape are marked at the beginning of the computation. These functions were defined by Hartmanis, Immerman, and Ma- haney [34] to study the complete sets for the class L. They also ob- served that the “natural” NP-complete sets are also complete under 1-L- reductions. Structure of complete sets under 1-L-reductions attracted a lot of attention, and the first result was obtained by Allender [14]: Theorem 2.13. (Allender) For the classes PSPACE and EXP, complete sets under 1-L-reductions are p-isomorphic to each other. While this shows a strong structure of complete sets of some classes under 1-L-reductions, it does not answer the 1-L-Isomorphism Conjecture. After a number of extensions and improvements [10, 29, 37], the author [1] showed that, Theorem 2.14. (Agrawal) Let A be a 1-L-complete set for NP. Then for every B ∈ NP, B ≤1−L 1,si,i A. Proof Sketch. We first show that A is also complete under forgetful 1- L-reductions. Forgetful 1-L-reductions are computed by TMs that, imme- diately after reading a bit of the input, forget its value. This property is formalized by defining configurations: A configuration of a 1-L TM is a tuple hq, j, wi where q is a state of the TM, j its input head position, and w the contents of its worktape including the position of the worktape head. A forgetful TM, after reading a bit of the input and before reading the next bit, reaches a configuration which is independent of the value of the bit that is read.
- 45. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 32 M. Agrawal Let B ∈ NP, and define B̂ to be the set accepted by the following algorithm: Input x. Let x = y10b 1k . Reject if b is odd or |y| 6= tb for some integer t. Otherwise, let y = y1y2 · · · yt with |yi| = b. Let vi = 1 if yi = uu for some u, |u| = b 2 ; vi = 0 otherwise. Accept iff v1v2 · · · vt ∈ B. The set B̂ is a “coded” version of set B and reduces to B via a p- reduction. Hence, B̂ ∈ NP. Let f be a 1-L-reduction of B̂ to A computed by TM M. Consider the workings of M on inputs of size n. Since M has O(log n) space, the number of configurations of M will be bounded by a polynomial, say q(·), in n. Let b = k⌈log n⌉ such that 2b/2 q(n). Let C0 be the initial configuration of M. By the Pigeon Hole Principle, it follows that there exist two distinct strings u1 and u′ 1, |u1| = |u′ 1| = b 2 , such that M reaches the same configuration, after reading either of u1 and u′ 1. Let C1 be the configuration reached from this configuration after reading u1. Repeat the same argument starting from C1 to obtain strings u2, u′ 2, and configuration C2. Continuing this way, we get triples (ui, u′ i, Ci) for 1 ≤ i ≤ t = ⌊n−b−1 b ⌋. Let k = n − b − 1 − bt. It follows that the TM M will go through the configurations C0, C1, . . ., Ct on any input of the form y1y2 . . . yt10b 1k with yi ∈ {uiui, u′ iui}. Also, that the pair (ui, u′ i) can be computed in logspace without reading the input. Define a reduction g of B to B̂ as follows: On input v, |v| = t, compute b such that 2b/2 q(b + 1 + bt), and consider M on inputs of size b + 1 + bt. For each i, 1 ≤ i ≤ t, compute the pair (ui, u′ i) and output uiui if the ith bit of v is 1, output uiu′ i otherwise. It is easy to argue that the composition of f and g is a forgetful 1-L-reduction of B to A. Define another set B′ as accepted by the following algorithm: Input x. Reject if |x| is odd. Otherwise, let x = x1x2 · · · xns1s2 · · · sn. Accept if exactly one of s1, s2, . . ., sn, say sj , is zero and xj = 1. Accept if all of s1, s2, . . ., sn are one and x1x2 · · · xn ∈ B. Reject in all other cases. Set B′ ∈ NP. As argued above, there exists a forgetful 1-L-reduction of B′ to A, say h. Define a reduction g′ of B to B′ as: g′ (v) = v1|v| . It is easy to argue that h ◦ g′ is a size-increasing, 1-L-invertible, 1-L-reduction of B to A and h ◦ g′ is 1-1 on strings of size n for all n. Modifying this to get a reduction that is 1-1 everywhere is straightforward. The above result strongly suggests that the 1-L-Isomorphism Conjecture is true. However, the author [1] showed that,
- 46. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability The Isomorphism Conjecture for NP 33 Theorem 2.15. (Agrawal) 1-L-complete sets for NP are all 2-L- isomorphic to each other but not 1-L-isomorphic. The 2-L-isomorphism above is computed by logspace TMs that are al- lowed two left-to-right scans of their input. Thus, the 1-L-Isomorphism Conjecture fails and a little more work shows that the 1-L-Encrypted Com- plete Set Conjecture is true! However, the failure of the Isomorphism Con- jecture here is for a very different reason: it is because 1-L-reductions are not powerful enough to carry out the isomorphism construction as in The- orem 2.2. For a slightly more powerful reducibility, 1-NL-reductions, this is not the case. Definition 2.13. A 1-NL-function is computed by TMs satisfying the re- quirements of definition 2.12, but allowed to be non-deterministic. The non-deterministic TM must output the same string on all paths on which it does not abort the computation. For 1-NL-reductions, the author [1] showed, using proof ideas similar to the above one, that, Theorem 2.16. (Agrawal) 1-NL-complete sets for NP are all 1-NL- isomorphic to each other. The author [1] also showed similar results for c-L-reductions for constant c (functions that are allowed at most c left-to-right scans of the input). 2.6.2. Reducing space The second way of restricting logspace reductions is by allowing the TMs only sublogarithmic space, i.e., allowing the TM space o(log n) on input of size n; we call such reductions sublog-reductions. Under sublog-reductions, NP has no complete sets, and the reason is simple: Every sublog-reduction can be computed by deterministic TMs in time O(n2 ) and hence if there is a complete set for NP under sublog-reductions, then NTIME(nk+1 ) = NTIME(nk ) for some k 0, which is impossible [26]. On the other hand, each of the classes NTIME(nk ), k ≥ 1, has complete sets under sublog- reductions. The most restricted form for sublog-reductions is 2-DFA-reductions: Definition 2.14. A 2-DFA-function is computed by a TM with read-only input tape and no work tape.
- 47. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 34 M. Agrawal 2-DFA functions do not require any space for their computation, and therefore are very weak. Interestingly, the author [4] showed that sublog- reductions do not add any additional power for complete sets: Theorem 2.17. (Agrawal) For any k ≥ 1, sublog-complete sets for NTIME(nk ) are also 2-DFA-complete. For 2-DFA-reductions, the author and Venkatesh [12] proved that, Theorem 2.18. (Agrawal-Venkatesh) Let A be a 2-DFA-complete set for NTIME(nk ) for some k ≥ 1. Then, for every B ∈ NTIME(nk ), B ≤2DFA 1,si A via a reduction that is mu-DFA-invertible. muDFA-functions are computed by TMs with no space and multiple heads, each moving in a single direction only. The proof of this is also via forgetful TMs. The reductions in the theorem above are not 2-DFA- invertible, and in fact, it was shown in [12] that, Theorem 2.19. (Agrawal-Venkatesh) Let f(x) = xx. Function f is a 2-DFA-function and for any k ≥ 1, there is a 2-DFA-complete set A for NTIME(nk ) such that A 6≤2DFA 1,si,i f(A). The above theorem implies that 2-DFA-Encrypted Complete Set Con- jecture is true. 2.6.3. Reducing depth Logspace reductions can be computed by (unbounded fan-in) circuits of logarithmic depth.c Therefore, another type of restricted reducibility is obtained by further reducing the depth of the circuit family computing the reduction. Before proceeding further, let us define the basic notions of a circuit model. Definition 2.15. A circuit family is a set {Cn : n ∈ N} where each Cn is an acyclic circuit with n Boolean inputs x1, . . . , xn (as well as the constants 0 and 1 allowed as inputs) and some number of output gates y1, . . . , yr. {Cn} has size s(n) if each circuit Cn has at most s(n) gates; it has depth d(n) if the length of the longest path from input to output in Cn is at most d(n). A circuit family has a notion of uniformity associated with it: cFor a detailed discussion on the circuit model of computation, see [52].
- 48. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability The Isomorphism Conjecture for NP 35 Definition 2.16. A family C = {Cn} is uniform if the function n 7→ Cn is easy to compute in some sense. This can also be defined using the complexity of the connection set of the family: conn(C) = {(n, i, j, Ti, Tj) | the output of gate i of type Ti is input to gate j of type Tj in Cn}. Here, gate type Ti can be Input, Output, or some Boolean operator. Family C is Dlogtime-uniform [18] if conn(C) is accepted by a linear-time TM. It is p-uniform [15] if conn(C) is accepted by a exponential-time TM (equivalently, by a TM running in time bounded by a polynomial in the circuit size). If we assume nothing about the complexity of conn(C), then we say that the family is non-uniform. An important restriction of logspace functions is to functions computed by constant depth circuits. Definition 2.17. Function f is a u-uniform AC0 -function if there is a u- uniform circuit family {Cn} of size nO(1) and depth O(1) consisting of unbounded fan-in AND and OR and NOT gates such that for each input x of length n, the output of Cn on input x is f(x). Note that with this definition, an AC0 -function cannot map strings of equal size to strings of different sizes. To allow this freedom, we adopt the following convention: Each Cn will have nk +k log(n) output bits (for some k). The last k log n output bits will be viewed as a binary number r, and the output produced by the circuit will be the binary string contained in the first r output bits. It is worth noting that, with this definition, the class of Dlogtime- uniform AC0 -functions admits many alternative characterizations, includ- ing expressibility in first-order logic with {+, ×, ≤} [18, 47], the logspace- rudimentary reductions [17, 39], logarithmic-time alternating Turing ma- chines with O(1) alternations [18] etc. Moreover, almost all known NP- complete sets are also complete under Dlogtime-uniform AC0 -reductions (an exception is provided by [7]). We will refer to Dlogtime-uniform AC0 - functions also as first-order-functions. AC0 -reducibility is important for our purposes too, since the complete sets under the reductions of the previous two subsections are also complete under AC0 -reductions (with uniformity being Dlogtime- or p-uniform). This follows from the fact that these sets are also complete under some appro- priate notion of forgetful reductions. Therefore, the class of AC0 -complete sets for NP is larger than all of the previous classes of this section.
- 49. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 36 M. Agrawal The first result for depth-restricted functions was proved by Allender, Balcázar, and Immerman [16]: Theorem 2.20. (Allender–Balcázar–Immerman) Complete sets for NP under first-order projections are first-order-isomorphic to each other. First-order projections are computed by a very restricted kind of Dlogtime-uniform AC0 family in which no circuit has AND and OR gates. This result was generalized by the author and Allender [6] to NC0 -functions, which are functions computed by AC0 family in which the fan-in of every gate of every circuit is at most two. Theorem 2.21. (Agrawal–Allender) Let A be a non-uniform NC0 - complete set for NP. Then for any B ∈ NP, B non-uniform NC0 -reduces to A via a reduction that is 1-1, size-increasing, and non-uniform AC0 - invertible. Further, all non-uniform NC0 -complete sets for NP are non- uniform AC0 -isomorphic to each other where these isomorphisms can be computed and inverted by depth three non-uniform AC0 circuits. Proof Sketch. The proof we describe below is the one given in [3]. Let B ∈ NP, and define B̂ to be the set accepted by the following algorithm: On input y, let y = 1k 0z. If k does not divide |z|, then reject. Otherwise, break z into blocks of k consecutive bits each. Let these be u1u2u3 . . . up. Accept if there is an i, 1 ≤ i ≤ p, such that ui = 1k . Otherwise, reject if there is an i, 1 ≤ i ≤ p, such that ui = 0k . Otherwise, for each i, 1 ≤ i ≤ p, label ui as null if the number of ones in it is 2 modulo 3; as zero if the number of ones in it is 0 modulo 3; and as one otherwise. Let vi = ǫ if ui is null, 0 if ui is zero, and 1 otherwise. Let x = v1v2 · · · vp, and accept iff x ∈ B. Clearly, B̂ ∈ NP. Let {Cn} be the NC0 circuit family computing a reduction of B̂ to A. Fix size n and consider circuit Ck+1+n for k = 4⌈log n⌉. Let C be the circuit that results from setting the first input k + 1 bits of Ck+1+n to 1k 0. Randomly set each of the n input bits of C in the following way: With probability 1 2 , leave it unset; with probability 1 4 each, set it to 0 and 1 respectively. The probability that any block of k bits is completely set is at most 1 n4 . Similarly, the probability that there is a block that has at most three unset bits is at most 1 n , and therefore, with high probability, every block has at least four unset bits. Say that an output bit is good if, after the random assignment to the input bits described above is completed, the value of the output bit depends on exactly one unset input bit. Consider an output bit. Since C is an NC0
- 50. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability The Isomorphism Conjecture for NP 37 circuit, the value of this bit depends on at most a constant, say c, number of input bits. Therefore, the probability that this bit is good after the assignment is at least 1 2 · 1 4c−1 . Therefore, the expected number of good output bits is at least m 4c , where m is the number of output bits of C whose value depends on some input bit. Using the definition of set B̂, it can be argued that Ω(n) output bits depend on some input bit, and hence Ω(n) output bits are expected to be good after the assignment. Fix any assignment that does this, as well as leaves at least four unset bits in each block. Now set some more input bits so that each block that is completely set is null, each block that has exactly two unset bits has number of ones equal to 0 modulo 3, and there are no blocks with one, three, or more unset bits. Further, for at least one unset input bit in a block, there is a good output bit that depends on the bit, and there are Ω( n log n ) unset input bits. It is easy to see that all these conditions can be met. Now define a reduction of B to B̂ as: On input x, |x| = p, consider Ck+1+n such that the number of unset input bits in Ck+1+n after doing the above process is at least p. Now map the ith bit of x to the unset bit in a block that influences a good output bit and set the other unset input bit in the block to zero. This reduction can be computed by an NC0 circuit (in fact, the circuit does not need any AND or OR gate). Define a reduction of B to A given by the composition of the above two reductions. This reduction is a superprojection: it is computed by circuit family {Dp} with each Dp being an NC0 circuit such that for every input bit to Dp, there is an output bit that depends exactly on this input bit. A superprojection has the input written in certain bit positions of the output. Therefore, it is 1-1 and size-increasing. Inverting the function is also easy: Given string y, identify the locations where the input is written, and check if the circuit Dp (p = number of locations) on this input outputs y. This checking can be done by a depth two AC0 circuit. This gives a 1-1, size-increasing, AC0 -invertible, NC0 -reduction of B to A. The circuit family is non-uniform because it is not clear how to deterministically compute the settings of the input bits. Exploiting the fact that the input is present in the output of the reductions, an AC0 - isomorphism, computed by depth three circuits, can be constructed between two complete sets following [21] (see [8] for details). Soon after, the author, Allender, and Rudich [8] extended it to all AC0 -functions, proving the Isomorphism Conjecture for non-uniform AC0 - functions.
- 51. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 38 M. Agrawal Theorem 2.22. (Agrawal–Allender–Rudich) Non-uniform AC0 -com- plete sets for NP are non-uniform AC0 -isomorphic to each other. Fur- ther, these isomorphisms can be computed and inverted by depth three non- uniform AC0 circuits. Proof Sketch. The proof shows that complete sets for NP under AC0 - reductions are also complete under NC0 -reductions and invokes the above theorem for the rest. Let A be a complete set for NP under AC0 -reductions. Let B ∈ NP. Define set B̂ exactly as in the previous proof. Fix an AC0 - reduction of B̂ to A given by family {Cn}. Fix size n, and consider Ck+1+n for k = n1−ǫ for a suitable ǫ 0 to be fixed later. Let D be the circuit that results from setting the first k + 1 input bits of Ck+1+n to 1k 0. Set each input bit of D to 0 and 1 with probability 1 2 − 1 2n1−2ǫ each and leave it unset with probability 1 n1−2ǫ . By the Switching Lemma of Furst, Saxe, and Sipser [28], the circuit D will reduce, with high probability, to an NC0 circuit on the unset input bits for a suitable choice of ǫ 0. In each block of k bits, the expected number of unset bits will be nǫ , and therefore, with high probability, each block has at least three unset bits. Fix any settings satisfying both of the above. Now define a reduction of B to B̂ that, on input x, |x| = p, identifies n for which the circuit D has at least p blocks, and then maps ith bit of input x to an unset bit of the ith block of the input to D, setting the remaining bits of the block so that the sum of ones in the block is 0 modulo 3. Unset bits in all remaining blocks are set so that the sum of ones in the block equals 2 modulo 3. The composition of the reduction of B to B̂ and B̂ to A is an NC0 - reduction of B to A. Again, it is non-uniform due to the problem of finding the right settings of the input bits. The focus then turned towards removing the non-uniformity in the above two reductions. In the proof of Theorem 2.21 given in [6], the uni- formity condition is p-uniform. In [7], the uniformity of 2.22 was improved to p-uniform by giving a polynomial-time algorithm that computes the cor- rect settings of input bits. Both the conditions were further improved to logspace-uniform in [3] by constructing a more efficient derandomization of the random assignments. And finally, in [2], the author obtained very efficient derandomizations to prove that, Theorem 2.23. (Agrawal) First-order-complete sets for NP are first- order-isomorphic.
- 52. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability The Isomorphism Conjecture for NP 39 The isomorphisms in the theorem above are no longer computable by depth three circuits; instead, their depth is a function of the depth of the circuits computing reductions between the two complete sets. 2.6.4. Discussion At first glance, the results for the weak reducibilities above seem to provide equal support to both the conjectures: The Isomorphism Conjecture is true for 1-NL and AC0 -reductions for any reasonable notion of uniformity, while the Encrypted Complete Set Conjecture is true for 1-L and 2-DFA reductions. However, on a closer look a pattern begins to emerge. First of all, we list a common feature of all the results above: Corollary 2.1. For r ∈ {1-L, 1-NL, 2-DFA, NC0 , AC0 }, r-complete sets for NP are also complete under 1-1, size-increasing, r-reductions. The differences arise in the resources required to invert the reductions and to construct the isomorphism. Some of the classes of reductions that we consider are so weak, that for a given function f in the class, there is no function in the class that can check, on input x and y, whether f(x) = y. For example, suppose f is an NC0 -function and one needs to construct a circuit that, on input x and y, outputs 1 if y = f(x), and outputs 0 otherwise. Given x and y, an NC0 circuit can compute f(x), and can check if the bits of f(x) are equal to the corresponding bits of y; however, it cannot output 1 if f(x) = y, since this requires taking an AND of |y| bits. Similarly, some of the reductions are too weak to construct the isomorphism between two sets given two 1-1, size-increasing, and invertible reductions between them. Theorems 2.14 and 2.15 show this for 1-L-reductions, and the same can be shown for NC0 -reductions too. Observe that p-reductions do not suffer from either of these two drawbacks. Hence we cannot read too much into the failure of the Isomorphism Conjecture for r-reductions. We now formulate another conjecture that seems better suited to getting around the above drawbacks of some of the weak reducibilities. This conjecture was made in [1]. Consider a 1-1, size-increasing r-function f for a resource bound r. Con- sider the problem of accepting the set range(f). A TM accepting this set will typically need to guess an x and then verify whether f(x) = y. It is, therefore, a non-deterministic TM with resource bound at least r. Let rrange ≥ r be the resource bound required by this TM. For a circuit accept- ing range(f), the non-determinism is provided as additional “guess bits”
- 53. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability 40 M. Agrawal and its output is 1 if the circuit evaluates to 1 on some settings of the guess bits. We can similarly define rrange to be the resource bound required by such a non-deterministic circuit to accept range(f). r-Complete Degree Conjecture. r-Complete sets for NP are also com- plete under 1-1, size-increasing, r-reductions that are rrange -invertible. Notice that the invertibility condition in the conjecture does not allow non-determinism. For p-reductions, Proposition 2.4. The p-Complete Degree Conjecture is equivalent to the Isomorphism Conjecture. Proof. Follows from the observation that prange = p as range of a p- function can be accepted in non-deterministic polynomial-time, and from Theorem 2.2. Moreover, for the weaker reducibilities that we have considered, one can show that, Theorem 2.24. For r ∈ {1-L, 1-NL, 2-DFA, NC0 , AC0 }, the r-Complete Degree Conjecture is true. Proof. It is an easy observation that for r ∈ {1-L, 1-NL, AC0 }, rrange = r. The conjecture follows from Theorems 2.14, 2.16, and 2.23. Accepting range of a 2-DFA-function requires verifying the output of 2-DFA TM on each of its constant number of passes on the input. The minimum resources required for this are to have multiple heads stationed at the beginning of the output of each pass, guess the input bit-by-bit, and verify the outputs on this bit for each pass simultaneously. Thus, the TM is a non-deterministic TM with no space and multiple heads, each moving in one direction only. So Theorem 2.18 proves the conjecture. Accepting range of an NC0 -function requires a non-deterministic AC0 circuit. Therefore, Theorems 2.21 and 2.23 prove the conjecture for r = NC0 . In addition to the reducibilities in the above theorem, the r-Complete Degree Conjecture was proven for some more reducibilities in [1]. These results provide evidence that r-Complete Degree Conjecture is true for all reasonable resource bounds; in fact, there is no known example of a reasonable reducibility for which the conjecture is false.
- 54. January 4, 2011 15:24 World Scientific Review Volume - 9in x 6in computability The Isomorphism Conjecture for NP 41 The results above also raise doubts about the intuition behind the En- crypted Complete Set Conjecture as we shall argue now. Consider AC0 - reductions. There exist functions computable by depth d, Dlogtime-uniform AC0 circuits that cannot be inverted on most of the strings by depth three, non-uniform AC0 circuits [35]. However, by Theorem 2.22, AC0 -complete sets are also complete under AC0 -reductions that are invertible by depth two, non-uniform AC0 circuits and the isomorphisms between all such sets are computable and invertible by depth three, non-uniform AC0 circuits. So, for every 1-1, size-increasing, AC0 -function, it is possible to efficiently find a dense subset on which the function is invertible by depth two AC0 circuits. Therefore, the results for weak reducibilities provide evidence that the Isomorphism Conjecture is true. 2.7. A New Conjecture In this section, we revert to the conjectures in their original form. The investigations for weak reducibilities provide some clues about the struc- ture of NP-complete sets. They strongly suggest that all NP-complete sets should also be complete under 1-1, size-increasing p-reductions. Proving this, of course, is hard as it implies DP 6= NP (Proposition 2.1). Can we prove this under a reasonable assumption? This question was addressed and partially answered by the author in [5], and subsequently improved by the author and Watanabe [13]: Theorem 2.25. (Agrawal–Watanabe) If there exists a 1-1, 2nǫ -secure one-way function for some ǫ 0, then all NP-complete sets are also com- plete under 1-1, and size-increasing, P/poly-reductions. In the above theorem, P/poly-functions are those computed by polynomial-size, non-uniform circuit families. Proof Sketch. Let A be an NP-complete set and let B ∈ NP. Let f0 be a 1-1, 2nǫ -secure one-way function. Recall that we have assumed that |f0(y)| is determined by |y| for all y. Håstad et al., [36], showed how to construct a pseudorandom generator using any one-way function. Pseudorandom gen- erators are size-increasing functions whose output cannot be distinguished from random strings by polynomial-time probabilistic TMs. Let G be the pseudorandom generator constructed from f0. Without loss of generality, we can assume that |G(y)| = 2|y| + 1 for all y. We also modify f0 to f as: f(y, r) = f0(y)rb where |r| = |y| and b = y · r, the inner product of
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- 56. Cetywayo; his Power and Character War Clouds Gathering Since the struggle with his brother in 1856, and the slaughter of the latter with about one-fourth of the Zulus of that time, Cetywayo had been the real ruler of his nation. In 1872, upon the death of Panda, he succeeded also to the nominal government and was approved by the British authorities. In appearance the great Zulu chief was, in these earlier years, handsome and dignified, besides being possessed of undoubted mental gifts. He was, however, pitiless and cruel in the extreme, as hard of heart as a piece of steel, and as regardless of human life as a lion or tiger in its native fastnesses. In organizing power he had the genius of Tshaka, and he brought out all that was best and all that was worst in the Zulu race--the most intelligent, fearless and active of South African Kaffirs, or Bantu. As time went on and Cetywayo drilled and exercised and trained his _impis_, it became evident that unpleasant results must follow and that, hemmed in as they were by the Transvaal, Natal and the sea, there were only two possible outlets for the fiery spirits of the growing Zulu force. Cetywayo would have found it hard to control them had he desired to do so. Like all native armies, and especially with such disciplined and ambitious soldiers as he now had, they were more than anxious to test their power, to wash their spears in blood and to taste of the fierce pleasures of war. In this connection Sir Bartle Frere wrote with vigor in a dispatch of January, 1879, justifying his instructions to Lord Chelmsford to advance into Zululand: Whether his (Cetywayo's) young men were trained into celibate gladiators as parts of a most efficient military machine, or allowed to become peaceable cattle herds; whether his young women were to be allowed to marry the young men, or to be assegaied by hundreds for disobeying the king's orders to marry effete veterans, might possibly be Zulu questions of political economy with which the British Government were not concerned to meddle; but they were part of the great recruiting system of a military organization which enabled the King to form, out of his comparatively small population, an army, at the very lowest estimate, of 25,000 perfectly trained and perfectly obedient soldiers, able to march three times as fast as we could, to dispense with commissariat of every kind and transport of every kind, and to fall upon this or any part of the neighboring colony (Natal) in such numbers and with such determination
- 57. The Zulus and the Boers that nothing but a fortified post could resist them; making no prisoners and sparing neither age nor sex. Demonstrations of aggressiveness were frequent. About the time when Sir Bartle Frere arrived at Cape Town a powerful Zulu force had, in the most menacing manner, paraded along the Natal frontier, and, in response to protests, was described as merely a hunting party. British officials, who had been sent into Zululand from time to time as envoys, were treated in the most contemptuous manner by the Zulu Idunas. On one occasion (in 1876) two native women were captured on Natal soil and carried back to punishment, which, in this case, meant death. Proofs were not wanting of Zulu attempts to create disturbance amongst other Bantu tribes in distant parts of the country, and, on December 10, 1878, Sir Bartle Frere wrote to the Colonial Secretary that: Whenever there has been disturbance and resistance to the authority of the Government between the Limpopo and the westernmost limits of Kaffir population, there we have found unmistakable evidence of a common purpose and a general understanding. The first embodiment of this fact was the Kaffir war already mentioned. Sandilli, leading the Gaika tribe, and Kreli the Galekas, had revolted in August, 1877, and only prompt military measures had saved the neighboring colonists from much suffering. As it was the tribes were not entirely subjugated until eight months after their first hostile action. The general effect, of course, was to still further encourage Cetywayo and his warriors in their aggressive ambitions. An additional factor to this end was the British annexation of the Transvaal in 1877. By placing their most hated enemy, the Boer, under British control it transferred the expression of that hatred to the new Government and the English people. A part of the general restlessness of the natives in the year of the annexation had been expressed in the war between Sekukuni, a Kaffir chief to the northeast, and the Boer Republic. The chief in question was a tool of Cetywayo's, and there is little doubt was egged on by him to hostilities which the latter intended as preliminary to a general attack upon the Transvaal; in which he was further encouraged by the defeat of the Boers and the retirement of President Burgers from his invasion of Sekukuni's
- 58. Zulu Declaration territory. But the British annexation temporarily averted the attack and the whole burden of Zulu hostility was practically assumed by the British; as well as the subsequent brunt of Zulu attack. The situation, therefore, was not a pleasant one for Sir Bartle Frere any more than it was for the colonists of Natal, or for the Boers of the Transvaal prior to their annexation. It had been anticipated by Sir George Grey, a quarter of a century before, when he had urged that the growth of the Zulu power be checked by the establishment of a protectorate, or watched by the placing of a permanent Resident at its capital. But his advice was disregarded, and, in 1876, when Sir Henry Bulwer, Governor of Natal, protested against some Zulu act of force upon the frontier, Cetywayo was able to reply with a temerity born of the possession of a splendidly developed fighting machine of many thousand men: I do kill; but do not consider yet I have done anything in the way of killing. Why do the white men start at nothing? I have not yet begun. I have yet to kill. It is the custom of our nation, and I shall not depart from it. In a dispatch to the Colonial Office on December 2, 1878, Sir Bartle Frere declared plainly that, as a result of these and other more practical manifestations, no one can really sleep in peace and security within a day's run of the Zulu border, save by sufferance of the Zulu Chief. In the end the war really came as a result of the Transvaal annexation, and, in the main, because of the bitter feeling between the Boers and the Zulus. During the month of September, 1878, Sir Bartle Frere, as High Commissioner for South Africa, visited Natal, and examined some territory in dispute between the Transvaal (then a British dependency) and Zululand. Finally he gave his decision as arbitrator in favor of the Zulu claim; but with a view to the general well-being of South Africa attached certain requirements to the announced Award. These included the disbandment of his army by Cetywayo, the reception of a British Resident at his capital of Ulundi, the surrender of certain persons guilty of an offence upon Natal territory, and the giving of specific guarantees for the better government of his people. The proposal obviously involved the establishment of a protectorate over Zulu territory, and the only possible alternative to its refusal was war. Knowing the ambitions of Cetywayo and his army, as Sir Bartle Frere did, he could hardly have expected the acceptance of these propositions or have supposed that there could be any other result than immediate hostilities. As a matter of fact no reply was received, and on
- 59. Advance into Zululand A Large Force Slaughtered January 10, 1879, Lieutenant-General Lord Chelmsford, who had commanded in the Kaffir War of the preceding year, crossed the Lower Tugela with a force which was small, but generally deemed sufficient, and marched into Zululand toward a place called Isandlhwana, where camp was formed for a few days. Colonel Pearson, with a flying column of 2,000 white troops and a similar number of blacks, marched on toward Ulundi, and got as far as Etshowe, after beating back a Zulu army of about his own number. A third column under Colonel Evelyn Wood marched from another direction toward the same objective point, reached a post called Kambula, and remained there for some time after duly fortifying it and defeating a persistent attack from a large Zulu army. Incidentally, one of his patrols was surprised by the enemy, and ninety-six of the party killed, including Colonel Weatherley and his son. Meanwhile Lord Chelmsford had moved the main body of his forces to the capture of a large kraal near Isandlhwana, leaving about a thousand British, Colonial and native troops to guard the camp. Despite the warnings of some Dutch farmers, no attempt had been made at protecting the place by trench, or embankment, or even by the traditional and easy laager of wagons. Danger was hardly dreamed of until, on January 22d, the horns of a Zulu army of twenty thousand men were found to be closing around the devoted troops. There was practically nothing to do but to die, and this the soldiers did with their faces to the foe, fighting as long as their ammunition lasted and killing over a thousand Zulus. A few irregular mounted troops escaped, as did the bulk of the natives; but seven hundred British regulars and over a hundred Colonial troops were slaughtered by an enemy who gave no quarter and from whom none was asked or expected. Not far away from this camp, on the Natal frontier and guarding the line of communication, was a small depot for provisions and hospital work under the charge of Lieutenants Chard and Bromhead with 130 soldiers. In the afternoon of the fateful day at Isandlhwana this little post of Rorke's Drift was attacked by a picked Zulu army of four thousand men, and for eleven hours was defended so desperately, behind hastily improvised fortifications of biscuit boxes and grain bags, that the enemy retired after leaving over 300 men dead on the field. The little garrison was saved, and, more important still, Natal was saved from a sweeping and devastating raid of savage warriors. Lord Chelmsford at once fell back upon his base of supplies in the Colony, and
- 60. Death of Prince Imperial Redress Necessary the other columns at Etshowe and Kambula, respectively, proceeded, as already stated, to fortify themselves and await events. Further movements were slow in arrangement and reinforcements slow in coming, but, finally, Lord Chelmsford advanced again into Zululand with 4,000 British and Colonial troops and a thousand natives, and on July 4th, after relieving Etshowe and beating back the enemy at Gungunhlovu, reached Ulundi, where he defeated a Zulu army of 20,000 men. Meantime Sir Garnet Wolseley had been sent out to supersede Lord Chelmsford and to administer the regions affected by the war. He arrived on the scene very soon after this decisive conflict, and was able to report to the War Office that Zululand was practically at peace again. A few months later Colonel Baker Creed Russell went to the further rescue of the Boers in their seemingly hopeless struggle with the Bapedis, and, on November 28th, stormed and captured Sekukuni's stronghold. One of the melancholy incidents of a most unpleasant little war was the death of the Prince Imperial of France. The Zulus must have lost ten thousand men, all told, and their power was absolutely shattered. Cetywayo, after remaining in concealment for a time, was eventually captured and sent to live in guarded comfort near Cape Town. A little later he was allowed to visit England, where he was well received, and proved himself a dignified savage, and in 1883 was re-established in Zululand after the practical failure of Sir Garnet Wolseley's attempt to govern that region through thirteen semi-independent chiefs. Civil war followed, Cetywayo died, his sons kept up the internal conflict, the Transvaal annexed what is now called the District of Vryheid, and in 1887 what remained of the country was proclaimed British territory. Thus, and finally, was settled a question which threatened the very existence of the thirty thousand white people of Natal—surrounded within their own territory by three hundred thousand Zulus and faced upon their border by a strong Zulu nation and its army of 25,000 to 40,000 men. Sir Bartle Frere was vigorously denounced for the war, for the disaster at Isandlhwana, and for everything connected with the matter. Yet it seems to the impartial judgment of later days that he only did what was wise in a most difficult and dangerous situation. There appears to be no doubt that Cetywayo was simply awaiting his chance to over-run the Transvaal and Natal. In writing to the
- 61. Colonial Office, on March 1, 1879, Sir Bartle Frere pointed out the necessity of taking immediate action, and the difficulty, or worse, of waiting two months—in days prior to cable communication—for exact authority to move in the matter of compelling redress, and added: The Zulus had violated British territory, slain persons under English protection, and had repeatedly refused the redress we demanded. Could a final demand for redress on this account be postponed? It seems to me clearly not, with any safety to Natal and its inhabitants. In another despatch to the Colonial Office, on January 13, 1880, the High Commissioner replied to some attacks from Mr. Gladstone by declaring that in the judgment of all military authorities, both before the war and since, it was absolutely impossible for Lord Chelmsford's force, acting on the defensive within the Natal boundary, to prevent a Zulu impi from entering Natal and repeating the same indiscriminate slaughter of all ages and sexes which they boast of having effected in Dingaan's other massacres of forty years ago. He defended Lord Chelmsford, and incidentally stated that the disaster at Isandlhwana was due to disregard of orders. South Africa was for a time, however, the grave of Sir Bartle Frere's reputation, both in this connection and that of the Transvaal, and his recall followed a few months after the writing of the above despatch. But historical retrospect is wiser than political opinion, and time has now revived the fame of a great man and a wise statesman, and declared that there was practical truth and justice in the farewell address presented to him by the people of Albany in the Colony of the Cape: We have watched with the most anxious interest your career during that eventful period when the affairs of the neighboring Colony of Natal were administered by you; we perfectly understand that at that crisis the deep-laid plans and cruel purposes of the savage and bloodthirsty king of the Zulus were just reaching their full development, and that his inevitable and long- expected encounter with the British power could no longer be averted; it was, no doubt, fortunate for that colony, and for the honor of the British name, that you were on the spot ready to sacrifice every personal consideration, and to undertake one of the heaviest and most tremendous responsibilities ever undertaken by a servant of the Crown. Your excellent plans, your steady determination, your unflagging perseverance, led to the downfall of a barbarous tyrant, the break-up of a most formidable and
- 62. Order in Natal and the Transvaal unwarrantable military power, and the establishment of peaceful relations, which, properly managed, might have ensured the lasting peace and prosperity which you have systematically desired to secure for South Africa. With the ending of this war and the temporary settlement of the Transvaal troubles there came to Natal a period of progress in both constitutional and material matters. The natives of the Province had always been well treated by the Imperial authorities, and there were none of the complexities of dual control so noticeable at the Cape; while the small number of Dutch settlers who remained after the forties were not important enough to create racial friction or to seriously antagonize the surrounding Zulus. The many privileges and immunities of the latter, and the possession of large tracts of land given and secured to them by the Colonial Office, seem to have made them a fairly satisfied people and to have prevented any organized effort at any time to join hands with their kin under Panda or Cetywayo. The experience of Englishmen with the Maori, the Red Indian, or the Kaffirs to the west of Natal, have not been repeated in that little Colony, and the small population of whites has lived in comparative security, though not without frequent fear, amidst the ever-increasing numbers of a savage race. Something of this has been due to the wise administration of the Colonial Governors and to their reasonable immunity from the influences which controlled the Cape and dragged the Colonial Office first one way and then the other. The local whites were also too few to claim constitutional government, to assert a right to control the natives, or to do more than occasionally protest against incidents such as the Transvaal slave-raids upon Kaffir tribes or hostility towards its general system of apprenticeship. In 1845 the first Lieutenant-Governor, under the jurisdiction of the Governor of Cape Colony, had been appointed in the person of Mr. Martin West. He was succeeded, in 1850, by Mr. Benjamin Pine, and, in 1856, by Mr. John Scott, who brought with him a Royal charter constituting the Colony, separating it from the Cape, and giving it an appointive Council. In 1866 an Assembly was created, with the same limitations as to responsible government which characterized all the Colonial Assemblies of that time.
- 63. An Uprising Threatened Mr. John Maclean, C.B., was appointed Lieutenant-Governor, and Mr. R. W. Keate became the first Governor of Natal in 1867. His successors were as follows, and their names mark several important incidents in South African history: 1872, Sir Anthony Musgrave, K.C.M.G. 1873, Sir Benjamin Pine, K.C.M.G. 1875, Major-General Sir Garnet Wolseley, B.C. 1875, Sir Henry E. Bulwer, K.C.M.G. 1880, General Sir Garnet Wolseley, G.C.B. 1880, Major-General Sir G. Pomeroy Colley. 1881, Brig.-General Sir H. Evelyn Wood. 1881, Lieut.-Colonel C. B. H. Mitchell, C.M.G. 1882, Sir Henry E. Bulwer, K.C.M.G. 1885, Sir Charles B. H. Mitchell, K.C.M.G. 1886, Sir Arthur E. Havelock, K.C.M.G. 1889, Sir Charles Mitchell, K.C.M.G. 1893, Sir W. F. Hely-Hutchinson, G.C.M.G. Under the régime of Sir Benjamin Pine occurred one of those native wars which illustrate at once the precarious tenure of peace with savage tribes and the danger of a Governor falling between the two stools of a weak white population demanding protection against the serried masses of native races and a Colonial Office controlled, to some extent, by missionary and religious influences with sympathies wider than their statecraft or knowledge. Langalibalele, Chief of the Hlubis in Natal—a tribe which was great and powerful in the days preceding Tshaka—had gradually strengthened his people in numbers and in training until he thought himself able to defy the Natal Government, and to send his young men into neighboring communities to purchase guns and ammunition in defiance of the regulations of the Colony. Messages were in vain sent from Pietermaritzburg demanding an account of the matter and his presence at the capital. Finally, a small party of volunteers was sent to compel his obedience, and met with the usual preliminary repulse. Then upon a thread seemed to hang the peace of South Africa. Langalibalele was known to be held in high respect by Kaffir tribes
- 64. Gen. Wolseley Arrives in State Government of Natal from the Caledon to the Fish River, and it was afterwards proved that he really had tried to effect a general rising. Prompt measures were taken, however, by all the Governments—even those of the Republics offering aid —and the Chief was surrounded by a large force of Natal and Cape Mounted Police, captured, tried by a special Court and sentenced to imprisonment for life. Meantime the influence of Bishop Colenso and the Aborigines Protection Society had made the Colonial Office doubtful of the justice of these steps. The Governor was recalled, sentences were commuted, and compensation was given from the Imperial Treasury to a tribe which had suffered through expressing sympathy with the rebels. The coming of Sir Garnet Wolseley, in 1875, amid much glitter of state and ceremony, marked the attempt of Lord Carnarvon to promote the federation of the Colonies; and the despatch of the same distinguished soldier, in 1880, was an effort to gather up the threads of military organization after the reverses and successes of the Zulu War. The death of Sir George Pomeroy Colley at Majuba Hill and the accession of Sir Evelyn Wood, with instructions to make peace with the Transvaal, are landmarks in the annals of the whole region; while the coming of Sir Walter Hely-Hutchinson in 1893, with extended powers as Governor of Natal and Zululand, marks the grant of complete responsible government to this miniature Colonial India, twenty years after it had been given to Cape Colony, and nearly fifty years after Canada had received it. Under this constitution there is now a Legislative Council of eleven members, nominated by the Governor-in-Council and appointed for ten years, and a Legislative Assembly of thirty-seven members, elected by popular constituencies—mainly white—for four years. The Ministry holds office by the same Parliamentary tenure as do all British Governments under free institutions, and, since 1893, the Prime Ministers have been Sir John Robinson, K.C.M.G., who held office until 1897; the Right Hon. Harry Escombe, P.C., who succeeded him and participated in the Queen's Diamond Jubilee; Sir Henry Binns, K.C.M.G., who died in 1899; and the present occupant of the position, Lieut.-Colonel Albert Henry Hime, C.M.G. The franchise of the Colony is liberal, and every European who is a British subject and possesses real property worth $250, occupies such property at an annual rental of not less than $50, or is in receipt of an income of $480 and upwards, can vote. He must, however, have resided in the Colony for three
- 65. years. Natives are entitled to vote under the same conditions after seven years' voluntary exemption from the action of the special native laws and the tribal system. One of the curious conditions of Natal, and which entitles the Colony to consideration as a sort of miniature India, has been elsewhere casually referred to. It was thought, at first, that in a country which combined tropical vegetation with a healthful climate and with a great reserve force of natives for local labor, immense development of production might be possible. Coffee, sugar, arrowroot, cotton and tea were all found to thrive in its fruitful soil. But European workers did not come in any number, and it was soon found that the natives would not work with the least bit of persistence or dependence. In this difficult situation planters and capitalists turned to the Eastern Empire, and coolies were engaged under contract for a term of years. And, when their term was up, these hired immigrants, as a rule, showed no desire to return, and settled down for good in a land which seemed to their minds greatly superior to the one they had left. Naturally, too, Indian traders followed, and, in time, a small but steady stream of immigrants flowed in from India, and through their cheap mode of living soon captured the bulk of retailing trade in the country, while also doing most of the cheaper labor. Of this class of settlers, now nearly equal in numbers to the white population, there were 17,000 in 1879, 41,000 in 1891 and 53,000 in 1898. They do not, through taxes, add greatly to the revenues of the country, or in any sense to its military strength, but they do add appreciably to its productive and industrial capabilities.
- 66. FIRST SERIOUS BOER-BRITISH BATTLE, MAJUBA HILL, 1881. In which the Boers defeated the English and gained internal independence.
- 67. BATTLE BETWEEN THE ENGLISH AND THE ZULUS, SOUTH AFRICA, 1879
- 68. Resources of Natal England's Wise and Generous Policy In this latter connection there were, in 1892, over four million dollars invested in the sugar industry, including 36 factories, with an output of 15,000 tons and employing 6,000 coolies. But, although great possibilities exist in this and other industrial directions, serious development had only just commenced when the present war broke out, and the central resource of the Colony was still sheep and cattle raising, together with a fair amount of straight agricultural work such as the cultivation of maize, oats, barley, potatoes and vegetables of various kinds. Fruit, such as pineapples, oranges, lemons, bananas, peaches, etc., were, of course, grown to any extent desired. That the general progress of production was fair is seen from the fact that the Natalian exports rose from $6,200,000 in 1893 to $8,100,000 in 1897. Other conditions were good. The imports, chiefly from Great Britain, advanced during the same period from $11,000,000 to $29,900,000, and the revenue from five millions to eleven millions. Durban became the port for a large transit trade to the interior States. The population as a whole grew from 361,000 in 1867 to 543,900 in 1891, and 829,000 in 1898—four hundred thousand of this increase being amongst the natives. Educational progress was excellent. In 1892 the regular attendance at Government and inspected schools was 6,000, while 2,200 attended private schools, and only some 200 children were reported as receiving no education. There were 74 schools for natives, with a total attendance of 4,050, and 24 schools for Indian children, with an attendance of 1,402. In 1897 there were 7,685 in regular attendance at Government and inspected schools, and 1,600 at the private schools. There were 159 native schools with an attendance of 8,542, and 30 Indian schools with 1,961 pupils. Upon the whole, the historic life of Natal since the days of Dutch and native turmoil has not, with the exception of the eventful period of 1876-81, been a stormy one. The Dutch are too much in the minority to cause much trouble, and a fair measure of good feeling seems to have prevailed locally. The whole white population are fairly well agreed upon franchise questions as the free British principle works out in the practical exclusion of the ignorant and tribal savage. They are at one upon tariff matters, and the present system is for revenue only and is very low—the ordinary ad valorum rate being five per cent. Politics have not been as bitter as in Cape Colony, owing to a practical, though not always expressed, recognition of the fact that good
- 69. Religious Intolerance of the Boers reasons existed for not giving complete control over an immense black population, involving in its results at times the whole Imperial policy and system in South Africa, into the hands of thirty, forty, or fifty thousand white men, women and children, all told. The wise handling of the native problem, the conciliation of the Kaffir and the careful local laws, did, however, make this finally possible, and the Government of the Colony since 1893 has been all that could be reasonably desired. There is some rivalry with Cape Colony, owing to the latter's annexation of Griqualand East and Pondoland which Natal had hoped to acquire, and also, in some measure, to the railway competition of the richer and stronger Colony. But Natal has been allowed to absorb Zululand and Tongaland on its eastern border, and to thus reach up to Portuguese territory. The people have also led an easy and tranquil life, and are as a rule comfortably off. Now, of course, this is all changed, and the little Colony is the scene of an Empire-making strife, while its fruitful soil, or beautiful valleys and picturesque hills, resound with the march of armed men and echo with the roar of artillery. A tardy measure of healthful progress has thus been suddenly and summarily arrested; but in the end it is probable that good will come of evil and the natural riches of a splendid region be more generally recognized and developed. CHAPTER XII. A Review of the South African Question. The South African War of 1899 grew out of racial conditions and national considerations far apart from, and long precedent to, the growth of Kimberley and Johannesburg or the discovery of diamonds and gold. It arose, primarily, from racial tendencies which had grown more and more opposed to each other as the climate and conditions of South Africa accentuated their peculiarities. History and tradition had early driven into the Boer's heart an intense intolerance of religious thought to which the isolation of the veldt added an almost incomprehensible ignorance. A wider survey of
- 70. Two Opposite Views of Liberty the world and a fuller grasp of the essentials of liberty had, meanwhile, developed in the Englishman's mind[1] a love for free religious thought and practice to which his belief in schools and his affection for literature and the press added strength and character. The Dutchman was nomadic in life, pastoral in pursuit, lazy and sluggish in disposition. The Englishman was at times restless in seeking wealth or pleasure, but upon the whole he liked to settle down in a permanent home and with surroundings which he could make his own in ever-increasing comfort and usefulness. He drew the line at no single occupation and made, as the case might be, a good farmer, or artisan, or labourer, or merchant. And he was usually of active mind as well as body. [1] I use the word Englishman here in a general sense, and inclusive of the Scotchman or Irishman. The Dutchman in South Africa wanted liberty to do as he liked and to live as he chose, but he did not wish to accord that liberty to inferior races, or to attempt the training of them in its use and application. The Englishman, on the other hand, loved liberty in a broad way, and wanted nothing better than to see it applied to others as freely and fully as to himself. The one race looked upon the negro as only fitted to be a human chattel and as not being even a possible subject for improvement, education or elevation. The other, in all parts of the world as well as in the Dark Continent, believed in the humanity of the coloured man, whether black, or red, or brown, and looked upon him as fitted for civilization, for Christianity and for freedom. He considered him as material for good government and for fair play. Both views, however, have been carried to an extreme in South Africa and upon either side evil resulted. The Boer treated the native from the standpoint of an intolerant and ignorant slave-owner. The Colonial Office tried to treat him solely from the standpoint of the sympathizing and often prejudiced missionary. Hence, in part, the Great Trek; hence some of the Kaffir raids and consequent sufferings of the early settlers; hence an addition to the growing racial antagonism.
- 71. Two Opposing Views of Government Boer Ideas of Democracy The principles of government believed in and practiced by the Dutch and British in South Africa have been and are diametrically opposed. The one took territory from the natives wherever and whenever he could and used it without scruple, and without return in the form of just government, for his own purposes. The latter, time and again, avoided the acquisition of territory; experienced war after war which might have been averted by the prompt expression of authority and strength; gave up regions to native chiefs which had afterwards to be conquered by force of arms; tried every phase of policy in the form of alliances, protectorates and buffer states in order to avoid increased responsibilities; gave up the Orange Free State to an independent existence under circumstances of almost incredible insistence; annexed the Transvaal with indifference, and gave it up without serious thought; in later days allowed German East Africa to be established, and at one time practically declined the acquisition of Delagoa Bay; permitted the Boers of the Transvaal to annex part of Zululand and to take almost the whole of Swaziland at the expense, even, of possible injustice to the natives. And all this from an honest though mistaken desire to avoid unnecessary expansion of authority or extension of territory. In those departments of Government which are apart from questions of acquiring or ruling dependent states there was the same antagonism. Equality being an unknown principle to the Boer, it was, perhaps, natural that he should endeavor to make his own language and laws and institutions the pivot of administration in any country under his control; that he should regard with suspicion and fear any attempt to raise the status of surrounding natives; and should reject with contempt, in the Transvaal at least, later efforts on the part of civilized aliens to obtain equality of political rights. The Dutchman in South Africa knew, in earlier days as well as at the present time, absolutely nothing of democracy in the British sense of the word. Republicanism, in the sense of Government by the majority, he does not even now understand—unless the majority be Dutch. To dream of convincing, or trying to convince others, by argument and discussion that some particular policy is better than another has always been far from his point of view. He has been too long accustomed to using the shot-gun or whip upon inferior races to deem such a policy either desirable or possible. The region these two races were destined to dominate was, and is, a splendid one. It had an infinite variety of resource and tropical production
- 72. Varied Opportunities for Settlers Statistics and Finances of South Africa and temperate growth. Within the million and a half square miles of South African territory were room and verge for a vastly greater white population than has yet touched its shores; while every racial peculiarity or pursuit could find a place in its towns and farms and mines and upon its rolling veldt. To the lover of quiet village life and retirement nothing could be more pleasant than parts of Natal and Cape Colony, and of the two Republics. To the keen business man, eager for gain and intent upon quick returns, the rapid and wealth-producing progress of the great mining towns gave all that could be desired. To the adventurous spirit, willing to suffer hardships and endure labor in its severest form for a possibly glittering return, the diamond and gold fields offered untold opportunities. To the hunter and tourist and traveller the myriad wild animals of the interior gave a pleasure only second to that felt by the Kaffir and the Boer when hunting the lion to his lair or the elephant in its native jungle. To the man fond of country life the vast plains, stretching in varied degrees of value and elevation from Cape Town to the Zambesi, afforded room for pastoral occupation and the raising of cattle and sheep upon a veritable thousand hills. To the seeker after new industries, ostrich farming, mohair, the feather industry and diamond mining have from time to time proved the greatest attraction. To the farmer or planter parts of the region were eminently fitted for the raising of wheat and other cereals, and the cultivation of tobacco, cotton, sugar and rice. To the restless and wandering Boer, South Africa seems to have given for a time everything that his spirit desired—isolation, land, wild animals to hunt, independence of control, freedom from the trammels of education and taxation and civilization. To the quieter Dutchman of Cape Colony has been given every element of British liberty and privilege of British equality; as well as land in plenty, and for thirty years, at least, the pledge of internal peace. According, also, to the latest figures[1] the material progress and recent position of all these countries has been good. Cape Colony, in 1897-98, had a revenue of $36,940,000, an expenditure of $34,250,000 and an indebtedness of $136,400,000; a tonnage of British vessels, entered and cleared, amounting to 12,137,000, together with 2,835 miles of railway and 6,609 miles of telegraph; exports of $108,300,000, and imports of $90,000,000; and 132,000 scholars in its schools. Natal and Zululand, combined, had a revenue of $11,065,000, an expenditure of $8,120,000 and
- 73. an indebtedness of $38,720,000; a tonnage of British vessels, entering and clearing, of 2,132,000, together with 487 miles of railway and 960 of telegraph; exports of $8,100,000 and imports of $30,000,000; and 19,222 scholars in its schools. The exports of Basutoland, under purely native control, had grown to $650,000 and its imports to half a million. The length of railway in the Bechuanaland Protectorate was 586 miles and in Rhodesia 1,086 miles; while the telegraph lines of the former region covered 1,856 miles. The South African Republic, or Transvaal, had a revenue of $22,400,000, an expenditure of $21,970,000 and an indebtedness of $13,350,000; announced imports of $107,575,000 and no declared exports; railways of 774 miles in total length and telegraph lines of 2,000 miles; and scholars numbering 11,552. The Orange Free State had a revenue of $2,010,000, an expenditure of $1,905,000 and an indebtedness of $200,000; imports of $6,155,000—chiefly from Cape Colony—and exports of $8,970,000, which were divided principally between Cape Colony and the Transvaal; 366 miles of railway, 1,762 miles of telegraph and 7,390 scholars in its schools. The following table[1] gives an easily comprehended view of South Africa as divided amongst its Kaffir, Dutch and English communities in respect to mode of government and measure of British responsibility: [1] British Empire Series. Vol. II. Kegan Paul, Trench, Trübner 4 Co., Limited. London, 1809. [1] South Africa. By W. Basil Worsfold, M.A. London, 1895. MODE OF GOVERNMENT. { Cape Colony } Responsible Government Three British Colonies { Natal } { { Bechuanaland } Crown Colony. { South African } Full internal freedom { Republic } within terms of
- 74. Two Republics { or } Conventions of 1852-54 { Transvaal } and 1881-84. { Free State. } { Basutoland, } Officers under High { Zululand, } Commissioner. { Tongaland, } Native Territories { { Transkei, } Officers under Cape { Tembuland, } Government. { Griqualand, } { Pondoland. } Territories of } { Administrator who the Chartered } . . . . . . . { represents the Directors Company } { and Secretary of State } { jointly. Yet, with all the varied advantages and evidences of substantial progress and prosperity given above, the present war has broken out in a result which could not have been different had the whites of South Africa been dwelling amidst limited areas, restricted resources, few liberties and a crowded population of competitive classes. Some of the reasons for this situation have been pointed out, and they include natural racial differences; a quality which Lord Wolseley described in a speech at the Author's Club on November 6, 1899, when he declared that of all the ignorant people in the world that I have ever been brought into contact with I will back the Boers of South Africa as the most ignorant; the inherent desire of the Dutch population for native slave labor and intense aversion to principles of racial equality; mistakes of administration and more important errors of judgment in territorial matters made by the British Colonial Office; a Dutch pride of race born from isolation, ignorance and prejudice and developed by various influences into an aggressive passion for national expansion and a vigorous determination to ultimately overwhelm the hated Englishman, as well as the despised Kaffir, and to thus dominate South Africa. Of the elements entering into this last and perhaps most important evolution the Afrikander Bund has
- 75. Afrikander Bund been the chief. The formation of this organization really marks an epoch in South African history, and has proved, in the end, to be one of the most effective and potent forces in the creation of the present situation. Nominally, it was organized in 1881 amongst the Dutch farmers of Cape Colony for the purpose of promoting agricultural improvement and co-operation and for the increase of their influence in public business and government. In 1883 it swallowed up the Farmer's Protective Association—also a Dutch organization. Practically, it was a product of the feeling of racial pride, which developed in the heart and mind of every Boer in South Africa as a result of Majuba Hill and the surrender of 1881. The openly asserted influence of their Transvaal brethren, and of this triumph, had prevailed with the Cape Boers to such an extent that the latter were able to compel the rejection of Lord Carnarvon's federation scheme although they did not at the time possess a large vote in the Cape Legislature or a single member in the Government. The same influence created a desire for racial organization, and the result was the Afrikander Bund. Its chief individual and local promoter was Mr. Jan Hendrik Hofmeyr, a man whose record is one of a loyalty to the British Crown which seems, in some peculiar fashion, to have equalled his loyalty to his race. In the beginning of the Bund, and during its earlier years, he could easily harmonize the two principles. How he could do so at a later period is one of the puzzles of history and of personal character. Incidentally, it may be said that Mr. Hofmeyr attended the Colonial Conference of 1887, in London, and contributed to its proceedings the then novel proposition that each part of the Empire should levy a certain duty upon foreign products—above that imposed upon goods produced in and exported to British dominions—and that the proceeds should be devoted to the maintenance and improvement of the Imperial Navy. He also attended the Colonial Conference at Ottawa in 1894, and had, consequently, received all the knowledge of Imperial development and power which travel and experience and association with the rulers of its various countries could afford. He has, since 1881, always declined office at the Cape, and it is, therefore, apparent that the solution of the personal problem must, in his case, be left to the future—with, perhaps, the further intimation that he is looked upon with great suspicion by local loyalists, and is considered to be the owner, or controlling influence, of Our Land, the chief anti-British organ in Cape Colony.
- 76. An Imperium in Imperio Mr. Reitz and the Present War From the first the Bund was regarded with suspicion by not only English politicians in the Colony, but by a few of the more sober and statesmanlike leaders amongst the Dutch. They were, however, won over, as time passed, except the President of the Orange Free State. Sir John Brand—he had accepted knighthood from the Queen as an evidence of his British sympathies—absolutely refused to have anything to do with it. I entertain, said he, grave doubts as to whether the path the Afrikander Bund has adopted is calculated to lead to that union and fraternization which is so indispensable for the bright future of South Africa. According to my conception the institution of the Bund appears to be desirous of exalting itself above the established Government and forming an imperium in imperio. But, wise and far-seeing as were these views, the Free State President could not hold back his own people from sharing in the movement. Mr. F. W. Reitz, then a Judge at Bloemfontein, afterwards President in succession to Sir John Brand, and, finally, State Secretary of the Transvaal under President Kruger, joined enthusiastically in its organization, and soon had many branches in the Free State itself. Of this period in the history of the Bund, Mr. Theodore Schreiner, son of a German missionary, brother of the Cape Premier and of Olive Schreiner—the bitter anti-British writer—has described an interesting incident in the Cape Times. He says that in 1882 Mr. Reitz earnestly endeavored to persuade him to join the organization, and that the conversation which took place upon his final refusal was so striking as to indelibly convince him that in the mind of Reitz and of other Dutch leaders it constituted, even then, a distinct and matured plot for the driving of British authority out of South Africa. During the seventeen years that have elapsed, says Mr. Schreiner, I have watched the propaganda for the overthrow of British power in South Africa being ceaselessly spread by every possible means—the press, the pulpit, the platform, the schools, the colleges, the Legislature—until it has culminated in the present war, of which Mr. Reitz and his co-workers are the origin and the cause. Believe me, sir, the day on which F. W. Reitz sat down to pen his Ultimatum to Great Britain was the proudest and happiest moment of his life, and one which has, for long years, been looked forward to by him with eager longing and expectation.
- 77. Dutch and English not Harmonious Branches of the Bund, within a few years, were established all over Cape Colony and the Free State, and, by 1888, the slow-moving mind of the Cape Dutch had grasped the racial idea thus presented with sufficient popular strength to warrant the holding of a large and general Congress. In his opening address the President spoke of a United South Africa under the British flag; but at the meeting held on March 4, 1889, at Middleburg, while much was said about the future Afrikander union, references to Britain and the flag were conveniently omitted. The platform, as finally and formally enunciated at this gathering, included the following paragraphs: 1. The Afrikander National Party acknowledge the guidance of Providence in the affairs of both lands and peoples. 2. They include, under the guidance of Providence, the formation of a pure nationality and the preparation of our people for the establishment of a United South Africa. 3. To this they consider belong— a. The establishment of a firm union between all the different European nationalities in South Africa. b. The promotion of South Africa's independence. There was also a clause of gratuitous impertinence towards the Imperial country—through whose grant of absolute self-government in 1872 the Bund was now beginning to aim, with practical effort, at the racial control of the Colony—in the declaration that outside interference with the domestic concerns of South Africa shall be opposed. Under the general principles of the platform these domestic concerns meant, of course, the relation of the different States toward each other, and the growing rivalry of Dutch and English in matters of Colonial Government, as well as the old- time question of native control and the newer one of territorial extension on the part of Cape Colony. So long as President Brand lived and ruled at
- 78. Mr. Cecil Rhodes to the Front Bloemfontein there remained, however, some check upon the Bund as well as upon President Kruger. If he had opposed the Bund actively, as he certainly did in a passive and deprecatory sense, the result might have been a serious hindrance to its progress. Brand's policy was to, indirectly and quietly, keep the Cape Colony and the Free State in harmonious and gradually closer co-operation instead of promoting that closer union of the two republics which was one of the ideals of the Bund leaders. He refused to accept Kruger's proposal of isolating their countries from the British possessions, and thus promoting the policy which, without doubt, had, since 1881, been shaping itself in the latter's mind. But, in 1888, Sir John Brand died, and was succeeded by F. W. Reitz. The influence of the new régime became at once visible in the platform above quoted, and in the whole succeeding policy of the Free State. It now assumed a more and more intimate alliance with the Transvaal, and frequently, during these years, the question of a union of the two countries was discussed. In 1896 Reitz resigned and accepted the State Secretaryship of the Transvaal—a position analogous in personal power, though not in the matter of responsibility to the people, with that of a Colonial Premier. Mr. M. T. Steyn became President of the Free State and the triumvirate of Kruger, Steyn and Reitz formed, with Mr. W. P. Schreiner and Mr. J. W. Sauer, in the Cape Parliament and Afrikander Bund, a very strong Dutch combination. Just where Mr. Hofmeyr stood it is hard to say now, but the probabilities are that, he was pretty well acquainted with the plots and schemes of these leaders. Meanwhile Mr. Cecil Rhodes had come to the front in mining, in speculation, in wealth, in financial organization, in politics, and in a great policy of Empire expansion. He had studied South Africa from the Cape to the Zambesi as few or no Englishmen have ever been able to do. He understood its Governments, its peoples and its racial complexities with the innate thoroughness of genius or of a woman's intuition. To him the looming menace of the Afrikander Bund was as clear AS it had been to President Brand, and, from the time when lie entered the Cape Parliament in 1880 and became Premier in 1890 until his retirement from the latter post in 1895, his whole heart and ambition was devoted to preventing Dutch expansion and to checkmating the new Dutch organization with its clever manipulators at Pretoria, Bloemfontein and Cape Town. To this end he founded the famous British South Africa Company, and, by acquiring control over the vast areas
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