![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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-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/1314267-250531014830-4da8c11a/85/Computability-In-Context-Computation-And-Logic-In-The-Real-World-Cooper-Sb-54-320.jpg)
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- 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. her back, holding one of her hands, perhaps; but Theresa was standing very straight—her back seemed unusually strong—and she was smiling faintly, while her hands were occupied in the swift removal of her gloves. There seemed no point at which she could be conveniently caressed, and Mrs. Morton sank into the chair beside the tea-table. You will be glad of tea, she said. Basil, won't you make Theresa sit down? She looks so tired. Now, dear, you would like some hot toast. Theresa was in an uncertain temper, and if she had not been very eager for buttered toast, she would have refused it as a form of contradiction; but the sight of it shining in the hearth overcame annoyance with desire. She foresaw, however, a quick starvation if Mrs. Morton continued to accompany offers of food with these firmly uttered statements. You had a tiring journey? There was just a redeeming tilt at the end of the sentence, and Theresa condescended to consider it a question. No thank you. I liked watching the country. But in winter time it is all so sad. But this is spring—almost! And I saw some lambs—the first. They're early here. And as she spoke she saw the green cleanliness of the earth when the snow has melted into it, and lambs, like little forgotten patches of that snow, leaping about the hills. She went on quickly. And there were pigs. I like them. They're so greedy, and they don't pretend to care for anything except their— except what they eat. The subject of pigs was not encouraged. Basil was handing her more toast, as though he wished it were a kingdom, and she knew he was too much engaged with the joy of her presence to listen to her babblings. It was right that he should be happy at seeing her in the home they were to share, yet, in that moment, he lost something
- 57. with which she once had dowered him. She eyed him critically. He was good to look at, and beauty always softened her; but his strongest appeal for her had been his distance, and here, among the teacups with his mother, he was too near, he almost seemed domestic. She realized the cold cruelty of her phase, she hoped it would not last, but she could do nothing to be rid of it. She was forced to her callous scrutiny, she was entirely shorn of any sense of possession, and while her mind told her she would recover her old sensations, her heart was like a dead thing in her breast. She knew the reason, for it lay on that heart which it had struck, and when she stirred she felt the sharp edge of Alexander's letter. She moved now, quickly. She wants a cushion, Mrs. Morton cooed, but Basil was already propping Theresa's back. She smiled at him, from the lips, trying to feel the kindness that lay crushed. You're lovely, he said, under cover of Mrs. Morton's manipulation of the tea things. She gave her emphatic half-shake of the head. She knew the wind had nipped her, that her hair fell in wisps about her face, and his loving blindness made her disloyalty the blacker. She would not be disloyal, but she questioned her love for him, she faced the possibility of resigning him, and at once she had an impulse to thrust herself into his arms. Instead, she put her hand in his and held it fast, and, like a gentle tide, she felt the return of tenderness. Alone in the pretty room prepared for her, and still with that determined loyalty upon her, she made to throw Alexander's letter in the fire; yet to do that, she argued, was to admit its power, and it had no power for anything but a disturbance that would pass. It came too late. A little while ago—she did not follow the thought, but she knew its path. She shut her eyes to it. She loved Basil. She could not picture life without him. After herself she belonged to him. She was proud to be his. He was good, and
- 58. true, and for all her self-esteem she wondered how he came to love her. After dinner, as they all sat in the drawing-room Theresa gazed at Mrs. Morton in a kind of wonder. She sat in her chair, crotcheting slowly, with frequent reference to an instruction book, and counting her stitches half aloud between her amiable sentences. In uttering commonplaces, she had a dignity which forced the listener to reach deeply or loftily for truth, and return from that vain pilgrimage with a sensation of having been robbed by the wayside. When she announced that their nearest neighbours, the Warings, were to have tea with them on the following day, Theresa waited anxiously for the something more implied in those pregnant tones. But Mrs. Morton serenely counted stitches. At length, You will like the Warings, she said. Theresa stared into the fire. She was prepared to hate anyone thus introduced. She was not far from hating Mrs. Morton. Her lips tightened, her idle hands pressed each other closely. Had this placid person ever been in love? Was she so obtuse that she could not feel the fret of Theresa's spirit? Did she not know that solitude is the great need of lovers, or realize that Basil had not yet so much as kissed her? The presence of the groom had prevented confidences on the drive, and in the house Mrs. Morton had shadowed her in excess of welcome. She looked at Basil, who was looking at her, and raised her eyebrows wearily. He raised his own, and they smiled in the delightful comradeship of annoyance shared. She wanted to talk to him, to make amends for the wickedness of her thoughts, and here they sat, all three, and her tongue was tied. She longed to tear the crotchet from Mrs. Morton's plump white hands; she felt the old anger of her childhood rising to her throat, and she pressed her hand to it and forced it back. Basil, Theresa's throat is sore. You shouldn't have driven her in the dog-cart on such a day. You shall have some sugared lemon, dear. Ring the bell, Basil.
- 59. Not for that, please! I haven't a sore throat. I—just happened to touch myself there—oh, really! There was a laughing anguish in her voice. Was she to be handcuffed as well as starved? Don't be afraid of giving trouble, dear. Theresa always tells the truth, Mother. Oh, of course! Very well. But she looked as if she had a sudden pain. I'm afraid it is a habit. That reminds me of an old lady I knew when I was young. I thought she had St. Vitus's dance, until her maid told me that she wore all her valuable jewellery on her—under her dress, and she was constantly touching herself to make sure it was all there. What were you hiding, Theresa? She lifted her chin to show him the pretty lines of her bare neck. Ah, your own beauty, he added softly. Something else, she said. Tell me. She shook her head. You must find out. Mrs. Morton's voice penetrated this happy murmur. You crotchet, Theresa? Morton had to shake the hand he held. Theresa, Mother asks you if you crotchet. Oh! No, I don't. That's very pretty. It is for you. Is it? Yes, a tray cloth. Thank you. How clever of you!
- 60. I'll teach you if you would care to learn. I don't think I could. I've got such stiff fingers for things like that. They're good enough for typing. Basil, did I tell you about that last woman of mine? It was during the recital of this tale that Mrs. Morton left the room. Theresa stopped and looked at the closing door. Was I saying anything wrong? she asked. I am so used to talking frankly to Mr. Smith and Jack, that I forget other people may not like it. Was I? No, dear, but the whole thing is rather disagreeable to her. But how? Well, you see—— Is it that she doesn't like you to marry a woman who has earned her own living? That, of course, was rather a shock. Darling, try to understand her attitude. She has old-fashioned notions of womanhood. She thinks you should not have been allowed to do the work you did, and I own that it seems unnatural to me, too. But you are wonderful, Theresa. You are the exceptional woman who can do these things. You are unscathed. She stood up and fell into that attitude in which he had first seen her. I am not unscathed, she said. If you drop down into hell, even another person's hell, you come back—scorched. And I have the marks. She turned to him quiveringly. Basil, have you ever suffered? I think so. My father was killed—I found him. And I—he was a great deal to me. Death! She flung back her head. Oh yes, yes, yes; death is so much worse, and so much better, than people fancy. But have you
- 61. felt your own heart shrivelling to a thing like a dried nut? Have you carried that about with you as—as some people do? And have you heard stories told by women whose eyes are dry because they have no tears left? I have. I have. Oh, shocking stories of sin, of things no girl should know the name of! She spoke more quietly. It's quite possible that I know more than you do of the world's evil, for you are the kind of person who never looks in the gutters: you keep your head high, but I look everywhere. And I want to see the gutter dirt: it's part of life, and the sun shines on that as well as on the flowers in the gardens. But I don't like it. You're not to think I like it. But you are to think I am very proud of having done that work. I suppose Mrs. Morton has not told your friends I am a working woman? She did not wish them to know. You must not think us snobs, Theresa, but in a place like this there are so many prejudices, and we do not want you to be hurt by them. I can't be hurt by foolishness, and I won't be in the conspiracy. And why should your mother feel like that? She is Mr. Smith's sister, and their father educated himself, and then made sweets. From her point of view isn't that as bad, worse even, than my honourable calling? You see, you are a woman, Theresa. Are we never to go unveiled and free? He smiled gently. Moreover, when my mother married my father she considered herself a member of his family rather than of her own. Oh! Some women do, you know. Oh! Don't hope for that from me, Basil. I won't be welded into anybody's family or anybody's nature. Darling,—his arms were round her—I never want you to be anything but yourself. She leaned back.
- 62. But is it a self you like? Are you satisfied with it? You know—she touched his chin lightly with her forefinger—we're going to have a lot of trouble. If we are together—— Because we are together. Oh, I can smell it afar off. I did directly I came into the house. Don't you like it? he asked, and released her gently. The house is beautiful—but we're not going to be alone in it, are we? Oh, I'm not complaining, but I rather wish we were going to have a semi-detached villa, and a maid like Bessie. Yet I hate housework! I'm afraid—I'm dreadfully afraid—I shall get annoyed. Her head was on one side, she twisted her fingers among his. Theresa, you will be considerate of my mother. Don't, don't, don't! If you put questions in the form of statements I shall go mad. There was patience in his look, but he redeemed it with a laugh. I beg your pardon. Theresa, will you be considerate of my mother? I'll try. I thought you prided yourself on your tact. I do. I have it highly developed, but the devil sometimes steals it. You are a little childish. Very! And my mother is dear to me. So was mine to me. She was—sweet, my mother was, but that didn't prevent my getting angry with her. I wish I didn't get so angry. Do you understand that you're engaged to a volcano, an active one? I'm beginning to. And I'm in eruption now. Be careful.
- 63. I love my volcano. She'll hurt you, often. Destroy you altogether, perhaps. Basil, I want to tell you something. There'll be times when I shall nearly hate you. Why? I don't know. It's just me. I'm cruel. But love me always, and I'll come back to you. I can't help loving you, dear, he said, and kissed her hair. But do you trust me? Darling, of course! She made herself more comfortable in his arms. Then I'll be worthy, if I can. Take care of me. She was happy that night when she went to bed, and, sitting by the fire with her softly slippered feet close to the blaze, she could take Alexander's letter from its place, and hold it easily in a hand on which Basil's diamonds sparkled. Only that morning the letter had been dropped into the hall as she stood there in her travelling coat, with the veil that swathed her little hat pushed up so that she might drink the hot milk Bessie offered. That'll be for the master, Bessie said. No, it's for you, Miss Terry. Now, drink the milk. I won't have people telling me you're thin. Of course, you're thin! You tell 'im I've given you hot milk every morning this last week. All right, Bessie, all right. He knows you take care of me. So 'e ought. She had held the letter in her pocket, stroking it with her thumb; and then Grace and the baby had come in to say good-bye, and not until she was in the train had she been able to read what Alexander wrote. Then she read it many times. Will you not be here to see it flush the hills? And the streams so fierce and heavy that it takes
- 64. your breath away. She wanted to be there. She thought she felt the cold spray on her face. She felt the air: passing through it was to be new-made. Her steps were buoyant, her eyes were washed and clean. She heard the water, she heard the larches singing, and her heart cried in her breast. She would dream to-night, and she longed for the darkness and feared it. She would see the lakeside and the black precipice, the water would be whispering at her feet, and she would be waiting, waiting. It was a long time since she had been there. But Alexander's letter roused her to more than this sickness of longing that she dared not analyse too closely. I've been waiting for that book of yours, he said. There would never be a book. And he was looking for it. She was hurt and shamed as by a promise broken to a child. Talking freely on that wonderful one day of theirs, she had told him what she meant to do, and he had given her that plunging look of response. How had she dared to talk like that, and then do nothing? She knew the answer. And now it was too late. She was to be a county lady. She had come to an age when she was no longer sure that she had the power she had always wanted; but she ought to have put it to the test, for she had told Alexander what she was going to do; she had told Alexander. The words came with such force that her lips framed them. She had told Alexander. She had another tale for him now. Oh yes, she said, you shall have a letter, and she quickly wrote it, sitting there with the firelight on her bare arms and her quick, thin hands. Dear Alexander, Thank you for your letter. It was like seeing the place. I didn't begin the book. I lost faith, and I'll never get it back. I'm weak, but perhaps it is a good thing and has saved the spilling of much ink. It was a young ambition of mine, and you know what Father is! So I'm going to be married instead, for that's a profession we all think we are fit for! I shall see you at Easter. It will be two years then.
- 65. Theresa. She felt like a penitent who has relieved her soul of sin and planted a dart in the breast of her confessor. CHAPTER XXIV As Theresa entered the drawing-room on the following afternoon, she felt the imminence of ceremony. Mrs. Morton had cast aside her crotchet and sat, in satin and old lace, awaiting the coming of her guests; and the room, softly and rosily shaded, seemed to Theresa like a temple raised to the social cult, with the tea-table for altar and Mrs. Morton for ministrant. She closed the door with a decorous quiet and advanced, her mouth curved into the faint smile that had some mobile quality though the lips were still. I thought you would be late, said Mrs. Morton. I did my hair three times. I wanted to look nice. You look charming, dear. I hope you are not feeling nervous. Oh no! I expect you are—a little. I remember my own introduction to the friends of Basil's father. It was in this room. It was a very anxious moment for me. One naturally wants to please, and I was very shy as a girl. You were younger than I am, perhaps. Only eighteen. Ah, I'm twenty-five. That makes a lot of difference. The picture of a maiden hearkening to the wisdom of the matron, she stood before
- 66. Mrs. Morton with her hands behind her back, her head bent to look and listen. But you are not married, dear. Mrs. Morton was finding it unexpectedly easy to talk to Theresa. And until a girl is married —— Yet I sometimes feel as though I have been married several times, she said. The words suggested a shocking fertility of imagination. My dear, what do you mean? Theresa laughed. Just that. One knows so much one hasn't actually experienced. I hope not! But I can't help it, she urged. It's how I happen to be made. Mrs. Morton moved uneasily. I'm afraid I don't know what you mean. I suppose I am very old-fashioned. She was disappointed at the very moment when she thought she was beginning to understand her son's love for this pale, quick girl with the watchful eyes, whose glances half-alarmed her. She was glad when the door was opened. Ah, here's Basil. Theresa turned to him. Basil, she said, have you ever been in a balloon? No. But you can imagine what it's like, can't you? Yes, I think so. Of course you can. She was eager, persuasive. You would have a feeling of having no inside, wouldn't you, and no feet? And you would feel like a little speck of dust, and because you were so small, it wouldn't seem to matter if you fell out into that enormous empty space? Would it?
- 67. He humoured her, smiling as he took in the radiance of her hair, the slimness of the green-clad body, the thin feet in their bronze- coloured shoes. Very likely, he said. You see! she exclaimed, laughing. Basil knows all about something he hasn't experienced. Why shouldn't I? Her lips changed their curve. Is it because I am a woman? Her little taunt was for him: she had forgotten his mother, on whose face there were small evidences of distress. What is it now, dear? he murmured, and led her to the window. Come and look at the trees against the sky. She went meekly, for the sake of the hand holding her; but she was shaken by inward laughter. Like a child she was being drawn out of mischief and enticed to look out of the window at the pretty sky. And later, when the guests had arrived, when Mr. and Mrs. Waring talked to her kindly and ponderously, and the three Misses Waring in the glow of their healthy young beauty asked who was her favourite author and if she liked the country, she knew that Mrs. Morton watched her nervously. She was annoyed by that suspicion of her manners, but stronger than her annoyance was her determination to please, not, like Mrs. Morton, for her lover's sake, but for her own. Her one sure talent cried loudly to be used, and as she listened to it, she felt a stir of physical pleasure in her breast. She, who had drawn the truth from unwilling lips, and brought back long-forgotten laughter, had no doubt of making what effect she chose on these amiable strangers. Sitting in a low chair, with folded hands on her knee, and looking younger than she was, she listened, smiled, and answered quietly while she studied the faces ringing her. She saw Mr. and Mrs. Waring deciding that she was a nice little thing, not pretty, not clever, but possessed of the vague niceness necessary for the complete young lady. That was not sufficient tribute for Theresa, and she awaited the opportunity to make Mr. Waring laugh. It came, she seized it with
- 68. some audacity, and the old gentleman's guffaw acknowledged her. Her lifted brows wondered at his amusement, but her mouth betrayed her. A pale flush of excitement was in her cheeks. Mrs. Waring and her daughters were smiling politely, while the head of the family leaned back in his chair to laugh, and, between his cackles, he repeated the joke to Morton. Morton, too, smiled politely; the humour did not reach him and, a little ashamed of his guest's clamour, he drew him on to agricultural matters; but those stiff smiles were Theresa's triumph, for the joke had been aimed at Mr. Waring alone, and it had hit the mark. The two matrons fell into talk, and, still wearing that gentle look of surprise, Theresa turned to the three young women: she seemed to ask for conversational help, and they gave it in the form of questions. Did she ride? No, she wished she did. She thought Basil was going to teach her. He rides perfectly. The second Miss Waring looked across the room to where he sat, and in that shy glance Theresa read renunciation, maidenly and empty of all bitterness. I expect you all do, she said. No, my sisters don't care for it. I love it. Basil taught her when she was small. She can ride anything, said the eldest sister proudly. They hunt together. We haven't lately, Rose, the other said, and blushed. Theresa leaned forward coaxingly. Oh, do go next time and let me see you both! she cried. It's splendid to see people doing things really well. Oh, do you think so? The second Miss Waring controlled a smile. Was she fond of gardening? This question was from the youngest beauty. No, she didn't know anything about it. They only had a patch
- 69. of rough grass at home, and an apple-tree. There was a pause. It was Rose who returned to the subject of books. I expect you are a great reader? Oh, more or less. I adore reading. And poetry. Whose poetry do you like best? I don't know, said Theresa slowly. She had assured them all of their superiority: they liked her; Mrs. Morton forgot to be nervous, Basil was glad to see her in that group of girls. Other visitors came and went. Two elderly sisters, adorned with large brooches and pendulous ear-rings, seated themselves before Theresa and told her anecdotes of Morton's childhood. Their voices defied her to rob him of his early virtues, and their looks prophesied her pernicious influence. She liked these ladies with their pleasant acidity: there was resistance in them; but it was with the arrival of Conrad Vincent that enjoyment brightened her eyes and loosed her tongue. He came in slowly and greeted his friends without haste, but when he stood before Theresa she felt the hurry of his mind. Behind the lazy glances of his eyes she saw the racing thoughts and warmed to him. He sat beside her, she turned to him as though at last she could greet a comrade, and the group broke up, leaving them alone. Do you know, Morton said, when his guests had gone, you talked to Vincent for a whole hour? Was it so long? It went in a flash. He is a good talker—provocative. I enjoyed it very much. You seemed to do so. Do you mind? No dear; but—— Was I rude?
- 70. Not rude. What then? You rather ignored the others. I really did my best, but when Mr. Vincent came I forgot them. I like him. I hope he'll come again. I should like to marry him for dull days when I've nothing to do, and you for all the rest. Don't, Theresa. I can't bear to hear you flippant about our love. It's the result of talking to him and of listening to the others. I wish Mr. Smith could have heard them. Did you hear the conversation about the thriftlessness of the agricultural labourer? They had the decency not to mention his wage. It was the eldest Miss Waring who was so eloquent. It seems she has been telling Jim somebody's wife how to spend her money! I wonder how much her own weekly bill of luxuries would come to. She is a charming girl. Yes, her complexion has been formed on fresh air, good food, pleasant exercise, and an easy conscience. I'm sure she's nice. I wonder what Mrs. Jim's complexion is like. And is she charming on a few shillings a week? Basil, while, in my professional manner, I was laughing at that ignorant young woman, I was searching my own conscience, and I thought, 'Can I—can I be going to live in this beautiful place while Mrs. Jim is so hungry?' And I don't think I can. What do you mean, Theresa? Is this—is this my dismissal? Not unless you make it that. Basil, I wish you would come out into the world. You are a good man: ever so much better than these dear souls who hunt, and ride, and shoot, and prop up the country. You tower above them. The nice hard lines of your face proclaim you! I wish you earned your living. I think I do. No one can call me idle. No, you are very busy.
- 71. And I employ a large number of men. Her lips twitched. I know. You are one of the props. But you have so much more than you need. Wouldn't you like to do something with it? Will you let me be another Simon Smith? I think his system of charity is pernicious! What's yours? Don't you give jellies to your Mrs. Jims? Yes. It is just the same thing. We shall never agree on these subjects, Theresa. No; they will be fruitful in discussion. Don't you want me to talk to you? Certainly. You're angry, aren't you? I hope not. Yes, you are! Look how good-tempered I am. Her eyes were alight with battle, her lips only parted for speech, and her hands were restless. Now she clasped them and swayed back and forward as she spoke. I should like to have four—no, five—hundred a year, and do good things with the rest of your income. Perhaps to-morrow I would rather have those pearls you want to give me, but I don't think so. Pearls do not become me! And to-day I want to build model cottages. We could let this house—— Theresa! Let us end this nonsense. We have lived here for generations. She laughed softly. I know, but somebody has to begin doing something else. And your workmen have lived in pigsties for generations. My workmen——! You don't know what you are talking about! The women of this house have never interfered in outside matters.
- 72. She banged her fist on the little tea-table. Don't talk to me as though I belonged to a harem! Don't be absurd, Theresa. He was very handsome when he was angry. I'm not absurd. If you say I'm not fit to know about your affairs— yes, and to interfere with them—I'm, I'm a chattel. He smiled. Nothing so peaceful, he assured her. If you wanted insignificance—— I didn't. I wanted you. I don't believe you knew what you were getting, she said, and left him. When she came downstairs for dinner, she found him awaiting her in the hall. Well? she said. Her eyes were very bright; she laughed at him. Have you forgiven me for the harem? Oh, hang the harem! Come into the smoking-room. She touched him on the arm. Basil, she said, you nearly swore. I wish—I wish you would really do it. I've no doubt there will be plenty of opportunity. Oh, I like you! she cried. I like you! He looked down at her. That's not enough. He saw her eyes darken, her mouth grow tremulous, but she controlled her lips and fortified herself against this new insistence. Then you must give me everything. I will. Theresa, forgive me. I've lived too long without you. And if you will come round the estate with me to-morrow, I'll show you where and how my people live. Bless you! Thank you. I really want to help, and, of course I'll come. She gave him his reward. Don't let us quarrel, because—I
- 73. love you. He caught her hands. Do you? Do you? Am I not proving it? I'm thrusting myself into a very uncomfortable place because of you. If you are not very nice I shan't be able to endure it. Mrs. Morton tells me you all dine regularly with each other once a month! This is a dreadful welding of opposites! But love— love is supposed to be a strong cement. And I love you more than ever, Theresa, more every day. He kissed her with a violence that hurt her lips. They parted painfully, and she looked up at him with a tiny crease between her brows, before she thrust her face into his coat, burrowing there, holding fearfully to his arm. Keep me, she said. Keep me. He had no words tender enough for her. The appeal swelled his love to a flood too full for turbulence, and he stroked her hair, drew her to his knee and rocked her there, so that she felt secure and was comforted like a child. But can you keep me? she said, sitting up with a jerk. Do you think you can? I mean to. But you won't if you lock me outside yourself. I don't feel that you have quite opened your doors. She hesitated, and spoke. Basil, I sometimes think there's an enemy of yours after me, and I'm hammering for you to let me in, and you're not quick enough. He laughed. Who is this enemy? Ah, do you think I dare turn round and face him? Open your doors, open your doors? They're wide, he said, and spread his arms. But it's rather a narrow wideness, she said, as she put her head on his shoulder. One might easily miss it in a hurry.
- 74. They were quiet for a little while, then Theresa spoke dreamily. I wish they wouldn't sound the dinner-gong. I never want to move again. Didn't I dress quickly? It was to get back to you. Basil, I like you in this mood. I'm not in a mood, dear. I'm always like this when you will let me be. No, she said positively, you are different. You were an indulgent potentate. Now you are a friend. You can't deceive me. I don't want to deceive you, but it is you who have changed. Oh, I hope not! she said heartily. He laughed: she was teaching him to do that, and the friendly sound mingled with the loud summons of the gong. She screwed up her eyes in merriment. I really believe you are beginning to appreciate me, she said, and hand in hand they went across the hall. I am going to show Theresa the plans of the estate, Mother, he said, during the progress of the stately meal. Certainly, dear. You will like that, Theresa. I am not at all sure that I shall, she said clearly. Then don't worry her, Basil, if she doesn't want to see them. But I do! And if I didn't I would! Well, don't get tired, dear. I'm afraid it will make your back ache. Oh, my back! That was suppled long ago, by a typewriter. Poor little Theresa, Mrs. Morton murmured, for the servants had left the room. Theresa cracked a nut as though it had been the lady's head. She cast a hot glance at Morton, who was delicately peeling an apple. He looked softly at her. In his eyes there was the tenderness of a pity
- 75. more understanding and deeper than his mother's: it was pity for all the laborious, independent women in a hard world. The lift of Theresa's head was a signal that Mrs. Morton was growing to fear. You needn't be sorry for me. You're sorry and half ashamed. Why? Why? Why? She held in her voice, and spoke with a breaking strain in it. And I resent being pitied. Why, as soon as I knew anything, I was trying to decide what I should be when I grew up. Mrs. Morton was propitiatory. It was very sweet and brave of you, my dear. No, it was just as natural as eating. And if I were the wife of Croesus, my daughters should have professions. She had a vision of those daughters: they were bright and eager, and they were her own, and for a moment the sight of them matured her impulsive and intolerant youth. She warmed to them: she felt a spreading as of wings, a softening of all her being, and her hands and lips were quieted and strong. She laughed as water laughs, trickling through the moss. She smiled from one end of the table to the other. I'm sorry I get so vehement, she said. I can't help it. I hope I wasn't rude. An apology from Theresa was almost more alarming than a scolding. No, no, dear, I quite understand, Mrs. Morton said in haste, while Basil smiled slowly, a little stiffly, conquering uneasiness with love. In the smoking-room, Theresa sat down emphatically and spoke with great decision. I'm horrid to your mother, she said. You are not very nice. She raises the devil in me! Theresa!
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