![xviii Acronyms and Abbreviations
LFSR linear feedback shift register
LP linear programming
LTL linear temporal logic
MTBDD multiterminal binary decision diagram
NNF numerical normal form
OBDD ordered binary decision diagram
PAC probably approximately correct
PBDD partitioned binary decision diagram
PC propagation criterion
QBF quantified Boolean formula
ROBDD reduced ordered binary decision diagram
RTL register-transfer level
SAC strict avalanche criterion
SAT satisfiability [not an acronym]
SBS stochastic binary system
SCC strongly connected component
SEC sequential equivalence checking
SEM sample error minimization
SOP sum-of-product
SQ statistical query
STG state transition graph
UBQ unreliable boundary query
UMC unbounded model checking
VC Vapnik-Chervonenkis
XBDD extended binary decision diagram](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-23-320.jpg)
![1
Compositions and Clones of Boolean
Functions
Reinhard Pöschel and Ivo Rosenberg
1.1 Boolean Polynomials
The representations of Boolean functions are frequently based on the fundamental
operations {∨, ∧,
}, where the disjunction x ∨ y represents the logical OR, the
conjunction x ∧ y represents the logical AND and is often denoted by x · y or
simply by the juxtaposition xy, and x
stands for the negation, or complement,
of x and is often denoted by x. This system naturally appeals to logicians and,
for some reasons, also to electrical engineers, as illustrated by many chapters of
this volume and by the monograph [7]. Its popularity may be explained by the
validity of many identities or laws: for example, the associativity, commutativity,
idempotence, distributive, and De Morgan laws making B := B; ∨, ∧,
, 0, 1
a Boolean algebra, where B = {0, 1}; in fact, B is the least nontrivial Boolean
algebra.
It is natural to ask whether there is a system of basic Boolean functions other
than {∨, ∧,
}, but equally powerful in the sense that each Boolean function may
be represented over this system. To get such a system, we introduce the following
binary (i.e., two-variable) Boolean function +̇ defined by setting x+̇y = 0 if x = y
and x+̇y = 1 if x = y; its truth table is
x y x+̇y
0 0 0
0 1 1
1 0 1
1 1 0
Clearly x+̇y = 1 if and only if the arithmetical sum x + y is odd, and for this
reason +̇ is also referred to as the sum mod 2. It corresponds to the “exclusive or”
of logic, whereby the “exclusive or” of two statements P and Q is true if “either
P or Q” is true. Notice that some natural languages, such as, French, distinguish
“or” from “exclusive or” (“ou” et “soit”), whereas most natural languages are less
3](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-28-320.jpg)
![4 Reinhard Pöschel and Ivo Rosenberg
precise. The function +̇ is also denoted ⊕ or + and, in more recent engineering
literature, by EXOR.
Consider the system {+̇, ·, 0, 1}, where 0 and 1 are constants. A reader familiar
with groups will notice that B; +̇ is an abelian group with neutral element 0
satisfying x+̇x ≈ 0 (also called an elementary 2-group), where ≈ stands for an
identity on B. Moreover, GF(2) := B; +̇, ·, 0, 1 is a field, called a Galois field
and denoted Z2 or F2. Thus, in GF(2) we may use all the arithmetic properties
valid in familiar fields (such as the fields Q, R, and C of all rational, real, and
complex numbers) but not their order or topological properties. In addition, GF(2)
also satisfies x+̇x ≈ 0 and x2
≈ x. Clearly GF(2) is the field of the least pos-
sible size, and so one may be inclined to dismiss it as a trivial and unimportant
object. Surprisingly, it has serious applications. A practical one is in cryptogra-
phy and coding theory (for secret or secure data transmission, for example, for
governments, banks, or from satellites; see Chapters 8 and 9 in this volume).
Denote x1+̇ · · · +̇xn by
r n
i=1xi . Let f be an n-ary Boolean function distinct
from the constant cn
0 (which is the n-variable Boolean function with constant
value 0). In its complete disjunctive normal form (DNF), replace the disjunction
∨ (of complete elementary conjunctions) by their sum mod 2
r
. This is still
a representation of f because for every (a1, . . . , an) ∈ Bn
, at most one of the
elementary conjunctions takes value 1 and 1+̇0+̇ · · · +̇0 = 1 and 0+̇0+̇ · · · +̇0 =
0. Using x0
≈ x
≈ 1+̇x and x1
≈ x throughout, we obtain a representation of
f over {+̇, ·, 0, 1}. The following proposition makes this more precise. Here the
symbol
denotes the usual arithmetical product. We make the usual convention
that in expressions involving +̇, · and 1 products are calculated before sums, for
example, xy+̇z stands for (xy)+̇z, and that
and
r
over the empty set are 1 and
0, respectively.
Proposition 1.1. [28] For every n-ary Boolean function f , there exists a unique
family F of subsets of N = {1, . . . , n} such that
f (x1, . . . , xn) ≈
r
I∈F
i∈I
xi . (1.1)
For example, x1 ∧ x2 ≈ x1+̇x2+̇x1x2 with F = {{1}, {2}, {1, 2}} (direct verifi-
cation). Call the right-hand side of (1.1) a Boolean polynomial.
Proof. Let f be an n-ary Boolean function. If f = cn
0, take F = ∅. Thus, let f =
cn
0. In the discussion leading to the proposition, we saw that f may be represented
over {+̇, ·, 0, 1}. Multiplying out the parentheses, we obtain a representation of f
as a polynomial in variables x1, . . . , xn over GF(2). In view of x2
≈ x, it may be
reduced to a sum of square-free monomials. From x+̇x ≈ 0, it follows that it may
be further reduced to such a sum in which every monomial appears at most once,
proving the representability of f by a Boolean polynomial. It remains to prove
the uniqueness. For every F ⊆ P(N), that is, a family of subsets of N, denote](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-29-320.jpg)
![6 Reinhard Pöschel and Ivo Rosenberg
and
Hi+1 = {a ∈ Si+1 : f (a) + |Ta| is odd}.
For (a1, . . . , an) ∈ Bn
, set
χ(a) = {1 ≤ i ≤ n : ai = 1}
and set F = χ(H0 ∪ · · · ∪ Hn). A straightforward proof shows that
f (x) ≈
r
I∈F
i∈I
xi .
For example, if n = 2 and f is the implication →, we get H0 = {(0, 0)}, H1 =
{(1, 0)}, H2 = {(1, 1)}, F = {∅, {1}, {1, 2}}, and x1 → x2 ≈ 1+̇x1+̇x1x2.
Remark 1.4. Contrary to disjunctive normal forms, there is no minimization
problem for representations of the form (1.1). However, the Boolean polynomials
of some Boolean functions are long. The worst one is the Boolean polynomial
of f (x1, . . . , xn) ≈ x
1 . . . x
n, whose family F is the whole P(N); for example,
for n = 2 we have x
1x
2 ≈ 1+̇x1+̇x2+̇x1x2. If, along with +̇, ·, 1, we also allow
the negation, certain Boolean polynomials may be shortened. Here we can use
some additional rules such as x
y ≈ y+̇xy and xx
≈ 0. For these more general
polynomials, we face minimization problems. It seems that these and the systematic
use of the more general polynomials have not been investigated in depth.
Remark 1.5. As the reader may suspect, Proposition 1.1 can be extended to any
finite field F (e.g., Z3) and maps f : Fn
→ F.
Remark 1.6. A long list of practical problems (mostly from operations research)
leads to the following problem. Let f be a map from Bn
into the set Z of integers
and assume that we want to find the minimum value of f on Bn
. The potential
of Boolean polynomials for this problem was realized quite early in [6]. A more
direct variant and a related duality are in [23, 24], but generally this approach
seems to be dormant.
1.2 Completeness and Maximal Clones
It is well known that every Boolean function may be represented over {∧, ·,
}
(e.g., through a DNF or a conjunctive normal form, CNF). In Section 1.1, we saw a
representation of Boolean functions over {+̇, ·, 1} (through Boolean polynomials).
In general, call a set F of Boolean functions complete if every Boolean function
is a composition of functions from F. In some older and East European literature,
a complete set is often referred to as functionally complete. In universal algebra,
the corresponding algebra B; F is termed primal.
Naturally we may ask about other complete sets and their size. The two examples
we have seen so far consist of three functions each. However, the set {∧, ·,
} is](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-31-320.jpg)
![1 Compositions and Clones of Boolean Functions 7
actually redundant. Indeed, {∧,
} is also complete as x1x2 ≈ (x
1 ∧ x
2)
(this
identity is known as one of the De Morgan laws).
A Boolean function f is Sheffer if the singleton set { f } is complete. Clearly a
Sheffer function is at least binary. The following functions NAND and NOR are
Sheffer:
x NAND y :≈ x
∨ y
, x NOR y :≈ x
∧ y
.
The suggestive symbols NAND and NOR were adopted by electrical engineers to
describe the functioning of certain transistor gates after these were invented in the
1950s. However, the two functions were introduced in logic long ago by Sheffer
[25] and Nicod [19]; they are known as Sheffer strokes or Nicod connectives and
are variously denoted by |, ⊥, , ↑, ↓, and so forth. The fact that NAND is Sheffer
follows from
x
≈ x NAND x, x ∧ y ≈ x
NAND y
;
consequently, the complete set {∧,
} can be constructed from NAND alone, and
hence NAND is Sheffer. The proof for NOR is similar. The fact that NAND and
NOR are the only binary Sheffer functions will follow from Corollary 1.5, which
provides a complete characterization of Sheffer functions.
Having seen a few complete sets of Boolean functions, we may ask about other
complete sets or – even better – for a completeness criterion, that is, for a necessary
and sufficient condition for a set F of Boolean functions to be complete. E. L. Post
found such a criterion in [20]. To formulate the criterion in a modern way, we need
two crucial concepts. For 1 ≤ i ≤ n, the ith n-ary projection is the Boolean n-ary
function en
i satisfying en
i (x1, . . . , xn) ≈ xi . Notice that en
i just replicates its ith
argument and ignores all other arguments. The composition of an m-ary Boolean
function f with the n-ary Boolean functions g1, . . . , gm is the n-ary Boolean func-
tion h, defined by setting h(x1, . . . , xn) = f (g1(x1, . . . , xn), . . . , gm(x1, . . . , xn))
for all n-tuples (x1, . . . , xn) in Bn
.
Now, a set of Boolean functions is a clone if it is composition-closed and
contains all projections. Thus, a clone is rich enough in the sense that we cannot
exit from it via any composition of its members (each possibly used several times).
The concept is similar to the concept of a transformation monoid (whereby the set
of all projections plays a role analogous to that of the identity selfmap in monoids).
It is not difficult to check that the intersection of an arbitrary set of clones is
again a clone. Thus, for every set F of Boolean functions, there exists a unique
least clone containing F. This clone, denoted by F, is the clone generated by F.
The clone F can be alternatively interpreted in terms of combinatorial switch-
ing circuits. Suppose that for each n-ary f ∈ F there is a gate (switching device,
typically a transistor) with n inputs and a single output realizing f (a1, . . . , an) on
the output whenever, for i = 1, . . . , n, the zero-one input ai is applied to the ith
input. An F-based combinatorial circuit is obtained from gates realizing functions
from F by attaching the output of each gate (with a single exception) to an input of
another gate so that (1) the circuit has a single external output; (2) to each input of](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-32-320.jpg)
![1 Compositions and Clones of Boolean Functions 9
Most clones and all maximal clones may be described by relations in the
following way. Recall that for a positive integer h, an h-ary relation on B is a
subset ρ of Bh
(i.e., a set of zero-one h-tuples; some logicians prefer to view it as
a map ϕ from Bh
into {+, −}, whereby ρ = ϕ−1
(+), or to call it a predicate). For
h = 1, the relation is called unary and is just a subset of B, whereas for h = 2, the
relation is binary.
An n-ary Boolean function f preserves an h-ary relation ρ if for every h × n
matrix A whose column vectors all belong to ρ, the h-tuple of the values of f on
the rows of A belongs to ρ. In symbols, if A = [ai j ] with (a1 j , . . . , ahj ) ∈ ρ for
all j = 1, . . . , n, then
( f (a11, . . . , a1n), . . . , f (ah1, . . . , ahn)) ∈ ρ.
Notice the “rows versus columns” type of the definition. Several examples are given
next. In the universal algebra terminology, f preserves ρ if ρ is a subuniverse of
B; f h
. The first impression may be that the definition is too artificial or covers
only special cases. It turns out that it is the right one not only for clones of
Boolean functions but also for clones on any finite universe. It seems that it was
first explicitly formulated in [15], and it has been reinvented under various names.
We illustrate this crucial concept using a few examples needed for the promised
completeness criterion.
Example 1.1. Let h = 1 and ρ = {0}. Then A = [0 · · · 0] (a 1 × n matrix), and
f preserves {0} if and only if f (0, . . . , 0) = 0 (in universal algebra terminology,
if and only if {0} is a subuniverse of B; f ).
Example 1.2. Similarly, a Boolean function f preserves the unary relation {1} if
and only if f (1, . . . , 1) = 1.
Example 1.3. Consider the binary relation σ := {(0, 1), (1, 0)} on B. Note that
σ is the diagram (or graph) π0
of the permutation π : x → x
. It can also be
viewed as a graph on B with the single edge {0, 1}. A 2 × n matrix A = [ai j ] has
all columns in σ if and only if a2 j = a
1 j holds for all j = 1, . . . , n. Writing aj for
a1 j , we obtain that f preserves σ if and only if
( f (a1, . . . , an), f (a
1, . . . , a
n)) ∈ σ
holds for all a1, . . . , an ∈ B. This is equivalent to the identity
f (x
1, . . . , x
n) ≈ f (x1, . . . , xn)
.
The standard name for a Boolean function satisfying this identity is selfdual
(see [7] and Section 1.3). Expressed algebraically, f is selfdual if and only if
the negation
is an automorphism of the algebra B; f . A selfdual f ∈ O(n)
is fully determined by its values f (0, a2, . . . , an) with a2, . . . , an ∈ B (because
f (1, b
2, . . . , b
n) = f
(0, b
2, . . . , b
n)). Thus, there are exactly 22n−1
selfdual n-ary
Boolean functions, and the probability that a randomly chosen f ∈ O(n)
is selfdual
is the low value 2−2n−1
: for example, it is approximately 0.0000152 for n = 5.](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-34-320.jpg)
![10 Reinhard Pöschel and Ivo Rosenberg
Example 1.4. Next consider ρ = {(0, 0), (0, 1), (1, 1)}. Notice that (x, y) ∈ ρ if
and only if x ≤ y (where ≤ is the natural order on B), and so we write x ≤ y
instead of (x, y) ∈ ρ. A 2 × n matrix A = [ai j ] has all columns in ≤ whenever
a1 j ≤ a2 j holds for all j = 1, . . . , n, and therefore f preserves ≤ if and only if
a1 ≤ b1, . . . , an ≤ bn ⇒ f (a1, . . . , an) ≤ f (b1, . . . , bn);
that is, if every argument is kept the same or increased, then the value is the
same or increases. This is the standard definition of a monotone (also isotone,
order-respecting, or order-compatible) Boolean function; see [7].
Example 1.5. As a final example, consider the 4-ary (or quaternary) relation
λ = {(x1, x2, x3, x1+̇x2+̇x3) : x1, x2, x3 ∈ B},
where +̇ is the sum mod 2 introduced in Section 1.1. Expressed differently, the last
coordinate in a 4-tuple from λ is exactly the parity check (making the coordinate
sum even), a basic error check used in computer hardware and other digital
devices; equivalently, a 4-tuple belongs to λ if and only if it contains an even
number of 1s. The description of the functions preserving λ is not as transparent
as in the preceding examples. In order to establish it, let us say that a Boolean
function f ∈ O(n)
is linear (or affine) if there are c ∈ B and 1 ≤ i1 . . . ik ≤ n
such that f (x1, . . . , xn) ≈ c+̇xi1
+̇ · · · +̇xik
.
Fact 2. A Boolean function preserves λ if and only if it is linear.
Proof. (⇒) Let f ∈ O(n)
preserve λ. Then
f (x1, . . . , xn) ≈ f (x1, 0, . . . , 0)+̇ f (0, x2, . . . , xn)+̇ f (0, . . . , 0). (1.2)
Here f (0, x2, . . . , xn) ∈ O(n−1)
also preserves λ, and so we can apply (1.2) to it.
Continuing in this fashion, we obtain
f (x1, . . . , xn) ≈
r
n
i=1
f (0, . . . , 0, xi , 0, . . . , 0)+̇d (1.3)
where d = f (0, . . . , 0)+̇ . . . +̇ f (0, . . . , 0) (n times). Each unary Boolean func-
tion f (0, . . . , 0, xi , 0, . . . , 0) is of the form ai x+̇bi for some ai , bi ∈ B, and thus
the right-hand side of (1.3) is a1x1+̇ · · · +̇an xn+̇c where c = b1+̇ · · · +̇bn+̇d. This
proves that f is linear.
(⇐) It can be easily verified that every linear function preserves λ.
The set of Boolean functions preserving a given h-ary relation ρ on B is denoted
by Pol ρ. It is easy to verify that Pol ρ is a clone by showing that it contains all
projections and that it is composition closed.
Now we are ready for the promised completeness criterion due to Post [20].](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-35-320.jpg)
![1 Compositions and Clones of Boolean Functions 11
Theorem 1.2. (Completeness Criterion.)
(1) A set F of Boolean functions is complete if and only if F is contained in
none of the clones
Pol{0}, Pol{1}, Polσ, Pol ≤, Polλ. (1.4)
(2) Each clone distinct from O extends to a maximal one, and the foregoing
five clones are exactly all maximal clones.
Remark 1.7. Part (2) is just a rephrasing of part (1). The criterion is often given
in the following equivalent form: a set F of Boolean functions is complete if and
only if there exist f1, . . . , f5 ∈ F such that
f1(0, . . . , 0) = 1, f2(1, . . . , 1) = 0,
f3 is not selfdual, f4 is not monotone, and f5 is not linear (here f1, . . . , f5 need
not be pairwise distinct; e.g., they are all equal for F = { f }, where f is Sheffer).
The proof given next follows A. V. Kuznetsov’s proof (see [10], pp. 18–20).
The relatively direct proof is of some interest due to a Slupecki-type criterion
(Lemma 1.4 later), but it does not reveal how the completeness criterion was
discovered.
Proof. The necessity is obvious, as all five clones listed are distinct from O. As
for sufficiency, in the following lemma we start by characterizing clones included
in neither of the first two clones in (1.4) and end up with the same for all clones
except the last. Remarkably, all this is done through the unary operations of the
clone. Recall that O(n)
denotes the set of n-ary Boolean functions.
Lemma 1.3. If C is a clone and f ∈ O(n)
, then:
(i) C ⊆ Pol{0} if and only if C contains the unary constant c1 or C contains
the negation.
(ii) C ⊆ Pol{1} if and only if C contains the unary constant c0 or C contains
the negation.
(iii) f ∈ Pol ≤ if and only if
f (a1, . . . , ai−1, 0, ai+1, . . . , an) f (a1, . . . , ai−1, 1, ai+1, . . . , an)
(1.5)
for some 1 ≤ i ≤ n and some a1, . . . , an ∈ B.
(iv) C ⊆ Pol{0}, C ⊆ Pol{1}, and C ⊆ Pol ≤ if and only if C contains the
negation.
(v) C is a subclone of none of Pol{0}, Pol{1}, Pol ≤, and Polσ if and only if
C ⊇ O(1)
, that is, C contains all four unary Boolean functions.](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-36-320.jpg)
![16 Reinhard Pöschel and Ivo Rosenberg
number increases rather rapidly with n. Thus we should rather ask about the
proportion τn of n-ary Sheffer functions among all n-ary Boolean functions. From
Corollary 1.6(iii), we get
τn = |Sh(n)
|/|O(n)
| = 2−2
− 2−2n−1
−1
,
and so limn→∞τn = 1/4. The numbers τn grow very fast to 1/4, for example, τ6
already shares the first 10 decimal places with 0.25. Thus, the probability that, for
n big enough, a randomly chosen n-ary Boolean function is Sheffer is practically
0.25.
In switching circuits, the constant unary functions c0 and c1 are usually available
(as constant signals) or can be realized very cheaply. This leads to the following
definition.
Definition 1.1. A set F of Boolean functions is complete with constants (or
functionally complete) if F ∪ {c0, c1} is complete. Similarly, a Boolean function f
is Sheffer with constants if { f, c0, c1} is complete.
The two constant functions take care of the first three maximal clones in Post’s
completeness criterion (Theorem 1.2), and so we have the following corollary.
Corollary 1.7. A set of Boolean functions is complete with constants if and only
if it contains a nonmonotone function and a nonlinear function. In particular, a
Boolean function f is Sheffer with constants if and only if f is neither monotone
nor linear.
Denote by γn the number of n-ary Boolean functions that are Sheffer with
constants. Further, let ϕn denote the number of n-ary monotone Boolean functions.
The number ϕn, called the Dedekind number, is of interest on its own. The relation
between γn and ϕn is given in [8].
Corollary 1.8.
(1) γn = 22n
− 2n+1
− ϕn + n + 2.
(2) limn→∞ γn/|O(n)
| = 1.
Proof. We start with two claims:
Claims.
(i) en
1, . . . , en
n, cn
0, cn
1 are the only n-ary monotone and linear Boolean func-
tions.
(ii) There are 2n+1
linear n-ary functions.
To prove the claims, consider a linear and monotone function f ∈ O(n)
. Then
f (x1, . . . , xn) = b+̇xi1
+̇ · · · +̇xik
for some b ∈ B, k ≥ 0 and 1 ≤ i1 · · · ik ≤
n. If k = 0 then clearly f = cn
0 or f = cn
1. Thus let k ≥ 1. For notational simplicity,
suppose that i j = j for j = 1, . . . , k. First, b = 0, because b = 1 leads to
1 = f (0, . . . , 0) f (1, 0, . . . , 0) = 0,](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-41-320.jpg)
![1 Compositions and Clones of Boolean Functions 17
which contradicts monotonicity. Finally, k = 1, because for k ≥ 2 we get a con-
tradiction from
1 = f (1, 0, . . . , 0) f (1, 1, 0, . . . , 0) = 0.
Now, for k = 1, clearly f = en
i1
. It is immediate that en
1, . . . , en
n, cn
0, cn
1 are both
monotone and linear, proving (i).
The number of linear n-ary Boolean functions follows from their general form
and from the unicity of their representation (Proposition 1.1). This proves the
claims.
Now (1) follows from Corollary 1.7 and the claims (by inclusion-exclusion),
and (2) is a consequence of (1) and an asymptotic for ϕn from [13].
Remark 1.12. The last argument slightly cuts corners, because the asymptotic
given in [13] is not in a form showing immediately that ϕn2−2n
goes to 0. Because
(2) is not our main objective, we are not providing a detailed proof. The known
values of ϕn for n = 1, . . . , 8 are 3, 6, 20, 168, 7581, 7828354, 2414682040998,
56130437228687557907788 [4, 27]. The first three values can be verified directly.
The differences 1 − γn2−2n
n = 1, . . . , 7 are approximately
1, 0.625, 0.121, 0.0029, 1.7 × 10−6
, 4.243 × 10−3
, 7 × 10−27
.
This indicates that the proportion of n-ary functions that are Sheffer with constants
indeed goes very fast to 1. This may be interpreted as follows. For a large n, an
n-ary Boolean function picked at random is almost surely Sheffer with constants.
Consider now a set F of Boolean functions, f ∈ F, and a Boolean function g.
Suppose
f ∈ M ⇔ g ∈ M
holds for M running through the five maximal clones. Then clearly F is complete
if and only if (F { f }) ∪ {g} is complete. This leads to the following more formal
definition.
Definition 1.2. The characteristic set of a Boolean function f is the subset f ∗
of
{1, . . . , 5} such that
1 ∈ f ∗
⇔ f /
∈ Pol{0},
2 ∈ f ∗
⇔ f /
∈ Pol{1},
3 ∈ f ∗
⇔ f /
∈ Polσ,
4 ∈ f ∗
⇔ f /
∈ Pol ≤,
5 ∈ f ∗
⇔ f /
∈ Polλ.
For example, f ∗
= {1, 3, 5} exactly if f (0, . . . , 0) = f (1, . . . , 1) = 1, f is not
selfdual and f is monotone but not linear. Boolean functions f and g are equivalent
– in symbols, f ∼ g – if f ∗
= g∗
: in other words, if for each maximal clone M](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-42-320.jpg)
![1 Compositions and Clones of Boolean Functions 21
Remark 1.15. The sizes of certain
(n)
i (= the n-ary functions from i ) were given
in [14]:
|
(n)
2 | = 2n−1
− n,
|
(n)
12 | = |
(n)
20 | = 1,
|
(n)
14 | = |
(n)
22 | = 2n−1
− 1,
|
(n)
15 | = |
(n)
23 | = 22n−2
− 2n−1
,
|
(n)
26 | = 2n−1
,
|
(n)
27 | = 22n−1
− 2n−1
.
Because 0 is the clone of all projections, we have |
(n)
0 | = n. Next, |
(n)
31 | was
given in Corollary 1.6 (iii). Finally,
|
(n)
1 | + |
(n)
3 | = 22n−1
−1
− 2n−1
,
|
(n)
5 | + |
(n)
7 | = 22n
−2
− 22n−1
−1
− 2n−1
.
Definition 1.3. A complete set F of Boolean functions is a basis if no proper
subset of F is complete, that is, if F is irredundant (or irreducible) with respect
to completeness.
The characteristic sets (see Definition 1.2) provide a tool for the description of
bases. According to the Completeness Criterion, a set F of Boolean functions is
a basis exactly if F∗
:= { f ∗
∈ F} is an irredundant cover of {1, . . . , 5} (i.e., F∗
is a cover of {1, . . . , 5} such that each proper subfamily of F∗
misses at least one
element of {1, . . . , 5}). Each Boolean function belongs to exactly one of the 15
sets listed in Proposition 1.10. The corresponding sets are ∅ and
{5}, {4}, {4, 5}, {3, 5}, {3, 4, 5}, {2, 3}, {2, 3, 4}, {2, 3, 4, 5},
{1, 3}, {1, 3, 4}, {1, 3, 4, 5}, {1, 2, 4}, {1, 2, 4, 5}, {1, 2, 3, 4, 5} (1.8)
(for example, ∗
14 = {2, 3, 4} because 14 = 23
+ 22
+ 21
). The determination of
all possible types of bases amounts to the problem of finding all irredundant covers
of {1, . . . , 5} formed from the 14 sets listed in (1.8). This is a purely technical
problem that can be solved by a simple computer program. It is so small that it
was solved by hand [9, 12, 14]. There are exactly 1, 17, 22 and 2 irredundant
covers consisting of 1, 2, 3 and 4 sets, respectively. For example, the cover by a
single set is {1, . . . , 5}, which evidently corresponds to the set 31 of all Sheffer
functions. An example of an irredundant 2-set cover is {1, 2, 4, 5}, {2, 3, 4, 5}.
A corresponding basis is { f, g} with f ∈ 27 and g ∈ 15, that is, f is selfdual
but belongs to no other maximal clone and g preserves 0 but belongs to no other
maximal clone.
The total number of bases consisting of two n-ary functions is given in [14].
From this it follows that the proportion of such bases among all pairs of n-ary](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-46-320.jpg)
![22 Reinhard Pöschel and Ivo Rosenberg
functions goes fast to 1
8
as n → ∞. Together with Corollary 1.6(iii), this shows that
a randomly chosen pair of n-ary Boolean functions is complete with probability
practically equal to 0.375.
Two bases F and G are of the same type if F = { f1, . . . , fn} and G =
{g1, . . . , gn} where f1 ∼ g1, . . . , fn ∼ gn. The result quoted earlier states: There
are exactly 42 types of bases, and each basis consists of at most four functions. The
latter statement ([10] p. 20) follows from the fact that f (0, . . . , 0) = 1 for some
f in the basis, and so either f (1, . . . , 1) = 0 or f is not selfdual. A four-element
basis is {c0, c1, ·, s3} where s3(x1, x2, x3) ≈ x1+̇x2+̇x3.
1.3 A Description of the Post Lattice
1.3.1 Definition of the Post Lattice
In the previous section, we saw the role of the five maximal clones of Boolean
functions and their intersections. It is natural to ask about the other clones of
Boolean functions and their inclusions; this leads to the following problems.
Denote by C the set of clones. One of the first questions about C may be its size.
As clones are subsets of the countably infinite set O (of all Boolean functions),
a priori |C| could be any cardinal less than or equal to 2ℵ0
, the cardinality of the
set of reals. Twenty years after his completeness paper, E. L. Post showed in [21]
that |C| = ℵ0
. (It turned out later that this is exceptional because almost all other
infinite variants are of continuum cardinality; a fact that, e.g., distinguishes the
classical two-valued logic from many-valued logics.)
Consider a nonvoid subset {Ci : i ∈ I} of C. It is easy to see that
C =
i∈I
Ci
is the greatest clone contained in every Ci . The clone C is called the meet (or
infimum) of {Ci : i ∈ I}.
Consider now any subset F of O and set
C(F) = {X ∈ C : X ⊇ F}.
Clearly C(F), the family of all clones containing F, is nonvoid because it contains
the greatest clone O. Thus
X∈C(F) X is a clone, denoted by F, and called the
clone generated by F. Clearly F contains F; hence F is a member of C(F),
and thus F is the least clone containing F. (The argument we just presented
is the standard one for the existence of the vector subspace spanned by a set of
vectors, the subgroup generated by a subset of a group, etc.) It follows that
i∈I
Ci
is the least clone containing all clones Ci (i ∈ I). It is called the join (or supremum)
of {Ci : i ∈ I}.](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-47-320.jpg)
![1 Compositions and Clones of Boolean Functions 23
Because of the existence of meets and joins, the ordered set L = (C, ⊆), where
⊆ is the set inclusion (or containment), is a so-called complete lattice. In his
landmark 100-page paper [21], Post gave a full description of the lattice L (today
called the Post lattice). Post actually characterized all composition-closed sets of
Boolean functions: however, there are only seven such sets that are not clones –
that is, do not contain all projections – and these very small sets can be easily
described.
1.3.2 Duality
We start with a helpful symmetry of the lattice L.
The dual of an n-ary Boolean function f is the Boolean function f ∂
defined
by setting
f ∂
(x1, . . . , xn) ≈ ( f (x
1, . . . , x
n))
(where x
is the negation). It can be checked that f ∂∂
= f . The duals of ∨ and
+̇ are · and ↔, respectively. Also, f is selfdual (see Example 1.3) if and only if
f = f ∂
. To every F ⊆ O, assign F∂
:= { f ∂
: f ∈ F}. If F is a clone, then F∂
is
also a clone and F → F∂
is a lattice automorphism of L, that is, a bijection (or
1-1 and onto selfmap) of C onto itself respecting the lattice joins and meets (to see
it, check that f → f ∂
respects the composition).
A finite ordered set is usually represented by its (Hasse) diagram. In such a
drawing, vertices correspond to the elements of the set and two vertices are joined
by a line segment exactly if the element corresponding to the vertex drawn higher
on the page covers the element corresponding to the vertex drawn lower on the
page. Here a covers b means that a b but a c b holds for no c. In certain
cases we also can draw diagrams of infinite orders.
In view of the automorphism F → F∂
of L, we can draw a diagram of L so
that it is symmetric with respect to the central vertical line: in other words, so
that the left-hand half is the mirror image of the right-hand half; that is, F → F∂
acts horizontally. This symmetry is tied to the relational description of clones as
follows.
Fact 3. If ρ is an h-ary relation on B and
ρ
:= {(a
1, . . . , a
n) : (a1, . . . , an) ∈ ρ},
then (Polρ)∂
= Polρ
.
For example, (Pol{0})∂
= Pol{0
} = Pol{1}, (Pol{1})∂
= Pol{0}, while F∂
= F
for the remaining three maximal clones because ≤∂
, σ∂
and λ∂
equal ≥, σ and λ,
respectively.
Proof. First we show
(Polρ)∂
⊆ Polρ
. (1.9)](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-48-320.jpg)
![1 Compositions and Clones of Boolean Functions 25
example, ∨ is the clone generated by the disjunction; and (3) special notations:
T0 = Pol{0}, T1 = Pol{1}, S = Polσ, M = Pol ≤ and L = Polλ for the maximal
clones and I for T0 ∩ T1. We abbreviate S ∩ M by SM (i.e., SM is the clone
of all functions that are both selfdual and monotone), and similarly for other
intersections of clones.
In order to find all clones, we start with the five maximal clones and their inter-
sections. In principle Proposition 1.10 (see Figure 1.2) may be used to determine
all the intersections of maximal clones. Nevertheless, we provide more details now
and draw the diagrams of certain easily describable intervals.
We start with a trivial case. Recall that a Boolean function f is idempo-
tent if f (x, . . . , x) ≈ x. Clearly I := T0T1(= T0 ∩ T1) is the clone of idempotent
functions.
1.3.4 The Clone SM
The clone SM is the intersection of the maximal clones S = Polσ and M = Pol ≤.
This clone consists of selfdual monotone functions. A quick check shows that
among the unary and binary functions, only projections are selfdual and monotone,
and this may lead to the impression that perhaps SM is the least clone J of all
projections. But SM contains a nontrivial ternary function
m(x1, x2, x3) :≈ x1x2 ∨ x1x3 ∨ x2x3,
called the majority or median function. It is clearly monotone. According to the
De Morgan laws, its dual m∂
satisfies
m∂
(x1, x2, x3) ≈ (x
1x
2 ∨ x
1x
3 ∨ x
2x
3)
≈ (x
1 ∨ x
2 )(x
1 ∨ x
3 )(x
2 ∨ x
3 )
≈ (x1 ∨ x2)(x1 ∨ x3)(x2 ∨ x3).
Applying the distributive law, it is easy to verify that
(x1 ∨ x2)(x1 ∨ x3)(x2 ∨ x3) ≈ x1x2 ∨ x1x3 ∨ x2x3,
proving m∂
= m and m ∈ S. (Actually, the last identity is important because it
characterizes the distributive lattices among all lattices.) An easy check shows
that m and the projections are the only ternary functions in SM. However, SM
contains many interesting functions of higher arity. (Median algebras, see, e.g.,
[1], generalize the majority function and some other functions from SM and seem
to have some applications in social sciences.) It is immediate that every function
from SM is idempotent, and so SM is contained in all maximal clones except the
clone L of linear functions. Moreover, SM is an atom of the Post lattice. Here an
atom of L, called a minimal clone, is a clone properly containing exactly the clone
J. For A, B ∈ C, A ⊆ B, the sets
[A) = {X ∈ C : X ⊇ A}, [A, B] = {X ∈ C : A ⊆ X ⊆ B}](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-50-320.jpg)
![1 Compositions and Clones of Boolean Functions 27
Figure 1.5. The filter [I L).
Indeed, for k 1,
x1+̇ · · · +̇x2k+1 ≈ (· · · ((x1+̇x2+̇x3)+̇x4+̇x5)+̇ · · · )+̇x2k+̇x2k+1.
Conversely,
x1+̇x2+̇x3 ≈ x1+̇x2+̇x3+̇x3+̇ · · · +̇x3.
This shows that I L is a minimal clone. The interval [I L) is shown in Figure 1.5.
1.3.6 The Clone ML
The last nontrivial intersection of two maximal clones is the clone ML := Pol ≤
∩Polλ of the monotone linear functions. We have already seen in Lemma 1.9(6)
that ML = c0, c1 is made up from the constants and the projections. Now ML is
contained in neither of T0, T1 and S. Between ML and M there are exactly the two
mutually dual clones ∨, c0, c1 and ·, c0, c1, and between ML and L there is
precisely the clone
, c0. However, ML is not minimal; in fact, ML is obviously
the join of the two minimal clones c0 and c1, and ML covers no other clone.
The interval [J, ML] and the filter [ML) are shown in Figure 1.6.
1.3.7 The Seven Minimal Clones
In Sections 1.3.4 through 1.3.6, we have seen all nontrivial intersections of maximal
clones (the intersection of all five maximal clones is obviously the trivial clone J](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-52-320.jpg)
![28 Reinhard Pöschel and Ivo Rosenberg
Figure 1.6. The interval [J, ML] and the filter [ML).
of all projections). Two of them, namely SM and I L, are even minimal clones,
and the third, namely ML, covers two minimal clones c0 and c1. At this point
the reader may be wondering about the full list of all minimal clones. Obviously
there is also the minimal clone
generated by the negation, but the existence
of other minimal clones is not obvious. In fact, there are only two more minimal
clones, namely ∨ and ·.
Proposition 1.11. There are exactly seven minimal clones:
c0, c1,
, ∨, ·, SM, I L
(where SM is generated by the majority function x1x2 ∨ x1x3 ∨ x2x3 and I L is
generated by x1+̇x2+̇x3).
The minimality of the clones ∨ and · is easily checked. We omit the more
complex proof that there are no other minimal clones. The various joins of the
seven minimal clones can be verified directly. Their diagram is drawn in Figure 1.7.](https://image.slidesharecdn.com/2345724-250526062358-4fd84fcc/85/Boolean-Models-And-Methods-In-Mathematics-Computer-Science-And-Engineering-1st-Edition-Yves-Crama-53-320.jpg)
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Boolean Models And Methods In Mathematics Computer Science And Engineering 1st Edition Yves Crama
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- 6. Boolean Models and Methods in Mathematics, Computer Science, and Engineering This collection of papers presents a series of in-depth examinations of a variety of ad- vanced topics related to Boolean functions and expressions. The chapters are written by some of the most prominent experts in their respective fields and cover topics ranging from algebra and propositional logic to learning theory, cryptography, computational complexity, electrical engineering, and reliability theory. Beyond the diversity of the questions raised and investigated in different chapters, a remarkable feature of the col- lection is the common thread created by the fundamental language, concepts, models, and tools provided by Boolean theory. Many readers will be surprised to discover the countless links between seemingly remote topics discussed in various chapters of the book. This text will help them draw on such connections to further their understanding of their own scientific discipline and to explore new avenues for research. Dr. Yves Crama is Professor of Operations Research and Production Management and former Dean of the HEC Management School of the University of Liège, Belgium. He is widely recognized as a prominent expert in the field of Boolean functions, combinatorial optimization, and operations research, and he has coauthored more than 70 papers and 3 books on these subjects. Dr. Crama is a member of the editorial board of Discrete Optimization, Journal of Scheduling, and 4OR – The Quarterly Journal of the Belgian, French and Italian Operations Research Societies. The late Peter L. Hammer (1936–2006) was a Professor of Operations Research, Mathematics, Computer Science, Management Science, and Information Systems at Rutgers University and the Director of the Rutgers University Center for Operations Research (RUTCOR). He was the founder and editor-in-chief of the journals Annals of Operations Research, Discrete Mathematics, Discrete Applied Mathematics, Discrete Optimization, and Electronic Notes in Discrete Mathematics. Dr. Hammer was the initiator of numerous pioneering investigations of the use of Boolean functions in operations research and related areas, of the theory of pseudo-Boolean functions, and of the logical analysis of data. He published more than 240 papers and 19 books on these topics.
- 7. encyclopedia of mathematics and its applications founding editor g.-c. rota Editorial Board R. Doran, P. Flajolet, M. Ismail, T.-Y. Lam, E. Lutwak The titles below, and earlier volumes in the series, are available from booksellers or from Cambridge University Press at www.cambridge.org. 100 E. Olivieri and M. Eulália Vares Large Deviations and Metastability 101 A. Kushner, V . Lychagin and V . Rubtsov Contact Geometry and Nonlinear Differential Equations 102 L. W. Beineke and R. J. Wilson (eds.) with P. J. Cameron Topics in Algebraic Graph Theory 103 O. J. Staffans Well-Posed Linear Systems 104 J. M. Lewis, S. Lakshmivarahan and S. K. Dhall Dynamic Data Assimilation 105 M. Lothaire Applied Combinatorics on Words 106 A. Markoe Analytic Tomography 107 P. A. Martin Multiple Scattering 108 R. A. Brualdi Combinatorial Matrix Classes 109 J. M. Borwein and J. D. Vanderwerff Convex Functions 110 M.-J. Lai and L. L. Schumaker Spline Functions on Triangulations 111 R. T. Curtis Symmetric Generation of Groups 112 H. Salzmann et al. The Classical Fields 113 S. Peszat and J. Zabczyk Stochastic Partial Differential Equations with Lévy Noise 114 J. Beck Combinatorial Games 115 L. Barreira and Y. Pesin Nonuniform Hyperbolicity 116 D. Z. Arov and H. Dym J-Contractive Matrix Valued Functions and Related Topics 117 R. Glowinski, J.-L. Lions and J. He Exact and Approximate Controllability for Distributed Parameter Systems 118 A. A. Borovkov and K. A. Borovkov Asymptotic Analysis of Random Walks 119 M. Deza and M. Dutour Sikirić Geometry of Chemical Graphs 120 T. Nishiura Absolute Measurable Spaces 121 M. Prest Purity Spectra and Localisation 122 S. Khrushchev Orthogonal Polynomials and Continued Fractions 123 H. Nagamochi and T. Ibaraki Algorithmic Aspects of Graph Connectivity 124 F. W. King Hilbert Transforms I 125 F. W. King Hilbert Transforms II 126 O. Calin and D.-C. Chang Sub-Riemannian Geometry 127 M. Grabisch et al. Aggregation Functions 128 L. W. Beineke and R. J. Wilson (eds.) with J. L. Gross and T. W. Tucker Topics in Topological Graph Theory 129 J. Berstel, D. Perrin and C. Reutenauer Codes and Automata 130 T. G. Faticoni Modules over Endomorphism Rings 131 H. Morimoto Stochastic Control and Mathematical Modeling 132 G. Schmidt Relational Mathematics 133 P. Kornerup and D. W. Matula Finite Precision Numbers Systems and Arithmetic
- 8. encyclopedia of mathematics and its applications Boolean Models and Methods in Mathematics, Computer Science, and Engineering Edited by YVES CRAMA Université de Liège PETER L. HAMMER
- 9. cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521847520 c Yves Crama and Peter L. Hammer 2010 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2010 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Boolean models and methods in mathematics, computer science, and engineering / edited by Yves Crama, Peter L. Hammer. p. cm. – (Encyclopedia of mathematics and its applications ; 134) Includes bibliographical references and index. ISBN 978-0-521-84752-0 1. Algebra, Boolean. 2. Probabilities. I. Crama, Yves, 1958– II. Hammer, P. L., 1936– III. Title. IV . Series. QA10.3.B658 2010 511.324–dc22 2010017816 ISBN 978-0-521-84752-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.
- 10. Contents Preface page vii Introduction ix Acknowledgments xiii Contributors xv Acronyms and Abbreviations xvii Part I Algebraic Structures 1 Compositions and Clones of Boolean Functions 3 Reinhard Pöschel and Ivo Rosenberg 2 Decomposition of Boolean Functions 39 Jan C. Bioch Part II Logic 3 Proof Theory 79 Alasdair Urquhart 4 Probabilistic Analysis of Satisfiability Algorithms 99 John Franco 5 Optimization Methods in Logic 160 John Hooker Part III Learning Theory and Cryptography 6 Probabilistic Learning and Boolean Functions 197 Martin Anthony 7 Learning Boolean Functions with Queries 221 Robert H. Sloan, Balázs Szörényi, and György Turán v
- 11. vi Contents 8 Boolean Functions for Cryptography and Error-Correcting Codes 257 Claude Carlet 9 Vectorial Boolean Functions for Cryptography 398 Claude Carlet Part IV Graph Representations and Efficient Computation Models 10 Binary Decision Diagrams 473 Beate Bollig, Martin Sauerhoff, Detlef Sieling, and Ingo Wegener 11 Circuit Complexity 506 Matthias Krause and Ingo Wegener 12 Fourier Transforms and Threshold Circuit Complexity 531 Jehoshua Bruck 13 Neural Networks and Boolean Functions 554 Martin Anthony 14 Decision Lists and Related Classes of Boolean Functions 577 Martin Anthony Part V Applications in Engineering 15 Hardware Equivalence and Property Verification 599 J.-H. Roland Jiang and Tiziano Villa 16 Synthesis of Multilevel Boolean Networks 675 Tiziano Villa, Robert K. Brayton, and Alberto L. Sangiovanni-Vincentelli 17 Boolean Aspects of Network Reliability 723 Charles J. Colbourn
- 12. Preface Boolean models and methods play a fundamental role in the analysis of a broad diversity of situations encountered in various branches of science. The objective of this collection of papers is to highlight the role of Boolean theory in a number of such areas, ranging from algebra and propositional logic to learning theory, cryptography, computational complexity, electrical engineering, and reliability theory. The chapters are written by some of the most prominent experts in their fields and are intended for advanced undergraduate or graduate students, as well as for researchers or engineers. Each chapter provides an introduction to the main questions investigated in a particular field of science, as well as an in-depth discussion of selected issues and a survey of numerous important or representative results. As such, the collection can be used in a variety of ways: some readers may simply skim some of the chapters in order to get the flavor of unfamiliar areas, whereas others may rely on them as authoritative references or as extensive surveys of fundamental results. Beyond the diversity of the questions raised and investigated in different chap- ters, a remarkable feature of the collection is the presence of an “Ariane’s thread” created by the common language, concepts, models, and tools of Boolean theory. Many readers will certainly be surprised to discover countless links between seem- ingly remote topics discussed in various chapters of the book. It is hoped that they will be able to draw on such connections to further their understanding of their own scientific disciplines and to explore new avenues for research. The collection intends to be a useful companion and complement to the mono- graph by Yves Crama and Peter L. Hammer, Boolean Functions: Theory, Algo- rithms, and Applications. Cambridge University Press, Cambridge, U.K., 2010, which provides the basic concepts and theoretical background for much of the material handled here. vii
- 14. Introduction The first part of the book, “Algebraic Structures,” deals with compositions and decompositions of Boolean functions. A set F of Boolean functions is called complete if every Boolean function is a composition of functions from F; it is a clone if it is composition-closed and contains all projections. In 1921, E. L. Post found a completeness criterion, that is, a necessary and sufficient condition for a set F of Boolean functions to be complete. Twenty years later, he gave a full description of the lattice of Boolean clones. Chapter 1, by Reinhard Pöschel and Ivo Rosenberg, provides an accessible and self-contained discussion of “Compositions and Clones of Boolean Functions” and of the classical results of Post. Functional decomposition of Boolean functions was introduced in switching theory in the late 1950s. In Chapter 2, “Decomposition of Boolean Functions,” Jan C. Bioch proposes a unified treatment of this topic. The chapter contains both a presentation of the main structural properties of modular decompositions and a discussion of the algorithmic aspects of decomposition. Part II of the collection covers topics in logic, where Boolean models find their historical roots. In Chapter 3, “Proof Theory,” Alasdair Urquhart briefly describes the more important proof systems for propositional logic, including a discussion of equa- tional calculus, of axiomatic proof systems, and of sequent calculus and resolution proofs. The author compares the relative computational efficiency of these differ- ent systems and concludes with a presentation of Haken’s classical result that resolution proofs have exponential length for certain families of formulas. The issue of the complexity of proof systems is further investigated by John Franco in Chapter 4, “Probabilistic Analysis of Satisfiability Algorithms.” Central questions addressed in this chapter are: How efficient is a particular algorithm when applied to a random satisfiability instance? And what distinguishes “hard” from “easy” instances? Franco provides a thorough analysis of these questions, starting with a presentation of the basic probabilistic tools and models and covering advanced results based on a broad range of approaches. ix
- 15. x Introduction In Chapter 5, “Optimization Methods in Logic,” John Hooker shows how math- ematical programming methods can be applied to the solution of Boolean inference and satisfiability problems. This line of research relies on the interpretation of the logical symbols 0 and 1 as numbers, rather than meaningless symbols. It leads both to fruitful algorithmic approaches and to the identification of tractable classes of problems. The remainder of the book is devoted to applications of Boolean models in various fields of computer science and engineering, starting with “Learning Theory and Cryptography” in Part III. In Chapter 6, “Probabilistic Learning and Boolean Functions,” Martin Anthony explains how an unknown Boolean function can be “correctly approximated,” in a probabilistic sense, when the only available information is the value of the function on a random sample of points. Questions investigated here relate to the quality of the approximation that can be attained as a function of the sample size, and to the algorithmic complexity of computing the approximating function. A different learning model is presented by Robert H. Sloan, Balázs Szörényi, and Gyorgy Turán in Chapter 7, “Learning Boolean Functions with Queries.” Here, the objective is to identify the unknown function exactly by asking questions about it. The efficiency of learning algorithms, in this context, depends on prior information available about the properties of the target function, about the type of representation that should be computed, about the nature of the queries that can be formulated, and so forth. Also, the notion of “efficiency” can be measured either by the number of queries required by the learning algorithm (information complexity) or by the total of amount of computational steps required by the algorithm (computational complexity). The chapter provides an introduction and surveys a large variety of results along these lines. In Chapter 8, Claude Carlet provides a very complete overview of the use of “Boolean Functions for Cryptography and Error-Correcting Codes.” Both cryp- tography and coding theory are fundamentally concerned with the transformation of binary strings into binary strings. It is only natural, therefore, that Boolean func- tions constitute a basic tool and object of study in these fields. Carlet discusses quality criteria that must be satisfied by error-correcting codes and by crypto- graphic functions (high algebraic degree, nonlinearity, balancedness, resiliency, immunity, etc.) and explains how these criteria relate to characteristics of Boolean functions and of their representations. He introduces several remarkable classes of functions such as bent functions, resilient functions, algebraically immune func- tions, and symmetric functions, and he explores the properties of these classes of functions with respect to the aforementioned criteria. In Chapter 9, “Vectorial Boolean Functions for Cryptography,” Carlet extends the discussion to functions with multiple outputs. Many of the notions introduced in Chapter 8 can be naturally generalized in this extended framework: families of representations, quality criteria, and special classes of functions are introduced and analyzed in a similar fashion.
- 16. Introduction xi Part IV concentrates on “Graph Representations and Efficient Computation Models” for Boolean functions. Beate Bollig, Martin Sauerhoff, Detlef Sieling, and the late Ingo Wegener discuss “Binary Decision Diagrams” (BDDs) in Chapter 10. A BDD for function f is a directed acyclic graph representation of f that allows efficient computation of the value of f (x) at any point x. Different types of BDDs can be defined by placing restrictions on the underlying digraph, by allowing probabilistic choices, and so forth. Questions surveyed in Chapter 10 are, among others: What is the size of a smallest BDD representation of a given function? How can a BDD be efficiently generated? How difficult is it to solve certain problems on Boolean functions (satisfiability, minimization, etc.) when the input is represented as a BDD? Matthias Krause and Ingo Wegener discuss a different type of graph represen- tations in Chapter 11, “Circuit Complexity.” Boolean circuits provide a convenient model for the hardware realization of Boolean functions. Krause and Wegener describe efficient circuits for simple arithmetic operations, such as addition and multiplication. Further, they investigate the possibility of realizing arbitrary func- tions by circuits with small size or small depth. Although lower bounds or upper bounds on these complexity measures can be derived under various assumptions on the structure of the circuit or on the properties of the function to be represented, the authors also underline the existence of many fundamental open questions on this challenging topic. Fourier transforms are a powerful tool of classical analysis. More recently, they have also proved useful for the investigation of complex problems in discrete math- ematics. In Chapter 12, “Fourier Transforms and Threshold Circuit Complexity,” Jehoshua Bruck provides an introduction to the basic techniques of Fourier anal- ysis as they apply to the investigation of Boolean functions and neural networks. He explains, in particular, how they can be used to derive bounds on the size of the weights and on the depth of Boolean circuits consisting of threshold units. The topic of “Neural Networks and Boolean Functions” is taken up again by Martin Anthony in Chapter 13. The author focuses first on the number and on the properties of individual threshold units, which can be viewed as linear, as nonlinear, or as “delayed” (spiking) threshold Boolean functions. He next discusses the expressive power of feed-forward artificial neural networks made up of threshold units. Martin Anthony considers yet another class of graph representations in Chap- ter 14, “Decision Lists and Related Classes of Boolean Functions.” A decision list for function f can be seen as a sequence of Boolean tests, the outcome of which determines the value of the function on a given point x. Every Boolean function can be represented as a decision list. However, when the type or the number of tests involved in the list is restricted, interesting subclasses of Boolean functions arise. Anthony investigates several such restrictions. He also considers the algorithmic complexity of problems on decision lists (recognition, learning, equivalence),
- 17. xii Introduction and he discusses various connections between threshold functions and decision lists. The last part of the book focuses on “Applications in Engineering.” Since the 1950s, electrical engineering has provided a main impetus for the development of Boolean logic. In Chapter 15, J.-H. Roland Jiang and Tiziano Villa survey the use of Boolean methods for “Hardware Equivalence and Property Verification.” A main objective, in this area of system design, is to verify that a synthesized digital circuit conforms to its intended design. The chapter introduces the reader to the problem of formal verification, examines the complexity of different versions of equivalence checking (“given two Boolean circuits, decide whether they are equivalent”), and describes approaches to this problem. For the solution of these engineering problems, the authors frequently refer to models and methods covered in earlier chapters of the book, such as satisfiability problems or binary decision diagrams. In Chapter 16, Tiziano Villa, Robert K. Brayton, and Alberto L. Sangiovanni- Vincentelli discuss the “Synthesis of Multilevel Boolean Networks.” A multilevel representation of a Boolean function is a circuit representation, similar to those considered in Chapter 11 or in Chapter 13. From the engineering viewpoint, the objective of multilevel implementations is to minimize the physical area occupied by the circuit, to reduce its depth, to improve its testability, and so on. Villa, Brayton, and Sangiovanni-Vincentelli survey efficient heuristic approaches for the solution of these hard computational problems. They describe, in particular, fac- toring and division procedures that can be implemented in “divide-and-conquer” algorithms for multilevel synthesis. The combinatorial structure of operating or failed states of a complex system can be reflected through a Boolean function, called the structure function of the system. The probability that the system operates is then simply the probability that the structure function takes value 1. In Chapter 17, Charles J. Colbourn explores in great detail the “Boolean Aspects of Network Reliability.” He reviews several exact methods for reliability computations, based either on “orthogonalization” or decomposition, or on inclusion-exclusion and domination. He also explains the intimate, though insufficiently explored, connections between Boolean models and combinatorial simplicial complexes, as they arise in deriving bounds on system reliability.
- 18. Acknowledgments The making of this book has been a long process, and it has benefited over the years from the help and advice provided by several individuals. The editors gratefully acknowledge the contribution of these colleagues to the success of the endeavor. First and foremost, all chapter contributors are to be thanked for the quality of the material that they have delivered, as well as for their patience and understanding during the editorial process. Several authors have contributed to the reviewing process by cross-reading each other’s work. Additional reviews, suggestions, and comments on early versions of the chapters have been kindly provided by Endre Boros, Nadia Creignou, Tibor Hegedűs, Lisa Hellerstein, Toshi Ibaraki, Jörg Keller, Michel Minoux, Rolf Möhring, Vera Pless, Gabor Rudolf, Mike Saks, Winfrid Schneeweiss, and Ewald Speckenmeyer. Special thanks are due to Endre Boros, who provided constant encouragement and tireless advice to the editors over the gestation period of the volume. Marty Golumbic gave a decisive push to the process by bringing most contributors to- gether in Haifa, in January 2008, on the occasion of the first meeting on “Boolean Functions: Theory, Algorithms, and Applications.” Terry Hart provided the effi- cient administrative assistance that allowed the editors to keep track of countless mail exchanges. Finally, I must thank my mentor, colleague, and friend, Peter L. Hammer, for helping me launch this ambitious editorial project, many years ago. Unfortunately, Peter did not live to see the outcome of our joint efforts. I am sure that he would have loved it, and that he would have been very proud of this contribution to the dissemination of Boolean models and methods. Yves Crama Liège, Belgium, January 2010 xiii
- 20. Contributors Martin Anthony Department of Mathematics London School of Economics and Political Science, UK Jan C. Bioch Department of Econometrics Erasmus University Rotterdam, The Netherlands Beate Bollig Department of Computer Science Technische Universität Dortmund, Germany Robert K. Brayton Department of Electrical Engineering Computer Sciences University of California at Berkeley, USA Jehoshua Bruck Computation and Neural Systems and Electrical Engineering California Institute of Technology, USA Claude Carlet Department of Mathematics University of Paris 8, France Charles J. Colbourn Computer Science and Engineering Arizona State University, USA John Franco Department of Computer Science University of Cincinnati, USA John Hooker Tepper School of Business Carnegie Mellon University, USA J.-H. Roland Jiang Department of Electrical Engineering National Taiwan University, Taiwan Matthias Krause Theoretical Computer Science Mannheim University, Germany Reinhard Pöschel Institut für Algebra Technische Universität Dresden, Germany Ivo Rosenberg Département de Mathématiques et de Statistique Université de Montréal, Canada xv
- 21. xvi Contributors Alberto L. Sangiovanni-Vincentelli Department of Electrical Engineering Computer Sciences University of California at Berkeley, USA Martin Sauerhoff Department of Computer Science Technische Universität Dortmund, Germany Detlef Sieling Department of Computer Science Technische Universität Dortmund, Germany Robert H. Sloan Department of Computer Science University of Illinois at Chicago, USA Balázs Szörényi Hungarian Academy of Sciences University of Szeged, Hungary György Turán Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago, USA Alasdair Urquhart Department of Philosophy University of Toronto, Canada Tiziano Villa Dipartimento d’Informatica University of Verona, Italy Ingo Wegener† Department of Computer Science Technische Universität Dortmund, Germany †Professor Wegener passed away in November 2008.
- 22. Acronyms and Abbreviations AB almost bent AIG AND-Inverter graph ANF algebraic normal form APN almost perfect nonlinear ATPG Automatic Test Pattern Generation (p. 698) BDD binary decision diagram BED Boolean Expression Diagram BMC bounded model checking BP branching program C-1-D complete-1-distinguishability CDMA code division multiple access CEC combinational equivalence checking CNF conjunctive normal form CQ complete quadratic CTL computation tree logic DD decision diagram DNF disjunctive normal form DPLL Davis-Putnam-Logemann-Loveland EDA electronic design automation FBDD free binary decision diagram FCSR feedback with carry shift register FFT fast Fourier transform FRAIG Functionally Reduced AIG FSM finite-state machine FSR feedback shift register GPS generalized partial spread HDL hardware description language HFSM hardware finite-state machine HSTG hardware state transition graph IBQ incomplete boundary query xvii
- 23. xviii Acronyms and Abbreviations LFSR linear feedback shift register LP linear programming LTL linear temporal logic MTBDD multiterminal binary decision diagram NNF numerical normal form OBDD ordered binary decision diagram PAC probably approximately correct PBDD partitioned binary decision diagram PC propagation criterion QBF quantified Boolean formula ROBDD reduced ordered binary decision diagram RTL register-transfer level SAC strict avalanche criterion SAT satisfiability [not an acronym] SBS stochastic binary system SCC strongly connected component SEC sequential equivalence checking SEM sample error minimization SOP sum-of-product SQ statistical query STG state transition graph UBQ unreliable boundary query UMC unbounded model checking VC Vapnik-Chervonenkis XBDD extended binary decision diagram
- 24. Boolean Models and Methods in Mathematics, Computer Science, and Engineering
- 28. 1 Compositions and Clones of Boolean Functions Reinhard Pöschel and Ivo Rosenberg 1.1 Boolean Polynomials The representations of Boolean functions are frequently based on the fundamental operations {∨, ∧, }, where the disjunction x ∨ y represents the logical OR, the conjunction x ∧ y represents the logical AND and is often denoted by x · y or simply by the juxtaposition xy, and x stands for the negation, or complement, of x and is often denoted by x. This system naturally appeals to logicians and, for some reasons, also to electrical engineers, as illustrated by many chapters of this volume and by the monograph [7]. Its popularity may be explained by the validity of many identities or laws: for example, the associativity, commutativity, idempotence, distributive, and De Morgan laws making B := B; ∨, ∧, , 0, 1 a Boolean algebra, where B = {0, 1}; in fact, B is the least nontrivial Boolean algebra. It is natural to ask whether there is a system of basic Boolean functions other than {∨, ∧, }, but equally powerful in the sense that each Boolean function may be represented over this system. To get such a system, we introduce the following binary (i.e., two-variable) Boolean function +̇ defined by setting x+̇y = 0 if x = y and x+̇y = 1 if x = y; its truth table is x y x+̇y 0 0 0 0 1 1 1 0 1 1 1 0 Clearly x+̇y = 1 if and only if the arithmetical sum x + y is odd, and for this reason +̇ is also referred to as the sum mod 2. It corresponds to the “exclusive or” of logic, whereby the “exclusive or” of two statements P and Q is true if “either P or Q” is true. Notice that some natural languages, such as, French, distinguish “or” from “exclusive or” (“ou” et “soit”), whereas most natural languages are less 3
- 29. 4 Reinhard Pöschel and Ivo Rosenberg precise. The function +̇ is also denoted ⊕ or + and, in more recent engineering literature, by EXOR. Consider the system {+̇, ·, 0, 1}, where 0 and 1 are constants. A reader familiar with groups will notice that B; +̇ is an abelian group with neutral element 0 satisfying x+̇x ≈ 0 (also called an elementary 2-group), where ≈ stands for an identity on B. Moreover, GF(2) := B; +̇, ·, 0, 1 is a field, called a Galois field and denoted Z2 or F2. Thus, in GF(2) we may use all the arithmetic properties valid in familiar fields (such as the fields Q, R, and C of all rational, real, and complex numbers) but not their order or topological properties. In addition, GF(2) also satisfies x+̇x ≈ 0 and x2 ≈ x. Clearly GF(2) is the field of the least pos- sible size, and so one may be inclined to dismiss it as a trivial and unimportant object. Surprisingly, it has serious applications. A practical one is in cryptogra- phy and coding theory (for secret or secure data transmission, for example, for governments, banks, or from satellites; see Chapters 8 and 9 in this volume). Denote x1+̇ · · · +̇xn by r n i=1xi . Let f be an n-ary Boolean function distinct from the constant cn 0 (which is the n-variable Boolean function with constant value 0). In its complete disjunctive normal form (DNF), replace the disjunction ∨ (of complete elementary conjunctions) by their sum mod 2 r . This is still a representation of f because for every (a1, . . . , an) ∈ Bn , at most one of the elementary conjunctions takes value 1 and 1+̇0+̇ · · · +̇0 = 1 and 0+̇0+̇ · · · +̇0 = 0. Using x0 ≈ x ≈ 1+̇x and x1 ≈ x throughout, we obtain a representation of f over {+̇, ·, 0, 1}. The following proposition makes this more precise. Here the symbol denotes the usual arithmetical product. We make the usual convention that in expressions involving +̇, · and 1 products are calculated before sums, for example, xy+̇z stands for (xy)+̇z, and that and r over the empty set are 1 and 0, respectively. Proposition 1.1. [28] For every n-ary Boolean function f , there exists a unique family F of subsets of N = {1, . . . , n} such that f (x1, . . . , xn) ≈ r I∈F i∈I xi . (1.1) For example, x1 ∧ x2 ≈ x1+̇x2+̇x1x2 with F = {{1}, {2}, {1, 2}} (direct verifi- cation). Call the right-hand side of (1.1) a Boolean polynomial. Proof. Let f be an n-ary Boolean function. If f = cn 0, take F = ∅. Thus, let f = cn 0. In the discussion leading to the proposition, we saw that f may be represented over {+̇, ·, 0, 1}. Multiplying out the parentheses, we obtain a representation of f as a polynomial in variables x1, . . . , xn over GF(2). In view of x2 ≈ x, it may be reduced to a sum of square-free monomials. From x+̇x ≈ 0, it follows that it may be further reduced to such a sum in which every monomial appears at most once, proving the representability of f by a Boolean polynomial. It remains to prove the uniqueness. For every F ⊆ P(N), that is, a family of subsets of N, denote
- 30. 1 Compositions and Clones of Boolean Functions 5 by ϕ(F) the corresponding Boolean polynomial. Clearly ϕ is a map from the set P(P(N)) of families of subsets of N into the set O(n) of n-ary Boolean functions. Claim. The map ϕ is injective. Indeed, by the way of contradiction, suppose ϕ(F) = ϕ(G) for some F, G ⊆ P(N) with F = G. Choose I ∈ (FG) ∪ (GF) of the least possible cardi- nality, say, I ∈ FG. Put ai = 1 for i ∈ I and ai = 0 otherwise. Then for a = (a1, . . . , an), it is easy to see that ϕ(F)(a) = ϕ(G)(a)+̇1 (as every subset of I is either in both families F and G or in neither). This contradiction shows G = F. Now |P(P(N))| = 22n = |O(n) |, and hence ϕ is a bijection from P(P(N)) onto O(n) , proving the uniqueness. Remark 1.1. The representation from Proposition 1.1 is sometimes referred to as the Reed-Muller expression or the algebraic normal form of f (see, e.g., Chapter 8). So far, Boolean polynomials have been less frequently used than the disjunctive and conjunctive normal forms, but they have proved indispensable in certain theoretical studies, such as enumeration or coding theory. More recently, electrical engineers have also become interested in Boolean polynomials. Remark 1.2. Boolean polynomials may be manipulated in a conceptually simple way. For example, using the representations x1 ∨ x2 ≈ x1+̇x2+̇x1x2 and x1 → x2 ≈ 1+̇x1+̇x1x2, we can compute (x1 ∧ x2)(x1 → x2) = (x1+̇x2+̇x1x2)(1+̇x1+̇x1x2) = x1+̇x2+̇x1x2+̇x2 1 +̇x1x2+̇x2 1 x2+̇x2 1 x2+̇x1x2 2 +̇x2 1 x2 2 = x1+̇x2+̇x1x2+̇x1+̇x1x2+̇x1x2+̇x1x2+̇x1x2+̇x1x2 = x2. This can be easily performed by a computer program, but we may face an explosion in the number of terms. Remark 1.3. Suppose that an n-ary Boolean function is given by a table. How do we find its Boolean polynomial or, equivalently, the corresponding family F? We could proceed via the complete DNF (as indicated earlier), but this again may produce a large number of monomials at the intermediate stages. A direct algorithm is as follows. Let f ∈ O(n) . On the hypercube Bn we have the standard (partial) order: (a1, . . . , an) ≤ (b1, . . . , bn) if a1 ≤ b1, . . . , an ≤ bn; for example; for n = 2 we have that (1, 0) ≤ (1, 1), but (1, 0) ≤ (0, 1) does not hold. As usual, we write a b if a ≤ b and a = b. For i = 1, . . . , n, set Si = {(a1, . . . , an) ∈ Bn : a1 + · · · + an = i}; hence, Si consists of all a ∈ Bn having exactly i coordinates equal to 1. Recursively, we construct Hi ⊆ Si (i = 0, . . . , n). Set H0 = S0 if f (0, . . . , 0) = 1, and H0 = ∅ otherwise. Suppose 0 ≤ i n and H0, . . . , Hi have been constructed. For a ∈ Si+1, set Ta = {b ∈ H0 ∪ · · · ∪ Hi : b ≤ a}
- 31. 6 Reinhard Pöschel and Ivo Rosenberg and Hi+1 = {a ∈ Si+1 : f (a) + |Ta| is odd}. For (a1, . . . , an) ∈ Bn , set χ(a) = {1 ≤ i ≤ n : ai = 1} and set F = χ(H0 ∪ · · · ∪ Hn). A straightforward proof shows that f (x) ≈ r I∈F i∈I xi . For example, if n = 2 and f is the implication →, we get H0 = {(0, 0)}, H1 = {(1, 0)}, H2 = {(1, 1)}, F = {∅, {1}, {1, 2}}, and x1 → x2 ≈ 1+̇x1+̇x1x2. Remark 1.4. Contrary to disjunctive normal forms, there is no minimization problem for representations of the form (1.1). However, the Boolean polynomials of some Boolean functions are long. The worst one is the Boolean polynomial of f (x1, . . . , xn) ≈ x 1 . . . x n, whose family F is the whole P(N); for example, for n = 2 we have x 1x 2 ≈ 1+̇x1+̇x2+̇x1x2. If, along with +̇, ·, 1, we also allow the negation, certain Boolean polynomials may be shortened. Here we can use some additional rules such as x y ≈ y+̇xy and xx ≈ 0. For these more general polynomials, we face minimization problems. It seems that these and the systematic use of the more general polynomials have not been investigated in depth. Remark 1.5. As the reader may suspect, Proposition 1.1 can be extended to any finite field F (e.g., Z3) and maps f : Fn → F. Remark 1.6. A long list of practical problems (mostly from operations research) leads to the following problem. Let f be a map from Bn into the set Z of integers and assume that we want to find the minimum value of f on Bn . The potential of Boolean polynomials for this problem was realized quite early in [6]. A more direct variant and a related duality are in [23, 24], but generally this approach seems to be dormant. 1.2 Completeness and Maximal Clones It is well known that every Boolean function may be represented over {∧, ·, } (e.g., through a DNF or a conjunctive normal form, CNF). In Section 1.1, we saw a representation of Boolean functions over {+̇, ·, 1} (through Boolean polynomials). In general, call a set F of Boolean functions complete if every Boolean function is a composition of functions from F. In some older and East European literature, a complete set is often referred to as functionally complete. In universal algebra, the corresponding algebra B; F is termed primal. Naturally we may ask about other complete sets and their size. The two examples we have seen so far consist of three functions each. However, the set {∧, ·, } is
- 32. 1 Compositions and Clones of Boolean Functions 7 actually redundant. Indeed, {∧, } is also complete as x1x2 ≈ (x 1 ∧ x 2) (this identity is known as one of the De Morgan laws). A Boolean function f is Sheffer if the singleton set { f } is complete. Clearly a Sheffer function is at least binary. The following functions NAND and NOR are Sheffer: x NAND y :≈ x ∨ y , x NOR y :≈ x ∧ y . The suggestive symbols NAND and NOR were adopted by electrical engineers to describe the functioning of certain transistor gates after these were invented in the 1950s. However, the two functions were introduced in logic long ago by Sheffer [25] and Nicod [19]; they are known as Sheffer strokes or Nicod connectives and are variously denoted by |, ⊥, , ↑, ↓, and so forth. The fact that NAND is Sheffer follows from x ≈ x NAND x, x ∧ y ≈ x NAND y ; consequently, the complete set {∧, } can be constructed from NAND alone, and hence NAND is Sheffer. The proof for NOR is similar. The fact that NAND and NOR are the only binary Sheffer functions will follow from Corollary 1.5, which provides a complete characterization of Sheffer functions. Having seen a few complete sets of Boolean functions, we may ask about other complete sets or – even better – for a completeness criterion, that is, for a necessary and sufficient condition for a set F of Boolean functions to be complete. E. L. Post found such a criterion in [20]. To formulate the criterion in a modern way, we need two crucial concepts. For 1 ≤ i ≤ n, the ith n-ary projection is the Boolean n-ary function en i satisfying en i (x1, . . . , xn) ≈ xi . Notice that en i just replicates its ith argument and ignores all other arguments. The composition of an m-ary Boolean function f with the n-ary Boolean functions g1, . . . , gm is the n-ary Boolean func- tion h, defined by setting h(x1, . . . , xn) = f (g1(x1, . . . , xn), . . . , gm(x1, . . . , xn)) for all n-tuples (x1, . . . , xn) in Bn . Now, a set of Boolean functions is a clone if it is composition-closed and contains all projections. Thus, a clone is rich enough in the sense that we cannot exit from it via any composition of its members (each possibly used several times). The concept is similar to the concept of a transformation monoid (whereby the set of all projections plays a role analogous to that of the identity selfmap in monoids). It is not difficult to check that the intersection of an arbitrary set of clones is again a clone. Thus, for every set F of Boolean functions, there exists a unique least clone containing F. This clone, denoted by F, is the clone generated by F. The clone F can be alternatively interpreted in terms of combinatorial switch- ing circuits. Suppose that for each n-ary f ∈ F there is a gate (switching device, typically a transistor) with n inputs and a single output realizing f (a1, . . . , an) on the output whenever, for i = 1, . . . , n, the zero-one input ai is applied to the ith input. An F-based combinatorial circuit is obtained from gates realizing functions from F by attaching the output of each gate (with a single exception) to an input of another gate so that (1) the circuit has a single external output; (2) to each input of
- 33. 8 Reinhard Pöschel and Ivo Rosenberg Figure 1.1. Combinatorial circuit realizing f (x1, x2, x3) ≈ x1x 2 ∨ x2x 3. a gate is attached a single external input or a single output of another gate but not both; and (3) the circuit is feedback-free (i.e., following the arcs from any external input, we always arrive at the external output without ever making a loop). An example is in Figure 1.1; see also Chapters 11 and 16. Now the clone F generated by F consists of all Boolean functions realizable by F-based combinatorial circuits. This interpretation also allows us to present an alternative definition of the concept of completeness: in this terminology, a set F of Boolean functions is complete exactly if F is the set O of all Boolean functions. Clearly O is the greatest clone (with respect to inclusion ⊆). A clone C distinct from O is maximal (also precomplete or preprimal) if C ⊂ D ⊂ O holds for no clone D: in other words, if O covers C in the containment relation ⊆. Thus, C is maximal exactly if C is incomplete, but C ∪ { f } is complete for each f ∈ O C. The concept is a direct analog of a maximal subgroup, a maximal subring, and so on. We prove in Theorem 1.2 that each clone distinct from O is contained in at least one maximal clone. This leads to the following almost immediate but still basic fact. Fact 1. A set F of Boolean functions is complete if and only if F is contained in no maximal clone. Proof. (⇒) By contradiction, if F is contained in some maximal clone M, then the clone F generated by F is included in M, and so F is not complete. (⇐) If F is incomplete, then F extends to a maximal clone M, and so F ⊆ F ⊆ M. Of course, Fact 1 is fully applicable only if we know the entire list of maximal clones. Then a set F of Boolean functions is complete if and only if for each maximal clone M (from the list) we can find some f ∈ F M.
- 34. 1 Compositions and Clones of Boolean Functions 9 Most clones and all maximal clones may be described by relations in the following way. Recall that for a positive integer h, an h-ary relation on B is a subset ρ of Bh (i.e., a set of zero-one h-tuples; some logicians prefer to view it as a map ϕ from Bh into {+, −}, whereby ρ = ϕ−1 (+), or to call it a predicate). For h = 1, the relation is called unary and is just a subset of B, whereas for h = 2, the relation is binary. An n-ary Boolean function f preserves an h-ary relation ρ if for every h × n matrix A whose column vectors all belong to ρ, the h-tuple of the values of f on the rows of A belongs to ρ. In symbols, if A = [ai j ] with (a1 j , . . . , ahj ) ∈ ρ for all j = 1, . . . , n, then ( f (a11, . . . , a1n), . . . , f (ah1, . . . , ahn)) ∈ ρ. Notice the “rows versus columns” type of the definition. Several examples are given next. In the universal algebra terminology, f preserves ρ if ρ is a subuniverse of B; f h . The first impression may be that the definition is too artificial or covers only special cases. It turns out that it is the right one not only for clones of Boolean functions but also for clones on any finite universe. It seems that it was first explicitly formulated in [15], and it has been reinvented under various names. We illustrate this crucial concept using a few examples needed for the promised completeness criterion. Example 1.1. Let h = 1 and ρ = {0}. Then A = [0 · · · 0] (a 1 × n matrix), and f preserves {0} if and only if f (0, . . . , 0) = 0 (in universal algebra terminology, if and only if {0} is a subuniverse of B; f ). Example 1.2. Similarly, a Boolean function f preserves the unary relation {1} if and only if f (1, . . . , 1) = 1. Example 1.3. Consider the binary relation σ := {(0, 1), (1, 0)} on B. Note that σ is the diagram (or graph) π0 of the permutation π : x → x . It can also be viewed as a graph on B with the single edge {0, 1}. A 2 × n matrix A = [ai j ] has all columns in σ if and only if a2 j = a 1 j holds for all j = 1, . . . , n. Writing aj for a1 j , we obtain that f preserves σ if and only if ( f (a1, . . . , an), f (a 1, . . . , a n)) ∈ σ holds for all a1, . . . , an ∈ B. This is equivalent to the identity f (x 1, . . . , x n) ≈ f (x1, . . . , xn) . The standard name for a Boolean function satisfying this identity is selfdual (see [7] and Section 1.3). Expressed algebraically, f is selfdual if and only if the negation is an automorphism of the algebra B; f . A selfdual f ∈ O(n) is fully determined by its values f (0, a2, . . . , an) with a2, . . . , an ∈ B (because f (1, b 2, . . . , b n) = f (0, b 2, . . . , b n)). Thus, there are exactly 22n−1 selfdual n-ary Boolean functions, and the probability that a randomly chosen f ∈ O(n) is selfdual is the low value 2−2n−1 : for example, it is approximately 0.0000152 for n = 5.
- 35. 10 Reinhard Pöschel and Ivo Rosenberg Example 1.4. Next consider ρ = {(0, 0), (0, 1), (1, 1)}. Notice that (x, y) ∈ ρ if and only if x ≤ y (where ≤ is the natural order on B), and so we write x ≤ y instead of (x, y) ∈ ρ. A 2 × n matrix A = [ai j ] has all columns in ≤ whenever a1 j ≤ a2 j holds for all j = 1, . . . , n, and therefore f preserves ≤ if and only if a1 ≤ b1, . . . , an ≤ bn ⇒ f (a1, . . . , an) ≤ f (b1, . . . , bn); that is, if every argument is kept the same or increased, then the value is the same or increases. This is the standard definition of a monotone (also isotone, order-respecting, or order-compatible) Boolean function; see [7]. Example 1.5. As a final example, consider the 4-ary (or quaternary) relation λ = {(x1, x2, x3, x1+̇x2+̇x3) : x1, x2, x3 ∈ B}, where +̇ is the sum mod 2 introduced in Section 1.1. Expressed differently, the last coordinate in a 4-tuple from λ is exactly the parity check (making the coordinate sum even), a basic error check used in computer hardware and other digital devices; equivalently, a 4-tuple belongs to λ if and only if it contains an even number of 1s. The description of the functions preserving λ is not as transparent as in the preceding examples. In order to establish it, let us say that a Boolean function f ∈ O(n) is linear (or affine) if there are c ∈ B and 1 ≤ i1 . . . ik ≤ n such that f (x1, . . . , xn) ≈ c+̇xi1 +̇ · · · +̇xik . Fact 2. A Boolean function preserves λ if and only if it is linear. Proof. (⇒) Let f ∈ O(n) preserve λ. Then f (x1, . . . , xn) ≈ f (x1, 0, . . . , 0)+̇ f (0, x2, . . . , xn)+̇ f (0, . . . , 0). (1.2) Here f (0, x2, . . . , xn) ∈ O(n−1) also preserves λ, and so we can apply (1.2) to it. Continuing in this fashion, we obtain f (x1, . . . , xn) ≈ r n i=1 f (0, . . . , 0, xi , 0, . . . , 0)+̇d (1.3) where d = f (0, . . . , 0)+̇ . . . +̇ f (0, . . . , 0) (n times). Each unary Boolean func- tion f (0, . . . , 0, xi , 0, . . . , 0) is of the form ai x+̇bi for some ai , bi ∈ B, and thus the right-hand side of (1.3) is a1x1+̇ · · · +̇an xn+̇c where c = b1+̇ · · · +̇bn+̇d. This proves that f is linear. (⇐) It can be easily verified that every linear function preserves λ. The set of Boolean functions preserving a given h-ary relation ρ on B is denoted by Pol ρ. It is easy to verify that Pol ρ is a clone by showing that it contains all projections and that it is composition closed. Now we are ready for the promised completeness criterion due to Post [20].
- 36. 1 Compositions and Clones of Boolean Functions 11 Theorem 1.2. (Completeness Criterion.) (1) A set F of Boolean functions is complete if and only if F is contained in none of the clones Pol{0}, Pol{1}, Polσ, Pol ≤, Polλ. (1.4) (2) Each clone distinct from O extends to a maximal one, and the foregoing five clones are exactly all maximal clones. Remark 1.7. Part (2) is just a rephrasing of part (1). The criterion is often given in the following equivalent form: a set F of Boolean functions is complete if and only if there exist f1, . . . , f5 ∈ F such that f1(0, . . . , 0) = 1, f2(1, . . . , 1) = 0, f3 is not selfdual, f4 is not monotone, and f5 is not linear (here f1, . . . , f5 need not be pairwise distinct; e.g., they are all equal for F = { f }, where f is Sheffer). The proof given next follows A. V. Kuznetsov’s proof (see [10], pp. 18–20). The relatively direct proof is of some interest due to a Slupecki-type criterion (Lemma 1.4 later), but it does not reveal how the completeness criterion was discovered. Proof. The necessity is obvious, as all five clones listed are distinct from O. As for sufficiency, in the following lemma we start by characterizing clones included in neither of the first two clones in (1.4) and end up with the same for all clones except the last. Remarkably, all this is done through the unary operations of the clone. Recall that O(n) denotes the set of n-ary Boolean functions. Lemma 1.3. If C is a clone and f ∈ O(n) , then: (i) C ⊆ Pol{0} if and only if C contains the unary constant c1 or C contains the negation. (ii) C ⊆ Pol{1} if and only if C contains the unary constant c0 or C contains the negation. (iii) f ∈ Pol ≤ if and only if f (a1, . . . , ai−1, 0, ai+1, . . . , an) f (a1, . . . , ai−1, 1, ai+1, . . . , an) (1.5) for some 1 ≤ i ≤ n and some a1, . . . , an ∈ B. (iv) C ⊆ Pol{0}, C ⊆ Pol{1}, and C ⊆ Pol ≤ if and only if C contains the negation. (v) C is a subclone of none of Pol{0}, Pol{1}, Pol ≤, and Polσ if and only if C ⊇ O(1) , that is, C contains all four unary Boolean functions.
- 37. 12 Reinhard Pöschel and Ivo Rosenberg Proof. (i) Let us first show that if C ⊆ Pol{0}, then C contains the unary constant c1 ≈ 1 or the negation. Because C ⊆ Pol{0}, clearly h(0, . . . , 0) = 1 for some h ∈ C. (a) Suppose h(1, . . . , 1) = 0. Then g(x) ≈ h(x, . . . , x) ≈ x . Here g ∈ C be- cause g(x, . . . , x) ≈ h(e1 1(x), . . . , e1 1(x)) where the clone C contains both h and the projection e1 1 and is composition closed. Thus, C contains the negation. (b) Suppose next that h(1, . . . , 1) = 1. By the same token as in (a), we get c1 ∈ C. This proves the statement. (ii) A similar argument as above applied to Pol{1} shows that C contains the unary constant c0 ≈ 0 or the negation. (iii) Let ≤ be the componentwise order on Bn introduced in Remark 1.3 of Section 1.1. From f ∈ Pol ≤, we obtain f (a) ≤ f (b) for some a ≤ b. Here already f (a) = 1 and f (b) = 0. For notational simplicity, we assume that a = (0, . . . , 0, ak+1, . . . , an), b = (1, . . . , 1, ak+1, . . . , an) for some 0 k n. For i = 0, . . . , k, set ai = (1, . . . , 1, 0, . . . , 0, ak+1, . . . , an), where the first i coordinates are 1 and the next k − i coordinates are 0. In view of f (a0) = f (a) = 1 and f (ak) = f (b) = 0, there exists 0 ≤ i k with f (ai ) = 1 and f (ai+1) = 0, proving (iii). (iv) By (i) and (ii), it suffices to consider the case of C containing both constants c0 and c1. Because C ⊆ Pol ≤, by (iii) there exists f ∈ C, satisfying (1.5). Define a unary Boolean function h by setting h(x) ≈ f (g1(x), . . . , gn(x)) where gi (x) ≈ e1 1(x) ≈ x and gj is the unary constant function with value aj for all j = i. Clearly h ∈ C and h(x) ≈ x . (v) By (iv), the clone C contains the negation. As C ⊆ Polσ, the clone C also contains a nonselfdual n-ary function f , and hence f (a1, . . . , an) = f (a 1, . . . , a n) for some a1, . . . , an ∈ B. Set x0 ≈ x and x1 ≈ x and define a unary g by g(x) ≈ f (xa1 , . . . , xan ). Clearly g is constant and g ∈ C because e1 1, the negation, and f are in the clone C. The other constant is g ∈ C. Thus, C ⊇ O(1) . This lemma leads to the question: what are the maximal clones containing all four unary operations? The following lemma asserts that the clone Polλ (of all linear functions, see Fact 2) is the unique maximal clone containing O(1) . Lemma 1.4. Let C be a clone such that C ⊇ O(1) . Then C = O if and only if C contains a nonlinear function. Proof. The necessity is obvious. For sufficiency, let f be a nonlinear function from C. In view of the De Morgan law: x ∧ y ≈ (x y ) , the set ·, is complete.
- 38. 1 Compositions and Clones of Boolean Functions 13 Now C already contains the negation, and so it suffices to show that the conjunction belongs to the clone O(1) ∪ { f } generated by the four unary operations and f . By Proposition 1.1, the function f can be represented by a Boolean polynomial r I∈F i∈I xi , which obviously is nonlinear, and so there exists J ∈ F with |J| 1. Choose such J of the least possible size, and for notational convenience let J = {1, . . . , j}. Define h ∈ O(2) by h(x1, x2) ≈ f (x1, x2, 1, . . . , 1, 0, . . . , 0), (1.6) where 1 appears j − 2 times. Clearly, h ∈ C. We obtain that h(x1, x2) ≈ x1x2+̇ax1+̇bx2+̇c (1.7) for some a, b, c ∈ B. Indeed, by the minimality of J, every I ∈ F {J} with |I| 1 meets the set { j + 1, . . . , n}, and so the corresponding product in (1.6) vanishes. Now h(x1+̇b, x2+̇a)+̇ab+̇c belongs to C because h, e1 1, and the negation belong to C. It can be verified that this Boolean function is actually the conjunction x1x2. To complete the proof of Theorem 1.2, simply combine Lemma 1.3(v) and Lemma 1.4. Remark 1.8. Post’s criterion was rediscovered at least ten times, but it would serve no purpose to list all the references. Its beauty lies in the simplicity of the five conditions, which may be verified directly on the given set F of Boolean functions; in particular, no construction or reference to another structure is necessary. In- deed, the first two conditions may be checked by inspecting the values f (0, . . . , 0) and f (1, . . . , 1) for f ∈ F. To find out whether a given n-ary Boolean function is selfdual, we need at most 2n−1 checks (see Example 1.3). Next, by Lemma 1.3(iii), the function f is monotone if and only if f (a1, . . . , an) ≤ f (b1, . . . , bn) whenever ai = 0, bi = 1 for a single i and aj = bj otherwise. Thus, we need to check at most n2n−1 pairs a, b. To check whether f is linear, put a0 = f (0, . . . , 0) and ai = (0, . . . , 0, 1, 0, . . . , 0), where 1 is at the ith place and i = 1, . . . , n. Clearly, f is linear exactly if f (x1, . . . , xn) ≈ a0+̇(a0+̇a1)x1+̇ · · · +̇(a0+̇an)xn, requiring at most 2n − n − 1 checks. Using these observations, one could write a computer program that could decide whether an arbitrary finite set F of Boolean function is complete. This would be executed in a priori bounded time where the (upper) bound depends on the sum of the arities of functions from F. In other words, there is an effective algorithm (in the foregoing sense) for the completeness problem.
- 39. 14 Reinhard Pöschel and Ivo Rosenberg Remark 1.9. Although it may take a little while to discover that a given set F of Boolean functions is complete, afterward a few values of at most five function from F are enough to convince anybody of its completeness. For an incomplete set F of Boolean functions, we can find all the maximal clones containing F. This information yields the necessary and sufficient conditions for the choice of an additional set G of Boolean functions capable of making F ∪ G complete (this may play a role in switching theory when F is the set of Boolean functions describing the functioning of new types of gates). Post’s completeness criterion yields an elegant characterization of Sheffer func- tions (i.e., functions f such that the singleton { f } is complete; see Section 1.2). Corollary 1.5. A Boolean function f ∈ O(n) is Sheffer if and only if (i) f (x, . . . , x) ≈ x , and (ii) f (a1, . . . , an) = f (a 1, . . . , a n) for some a1, . . . , an ∈ B. Proof. The condition (i) is equivalent to f ∈ Pol{0} ∪ Pol{1}, whereas the condi- tion (ii) says that f is not selfdual. (⇒) Both conditions are necessary by Post’s completeness criterion. (⇐) Let f satisfy (i) and (ii). Then f ∈ Pol{0} ∪ Pol{1} ∪ Polσ. Next we claim that Pol ≤ ⊆ Pol{0} ∪ Pol{1}. Indeed, by contraposition, let h ∈ Pol{0} ∪ Pol{1} be arbitrary. Then h(0, . . . , 0) = 1 0 = h(1, . . . , 1), proving h ∈ Pol ≤. In particular, f ∈ Pol ≤. Suppose now by contraposition that f is linear. In view of (i), clearly f (x1, . . . , xn) ≈ 1+̇a1x1+̇ · · · +̇an xn, where a1, . . . , an ∈ B satisfy a1+̇ · · · +̇an = 1. Now f (x 1, . . . , x n) ≈ 1+̇ f (1+̇x1, . . . , 1+̇xn) ≈ 1+̇1+̇a1+̇ · · · +̇an+̇a1x1+̇ · · · +̇an xn ≈ 1+̇a1x1+̇ · · · +̇an xn ≈ f (x1, . . . , xn), proving that f is selfdual. Thus f is nonlinear, and by Post’s completeness criterion { f } is complete. Remark 1.10. Corollary 1.5 may be rephrased. Denote by M the union of the five maximal clones. Then Sh := O M is the set of Sheffer functions. Corollary 1.5 in fact states that M = Pol{0} ∪ Pol{1} ∪ Polσ, and so the maximal clones Pol{0}, Pol{1}, and Polσ cover M. In fact, they also provide a unique irredundant cover of M (meaning that every cover of M by
- 40. 1 Compositions and Clones of Boolean Functions 15 maximal clones must include the foregoing three clones). To show this, it suffices to find in each of the three maximal clones a function belonging to no other maximal clone. The functions x y and x → y are such functions for Pol{0} and Pol{1}, whereas for Polσ, although there is no such binary function, there are four such ternary functions. We proceed to enumerate the n-ary Sheffer functions. For this we classify them by the “first” n-tuple (a1, . . . , an) in their table satisfying the condition (ii) of Corollary 1.5. More precisely, the weight w(a) of a = (a1, . . . , an) is defined by 2n−1 a1 + 2n−2 a2 + · · · + an, and the chain (also called a linear order or total order) on Bn induced by the weights is the lexicographic order defined by: a b if w(a) ≤ w(b). For i = 1, . . . , 2n−1 − 1, denote by Si the set of n-ary Sheffer functions f such that f (x, . . . , x) ≈ x and such that the condition (ii) from Corollary 1.5 holds for a ∈ Bn with w(a) = i, but not for any b ∈ Bn with w(b) i (i.e., a is the least element in the lexicographic order satisfying (ii)). From Corollary 1.5, we obtain the following corollary. Corollary 1.6. If n 1, then: (i) the sets S1, . . . , S2n−1−1 partition the set Sh(n) of n-ary Sheffer functions, (ii) |Si | = 22n −i−2 (i = 1, . . . , 2n−1 − 1), and (iii) there are exactly 22n −2 − 22n−1 −1 n-ary Sheffer functions. Proof. (i) The sets S1, . . . , S2n−1−1 cover Sh(n) by Corollary 1.5, and they are obvi- ously pairwise disjoint. (ii) Let 1 ≤ i ≤ 2n−1 − 1. For f ∈ Si and a = (a1, . . . , an) with w(a) i, the value f (a 1, . . . , a n) equals f (a) and so cannot be chosen freely. Moreover, f (0, . . . , 0) = 1 and f (1, . . . , 1) = 0, and hence we have exactly 22n −i−2 free choices, proving (ii). (iii) According to (i) and (ii), Sh(n) = 2n−1 −1 i=1 22n −i−2 = 22n −3 2n−1 −2 j=0 2− j . This is a finite geometric series with the quotient 2−1 , and a well-known formula yields (iii). Remark 1.11. According to Corollary 1.6(iii), there are 222 −2 − 221 −1 = 22 − 2 = 2 binary Sheffer functions. In fact, these are NOR and NAND, introduced earlier in this section. There are already 56 ternary Sheffer functions, and the
- 41. 16 Reinhard Pöschel and Ivo Rosenberg number increases rather rapidly with n. Thus we should rather ask about the proportion τn of n-ary Sheffer functions among all n-ary Boolean functions. From Corollary 1.6(iii), we get τn = |Sh(n) |/|O(n) | = 2−2 − 2−2n−1 −1 , and so limn→∞τn = 1/4. The numbers τn grow very fast to 1/4, for example, τ6 already shares the first 10 decimal places with 0.25. Thus, the probability that, for n big enough, a randomly chosen n-ary Boolean function is Sheffer is practically 0.25. In switching circuits, the constant unary functions c0 and c1 are usually available (as constant signals) or can be realized very cheaply. This leads to the following definition. Definition 1.1. A set F of Boolean functions is complete with constants (or functionally complete) if F ∪ {c0, c1} is complete. Similarly, a Boolean function f is Sheffer with constants if { f, c0, c1} is complete. The two constant functions take care of the first three maximal clones in Post’s completeness criterion (Theorem 1.2), and so we have the following corollary. Corollary 1.7. A set of Boolean functions is complete with constants if and only if it contains a nonmonotone function and a nonlinear function. In particular, a Boolean function f is Sheffer with constants if and only if f is neither monotone nor linear. Denote by γn the number of n-ary Boolean functions that are Sheffer with constants. Further, let ϕn denote the number of n-ary monotone Boolean functions. The number ϕn, called the Dedekind number, is of interest on its own. The relation between γn and ϕn is given in [8]. Corollary 1.8. (1) γn = 22n − 2n+1 − ϕn + n + 2. (2) limn→∞ γn/|O(n) | = 1. Proof. We start with two claims: Claims. (i) en 1, . . . , en n, cn 0, cn 1 are the only n-ary monotone and linear Boolean func- tions. (ii) There are 2n+1 linear n-ary functions. To prove the claims, consider a linear and monotone function f ∈ O(n) . Then f (x1, . . . , xn) = b+̇xi1 +̇ · · · +̇xik for some b ∈ B, k ≥ 0 and 1 ≤ i1 · · · ik ≤ n. If k = 0 then clearly f = cn 0 or f = cn 1. Thus let k ≥ 1. For notational simplicity, suppose that i j = j for j = 1, . . . , k. First, b = 0, because b = 1 leads to 1 = f (0, . . . , 0) f (1, 0, . . . , 0) = 0,
- 42. 1 Compositions and Clones of Boolean Functions 17 which contradicts monotonicity. Finally, k = 1, because for k ≥ 2 we get a con- tradiction from 1 = f (1, 0, . . . , 0) f (1, 1, 0, . . . , 0) = 0. Now, for k = 1, clearly f = en i1 . It is immediate that en 1, . . . , en n, cn 0, cn 1 are both monotone and linear, proving (i). The number of linear n-ary Boolean functions follows from their general form and from the unicity of their representation (Proposition 1.1). This proves the claims. Now (1) follows from Corollary 1.7 and the claims (by inclusion-exclusion), and (2) is a consequence of (1) and an asymptotic for ϕn from [13]. Remark 1.12. The last argument slightly cuts corners, because the asymptotic given in [13] is not in a form showing immediately that ϕn2−2n goes to 0. Because (2) is not our main objective, we are not providing a detailed proof. The known values of ϕn for n = 1, . . . , 8 are 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 [4, 27]. The first three values can be verified directly. The differences 1 − γn2−2n n = 1, . . . , 7 are approximately 1, 0.625, 0.121, 0.0029, 1.7 × 10−6 , 4.243 × 10−3 , 7 × 10−27 . This indicates that the proportion of n-ary functions that are Sheffer with constants indeed goes very fast to 1. This may be interpreted as follows. For a large n, an n-ary Boolean function picked at random is almost surely Sheffer with constants. Consider now a set F of Boolean functions, f ∈ F, and a Boolean function g. Suppose f ∈ M ⇔ g ∈ M holds for M running through the five maximal clones. Then clearly F is complete if and only if (F { f }) ∪ {g} is complete. This leads to the following more formal definition. Definition 1.2. The characteristic set of a Boolean function f is the subset f ∗ of {1, . . . , 5} such that 1 ∈ f ∗ ⇔ f / ∈ Pol{0}, 2 ∈ f ∗ ⇔ f / ∈ Pol{1}, 3 ∈ f ∗ ⇔ f / ∈ Polσ, 4 ∈ f ∗ ⇔ f / ∈ Pol ≤, 5 ∈ f ∗ ⇔ f / ∈ Polλ. For example, f ∗ = {1, 3, 5} exactly if f (0, . . . , 0) = f (1, . . . , 1) = 1, f is not selfdual and f is monotone but not linear. Boolean functions f and g are equivalent – in symbols, f ∼ g – if f ∗ = g∗ : in other words, if for each maximal clone M
- 43. 18 Reinhard Pöschel and Ivo Rosenberg either both f, g ∈ M or both f, g / ∈ M. As mentioned before, two equivalent functions are interchangeable with respect to completeness. Note that according to the Completeness Criterion, a set F of Boolean functions is complete if and only if F∗ := { f ∗ : f ∈ F} covers the set {1, . . . , 5}. Notice that the relation ∼ on O is the kernel of the map f → f ∗ and, as such, is an equivalence relation on O. Clearly the blocks (or equivalence classes) of ∼ are the (inclusion) minimal nonempty intersections of the five maximal clones and their complements (in O). In principle, there could be as many as 32 blocks (if the map were onto all subsets of {1, . . . , 5}), but in reality there are only 15. To derive this result, we need the following lemma, where C F stands for O F (the complement of F in O). Lemma 1.9. For i = 0, 1: (1) CPol{0} ∩ CPol{1} ⊆ CPol ≤, (2) Pol{i} ∩ Polσ ⊆ Pol{1 − i}, (3) CPol{0} ∩ CPol{1} ∩ Polλ ⊆ Polσ, (4) Pol{0} ∩ Pol{1} ∩ Polλ ⊆ Polσ, (5) Pol{i} ∩ CPol{1 − i} ∩ Pol(≤) = ci ⊆ Polλ, (6) Pol(≤) ∩ Polλ = c0, c1. Proof. (1) and (2): Immediate. (3) and (4): See the proof of Corollary 1.5. We only prove (5) for i = 0. Let an n-ary f belong to the left-hand side of (5), and let a1, . . . , an ∈ B. Then 0 = f (0, . . . , 0) ≤ f (a1, . . . , an) ≤ f (1, . . . , 1) = 0. Hence, f = cn 0 (= the n-ary constant 0), and so f ∈ c0 ⊂ Polλ. (6): See the Claims in the proof of Corollary 1.8. Now we are ready for the explicit list of the 15 blocks of ∼ or, equivalently, of the minimal nonempty intersections of the five maximal clones and their complements. We number the blocks as follows. With each subset A ⊆ {1, . . . , 5}, we associate w(A) := a∈A 25−a and we set w(A) := { f ∈ O : f ∗ = A}. Denote by J the clone of all projections and recall that Sh is the set of all Sheffer functions. Proposition 1.10. Among the sets 0, . . . , 31, exactly the following fifteen sets are nonempty: 0, 1, 2, 3, 5, 7, 12, 14, 15, 20, 22, 23, 26, 27, 31.
- 44. 1 Compositions and Clones of Boolean Functions 19 Moreover, 0 = J, 2 = {a1x1+̇ · · · +̇an xn : a1 + · · · + an 1 and odd}, 12 = c0 J, 14 = {a1x1+̇ · · · +̇an xn : a1 + · · · + an 0 and even}, 20 = c1 J, 22 = {1+̇a1x1+̇ · · · +̇an xn : a1 + · · · + an 0 and even}, 26 = {1+̇a1x1+̇ · · · +̇an xn : a1 + · · · + an 1 and odd}, 31 = Sh. Proof. Call f ∈ O idempotent if f ∈ Pol{0} ∩ Pol{1}, that is, f (x, . . . , x) ≈ x. (a) We show 0 = J. First 0 is the intersection of the five maximal clones, hence a clone by Section 1.2, and therefore 0 ⊇ J. Suppose to the contrary that there exists f ∈ 0 J. Then f is constant by Lemma 1.9(6), in contradiction to the idempotency of f . Thus, 0 = J. (b) The Boolean function f (x1, x2, x3) ≈ x1x2+̇x1x3+̇x2x3 belongs to 1. Indeed, f is idempotent, selfdual (because f (x1+̇1, x2+̇1, x3+̇1)+̇1 ≈ f (x1, x2, x3)), monotone (due to f (x1, x2, x3) = 1 ⇔ x1 + x2 + x3 ≥ 2), and clearly nonlinear. (c) 2 consists of idempotent, selfdual and nonmonotone functions, which translates into the condition stated in the second part of the proposition. The same applies to 26. (d) Let f (x, y, z) ≈ xy+̇xz+̇yz+̇x+̇y. Clearly f is idempotent and nonlinear. It is also selfdual and nonmonotone because, for example; f (1, 0, 0) = 1 0 = f (1, 0, 1). Thus, f ∈ 3. (e) The disjunction ∨ belongs to 5. (f) f (x, y, z) ≈ x+̇xy+̇xyz belongs to 7 because it is idempotent, nonself- dual ( f (1, 1, 0) = f (0, 0, 1) = 0), nonmonotone ( f (1, 0, 0) = 1 0 = f (1, 1, 0)), and nonlinear. (g) 12 and 20 are of the form indicated in the second part of the statement. (h) 14 and 22 are of the form indicated in the second part of the statement. (i) f (x, y) ≈ xy belongs to 15. (j) 31 is the set of Sheffer functions. The following table indicates which statement of Lemma 1.9 can be used to prove that i is void for the remaining values of i: i 4,6 8–11 13 16–19 21 24, 25, 28, 29 30 Statement # (4) (2) (5) (2) (5) (1) (3)
- 45. 20 Reinhard Pöschel and Ivo Rosenberg 27 26 20 0 2 1 23 3 7 5 12 14 15 22 31 Pol σ σ Pol Pol λ Pol{1} Pol{0} Pol (=) Figure 1.2. Venn diagram of the five maximal clones: Pol{0} = 0 ∪ · · · 3 ∪ 5 ∪ 7 ∪ 12 ∪ 14 ∪ 15, Pol{1} = 0 ∪ · · · 3 ∪ 5 ∪ 7 ∪ 20 ∪ 22 ∪ 23, Polσ = 0 ∪ · · · 3 ∪ 26 ∪ 27, Pol ≤ = 0 ∪ 1 ∪ 5 ∪ 12 ∪ 20, Polλ = 0 ∪ 2 ∪ 12 ∪ 14 ∪ 20 ∪ 22 ∪ 26. Remark 1.13. In the second part of the foregoing proposition, we have char- acterized only some of the i . The description of the other i can be ob- tained from their definitions. For example, 1 = (Pol{0} ∩ Pol{1} ∩ Polσ ∩ Pol ≤ ∩ CPolλ), is the set of all selfdual, monotone, and nonlinear f satisfying f (0, . . . , 0) = 0. Remark 1.14. The situation is depicted in Figure 1.2. In this Venn diagram of the five maximal clones, the ith region represents the set i . For example, the central region 0 represents the set 0 (of all projections), which is the intersection of all maximal clones.
- 46. 1 Compositions and Clones of Boolean Functions 21 Remark 1.15. The sizes of certain (n) i (= the n-ary functions from i ) were given in [14]: | (n) 2 | = 2n−1 − n, | (n) 12 | = | (n) 20 | = 1, | (n) 14 | = | (n) 22 | = 2n−1 − 1, | (n) 15 | = | (n) 23 | = 22n−2 − 2n−1 , | (n) 26 | = 2n−1 , | (n) 27 | = 22n−1 − 2n−1 . Because 0 is the clone of all projections, we have | (n) 0 | = n. Next, | (n) 31 | was given in Corollary 1.6 (iii). Finally, | (n) 1 | + | (n) 3 | = 22n−1 −1 − 2n−1 , | (n) 5 | + | (n) 7 | = 22n −2 − 22n−1 −1 − 2n−1 . Definition 1.3. A complete set F of Boolean functions is a basis if no proper subset of F is complete, that is, if F is irredundant (or irreducible) with respect to completeness. The characteristic sets (see Definition 1.2) provide a tool for the description of bases. According to the Completeness Criterion, a set F of Boolean functions is a basis exactly if F∗ := { f ∗ ∈ F} is an irredundant cover of {1, . . . , 5} (i.e., F∗ is a cover of {1, . . . , 5} such that each proper subfamily of F∗ misses at least one element of {1, . . . , 5}). Each Boolean function belongs to exactly one of the 15 sets listed in Proposition 1.10. The corresponding sets are ∅ and {5}, {4}, {4, 5}, {3, 5}, {3, 4, 5}, {2, 3}, {2, 3, 4}, {2, 3, 4, 5}, {1, 3}, {1, 3, 4}, {1, 3, 4, 5}, {1, 2, 4}, {1, 2, 4, 5}, {1, 2, 3, 4, 5} (1.8) (for example, ∗ 14 = {2, 3, 4} because 14 = 23 + 22 + 21 ). The determination of all possible types of bases amounts to the problem of finding all irredundant covers of {1, . . . , 5} formed from the 14 sets listed in (1.8). This is a purely technical problem that can be solved by a simple computer program. It is so small that it was solved by hand [9, 12, 14]. There are exactly 1, 17, 22 and 2 irredundant covers consisting of 1, 2, 3 and 4 sets, respectively. For example, the cover by a single set is {1, . . . , 5}, which evidently corresponds to the set 31 of all Sheffer functions. An example of an irredundant 2-set cover is {1, 2, 4, 5}, {2, 3, 4, 5}. A corresponding basis is { f, g} with f ∈ 27 and g ∈ 15, that is, f is selfdual but belongs to no other maximal clone and g preserves 0 but belongs to no other maximal clone. The total number of bases consisting of two n-ary functions is given in [14]. From this it follows that the proportion of such bases among all pairs of n-ary
- 47. 22 Reinhard Pöschel and Ivo Rosenberg functions goes fast to 1 8 as n → ∞. Together with Corollary 1.6(iii), this shows that a randomly chosen pair of n-ary Boolean functions is complete with probability practically equal to 0.375. Two bases F and G are of the same type if F = { f1, . . . , fn} and G = {g1, . . . , gn} where f1 ∼ g1, . . . , fn ∼ gn. The result quoted earlier states: There are exactly 42 types of bases, and each basis consists of at most four functions. The latter statement ([10] p. 20) follows from the fact that f (0, . . . , 0) = 1 for some f in the basis, and so either f (1, . . . , 1) = 0 or f is not selfdual. A four-element basis is {c0, c1, ·, s3} where s3(x1, x2, x3) ≈ x1+̇x2+̇x3. 1.3 A Description of the Post Lattice 1.3.1 Definition of the Post Lattice In the previous section, we saw the role of the five maximal clones of Boolean functions and their intersections. It is natural to ask about the other clones of Boolean functions and their inclusions; this leads to the following problems. Denote by C the set of clones. One of the first questions about C may be its size. As clones are subsets of the countably infinite set O (of all Boolean functions), a priori |C| could be any cardinal less than or equal to 2ℵ0 , the cardinality of the set of reals. Twenty years after his completeness paper, E. L. Post showed in [21] that |C| = ℵ0 . (It turned out later that this is exceptional because almost all other infinite variants are of continuum cardinality; a fact that, e.g., distinguishes the classical two-valued logic from many-valued logics.) Consider a nonvoid subset {Ci : i ∈ I} of C. It is easy to see that C = i∈I Ci is the greatest clone contained in every Ci . The clone C is called the meet (or infimum) of {Ci : i ∈ I}. Consider now any subset F of O and set C(F) = {X ∈ C : X ⊇ F}. Clearly C(F), the family of all clones containing F, is nonvoid because it contains the greatest clone O. Thus X∈C(F) X is a clone, denoted by F, and called the clone generated by F. Clearly F contains F; hence F is a member of C(F), and thus F is the least clone containing F. (The argument we just presented is the standard one for the existence of the vector subspace spanned by a set of vectors, the subgroup generated by a subset of a group, etc.) It follows that i∈I Ci is the least clone containing all clones Ci (i ∈ I). It is called the join (or supremum) of {Ci : i ∈ I}.
- 48. 1 Compositions and Clones of Boolean Functions 23 Because of the existence of meets and joins, the ordered set L = (C, ⊆), where ⊆ is the set inclusion (or containment), is a so-called complete lattice. In his landmark 100-page paper [21], Post gave a full description of the lattice L (today called the Post lattice). Post actually characterized all composition-closed sets of Boolean functions: however, there are only seven such sets that are not clones – that is, do not contain all projections – and these very small sets can be easily described. 1.3.2 Duality We start with a helpful symmetry of the lattice L. The dual of an n-ary Boolean function f is the Boolean function f ∂ defined by setting f ∂ (x1, . . . , xn) ≈ ( f (x 1, . . . , x n)) (where x is the negation). It can be checked that f ∂∂ = f . The duals of ∨ and +̇ are · and ↔, respectively. Also, f is selfdual (see Example 1.3) if and only if f = f ∂ . To every F ⊆ O, assign F∂ := { f ∂ : f ∈ F}. If F is a clone, then F∂ is also a clone and F → F∂ is a lattice automorphism of L, that is, a bijection (or 1-1 and onto selfmap) of C onto itself respecting the lattice joins and meets (to see it, check that f → f ∂ respects the composition). A finite ordered set is usually represented by its (Hasse) diagram. In such a drawing, vertices correspond to the elements of the set and two vertices are joined by a line segment exactly if the element corresponding to the vertex drawn higher on the page covers the element corresponding to the vertex drawn lower on the page. Here a covers b means that a b but a c b holds for no c. In certain cases we also can draw diagrams of infinite orders. In view of the automorphism F → F∂ of L, we can draw a diagram of L so that it is symmetric with respect to the central vertical line: in other words, so that the left-hand half is the mirror image of the right-hand half; that is, F → F∂ acts horizontally. This symmetry is tied to the relational description of clones as follows. Fact 3. If ρ is an h-ary relation on B and ρ := {(a 1, . . . , a n) : (a1, . . . , an) ∈ ρ}, then (Polρ)∂ = Polρ . For example, (Pol{0})∂ = Pol{0 } = Pol{1}, (Pol{1})∂ = Pol{0}, while F∂ = F for the remaining three maximal clones because ≤∂ , σ∂ and λ∂ equal ≥, σ and λ, respectively. Proof. First we show (Polρ)∂ ⊆ Polρ . (1.9)
- 50. 1 Compositions and Clones of Boolean Functions 25 example, ∨ is the clone generated by the disjunction; and (3) special notations: T0 = Pol{0}, T1 = Pol{1}, S = Polσ, M = Pol ≤ and L = Polλ for the maximal clones and I for T0 ∩ T1. We abbreviate S ∩ M by SM (i.e., SM is the clone of all functions that are both selfdual and monotone), and similarly for other intersections of clones. In order to find all clones, we start with the five maximal clones and their inter- sections. In principle Proposition 1.10 (see Figure 1.2) may be used to determine all the intersections of maximal clones. Nevertheless, we provide more details now and draw the diagrams of certain easily describable intervals. We start with a trivial case. Recall that a Boolean function f is idempo- tent if f (x, . . . , x) ≈ x. Clearly I := T0T1(= T0 ∩ T1) is the clone of idempotent functions. 1.3.4 The Clone SM The clone SM is the intersection of the maximal clones S = Polσ and M = Pol ≤. This clone consists of selfdual monotone functions. A quick check shows that among the unary and binary functions, only projections are selfdual and monotone, and this may lead to the impression that perhaps SM is the least clone J of all projections. But SM contains a nontrivial ternary function m(x1, x2, x3) :≈ x1x2 ∨ x1x3 ∨ x2x3, called the majority or median function. It is clearly monotone. According to the De Morgan laws, its dual m∂ satisfies m∂ (x1, x2, x3) ≈ (x 1x 2 ∨ x 1x 3 ∨ x 2x 3) ≈ (x 1 ∨ x 2 )(x 1 ∨ x 3 )(x 2 ∨ x 3 ) ≈ (x1 ∨ x2)(x1 ∨ x3)(x2 ∨ x3). Applying the distributive law, it is easy to verify that (x1 ∨ x2)(x1 ∨ x3)(x2 ∨ x3) ≈ x1x2 ∨ x1x3 ∨ x2x3, proving m∂ = m and m ∈ S. (Actually, the last identity is important because it characterizes the distributive lattices among all lattices.) An easy check shows that m and the projections are the only ternary functions in SM. However, SM contains many interesting functions of higher arity. (Median algebras, see, e.g., [1], generalize the majority function and some other functions from SM and seem to have some applications in social sciences.) It is immediate that every function from SM is idempotent, and so SM is contained in all maximal clones except the clone L of linear functions. Moreover, SM is an atom of the Post lattice. Here an atom of L, called a minimal clone, is a clone properly containing exactly the clone J. For A, B ∈ C, A ⊆ B, the sets [A) = {X ∈ C : X ⊇ A}, [A, B] = {X ∈ C : A ⊆ X ⊆ B}
- 51. 26 Reinhard Pöschel and Ivo Rosenberg Figure 1.4. The filter [SM). are called a filter and an interval of L, respectively. The diagram of the filter [SM) (of all clones containing the clone SM) is shown in Figure 1.4. Apart from the four maximal clones T0, T1, S, M, the filter consists of their intersections (e.g., I, T0 M, I M, and I S) and two additional clones (which are meet-irreducible in the lattice theory terminology). One is Polμ2 where μ2 := B2 {(0, 0)}, and the other is its dual Polμ 2 where μ 2 = B2 {(1, 1)}. 1.3.5 The Clone I L Another filter that can be easily described is [I L). The clone I L = Pol{0}Pol{1}Polλ consists of linear idempotent functions. An n-ary f ∈ I L is of the form a+̇xi1 +̇ · · · +̇xim with a ∈ B. Here a = 0 due to f (0, . . . , 0) = 0, and m is odd due to f (1, . . . , 1) = 1. Consequently, I L consists of functions of the form xi1 +̇ · · · +̇xi2k+1 with k ≥ 0. It is easy to verify that I L is the intersection of the four maximal clones T0, T1, L, and S. The clone I L is generated by each of its nontrivial functions (i.e., f ∈ I L J of the form x1+̇ · · · +̇x2k+1 with k 0).
- 52. 1 Compositions and Clones of Boolean Functions 27 Figure 1.5. The filter [I L). Indeed, for k 1, x1+̇ · · · +̇x2k+1 ≈ (· · · ((x1+̇x2+̇x3)+̇x4+̇x5)+̇ · · · )+̇x2k+̇x2k+1. Conversely, x1+̇x2+̇x3 ≈ x1+̇x2+̇x3+̇x3+̇ · · · +̇x3. This shows that I L is a minimal clone. The interval [I L) is shown in Figure 1.5. 1.3.6 The Clone ML The last nontrivial intersection of two maximal clones is the clone ML := Pol ≤ ∩Polλ of the monotone linear functions. We have already seen in Lemma 1.9(6) that ML = c0, c1 is made up from the constants and the projections. Now ML is contained in neither of T0, T1 and S. Between ML and M there are exactly the two mutually dual clones ∨, c0, c1 and ·, c0, c1, and between ML and L there is precisely the clone , c0. However, ML is not minimal; in fact, ML is obviously the join of the two minimal clones c0 and c1, and ML covers no other clone. The interval [J, ML] and the filter [ML) are shown in Figure 1.6. 1.3.7 The Seven Minimal Clones In Sections 1.3.4 through 1.3.6, we have seen all nontrivial intersections of maximal clones (the intersection of all five maximal clones is obviously the trivial clone J
- 53. 28 Reinhard Pöschel and Ivo Rosenberg Figure 1.6. The interval [J, ML] and the filter [ML). of all projections). Two of them, namely SM and I L, are even minimal clones, and the third, namely ML, covers two minimal clones c0 and c1. At this point the reader may be wondering about the full list of all minimal clones. Obviously there is also the minimal clone generated by the negation, but the existence of other minimal clones is not obvious. In fact, there are only two more minimal clones, namely ∨ and ·. Proposition 1.11. There are exactly seven minimal clones: c0, c1, , ∨, ·, SM, I L (where SM is generated by the majority function x1x2 ∨ x1x3 ∨ x2x3 and I L is generated by x1+̇x2+̇x3). The minimality of the clones ∨ and · is easily checked. We omit the more complex proof that there are no other minimal clones. The various joins of the seven minimal clones can be verified directly. Their diagram is drawn in Figure 1.7.
- 54. 1 Compositions and Clones of Boolean Functions 29 Figure 1.7. Joins of minimal clones. 1.3.8 An Infinite Descending Chain of Clones So far, starting from the top and the bottom, we have described a few clones, but a major part of the Post lattice is still missing. The key to it is a countably infinite descending chain of clones to be described in this section. For h ≥ 1, set μh = Bh {(0, . . . , 0)}. This very natural relation consists of all the 2h − 1 h-tuples having at least one coordinate 1. The fact that only (0, . . . , 0) does not belong to μh is behind the following “backward” formulation of f ∈ Polμh. An n-ary function f preserves μh if and only if the following holds: If the values of f on the rows of an h × n zero-one matrix A are all 0, then A has a zero column (i.e., a column (0, . . . , 0)T ). For example, c1, ∨ and → belong to every Polμh. Note that μ1 = B {0} = {1}. We show T1 = Polμ1 ⊃ Polμ2 ⊃ Polμ3 ⊃ · · · . Claim 1. Polμh ⊇ Polμh+1 for all h ≥ 1. Proof. Let f ∈ Polμh+1 be n-ary and let A be any h × n matrix over B with rows r1, . . . ,rh such that f (r1) = · · · = f (rh) = 0. Denote by A the (h + 1) × n matrix with rows r1, . . . ,rh,rh. Because f preserves μh+1, the matrix A has a zero
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- 62. This book is made possible by the generous co- operation of the officers of the West Point Manufacturing Company and Lanett Cotton Mills. It is the result of the combined efforts of the War Service Station in each mill locality to pay at least a feeble tribute to the gallant doughboy who enlisted in the cause of right and democracy. It is hoped that, as the years pass by, these crusaders and their posterity may find an increasing interest in this memorial to their heroism. Also, it has been thought advisable to preserve a record of the accomplishments of all those patriotic forces which contributed their part towards the successful termination of the greatest conflict in history. It would not be amiss to call particular attention to the War Service Stations, under whose leadership was fostered practically all of the patriotic work consummated by those at home. That these Stations
- 63. were a comfort to our boys—in their interest and solicitude for them—is attested by the letters reproduced.
- 64. The President’s War Message Delivered before Congress April 2, 1917 I have called the Congress into extraordinary session because there are serious, very serious, choices of policy to be made, and made immediately, which it was neither right nor constitutionally permissible that I should assume the responsibility of making. On the third of February last, I officially laid before you the extraordinary announcement of the Imperial German Government that on and after the first day of February it was its purpose to put aside all restraints of law or of humanity and use its submarines to sink every vessel that sought to approach either the ports of Great Britain and Ireland or the western coasts of Europe or any of the ports controlled by the enemies of Germany within the Mediterranean. That had seemed to be the object of the German submarine warfare earlier in the war; but since April of last year the Imperial Government had somewhat restrained the commanders of its undersea craft in conformity with its promise then given to us that passenger boats should not be sunk, and that due warning would be given to all other vessels which its submarines might seek to destroy, when no resistance was offered or escape attempted, and care taken that their crews were given at least a fair chance to save their lives in their open boats. The precautions taken were meager and haphazard enough, as was proved in distressing instance after instance in the progress of the cruel and unmanly business, but a certain degree of restraint was observed. The new policy has swept every restriction aside. Vessels of every kind, whatever their flag, their character, their cargo, their destination, their errand, have been ruthlessly sent to the bottom without warning and without thought of help or mercy for those on board—the vessels of friendly neutrals, along with belligerents. Even hospital ships and ships carrying relief to the sorely bereaved and stricken people of Belgium, though the latter were provided with safe conduct through the proscribed areas by the German Government itself and were distinguished by unmistakable marks of identity, have been sunk with the same reckless lack of compassion or of principle. I was for a little while unable to believe that such things would in fact be done by any government that had hitherto subscribed to the humane practices of civilized nations. International law had its origin in the attempt to set up some law which would be respected and observed upon the seas, where no nation had right of dominion and where lay the free highways of the world. By painful stage after stage has that law been built up, with meager enough results, indeed, after all was accomplished that could be accomplished, but always with a clear view, at least, of what the heart and conscience of mankind demanded. This minimum of right the German Government has swept aside under the plea of retaliation and necessity, and because it had no weapons which it could use at sea except these which it
- 65. is impossible to employ as it is employing them without throwing to the winds all scruples of humanity or of respect for the understandings that were supposed to underlie the intercourse of the world. I am not now thinking of the loss of property involved, immense and serious as that is, but only of the wanton and wholesale destruction of the lives of non-combatants, men, women and children, engaged in pursuits which have always, even in the darkest periods of modern history, been deemed innocent and legitimate. Property can be paid for; the lives of peaceful and innocent people cannot be. The present German submarine warfare against commerce is a warfare against mankind. It is a war against all nations. American ships have been sunk, American lives taken, in ways which it has stirred us very deeply to learn of, but the ships and people of other neutral and friendly nations have been sunk and overwhelmed in the waters in the same way. There has been no discrimination. The challenge is to all mankind. Each nation must decide for itself how it will meet it. The choice we make for ourselves must be made with a moderation of counsel and a temperateness of judgment befitting our character and our motives as a nation. We must put excited feeling away. Our motive will not be revenge or the victorious assertion of the physical might of the Nation, but only the vindication of right, of human right, of which we are only a single champion. When I addressed the Congress on the twenty-sixth of February last, I thought that it would suffice to assert our neutral rights with arms, our right to use the seas against unlawful interference, our right to keep our people safe against unlawful violence. But armed neutrality, it now appears, is impracticable. Because submarines are in effect outlaws when used as the German submarines have been used against merchant shipping, it is impossible to defend ships against their attacks as the law of nations has assumed that merchantmen would defend themselves against privateers or cruisers, visible craft giving chase upon the open sea. It is common prudence in such circumstances, grim necessity indeed, to endeavor to destroy them before they have shown their own intention. They must be dealt with upon sight, if dealt with at all. The German Government denies the right of neutrals to use arms at all within the areas of the sea which it has proscribed, even in the defense of rights which no modern publicist has ever before questioned their right to defend. The intimation is conveyed that the armed guards which we have placed on our merchant ships will be treated as beyond the pale of law and subject to be dealt with as pirates would be. Armed neutrality is ineffectual enough at best; in such circumstances and in the face of such pretensions it is worse than ineffectual; it is likely only to produce what it was meant to prevent; it is practically certain to draw us into the war without either the rights or the effectiveness of belligerents. There is one choice we cannot make, we are incapable of making: we will not choose the path of submission and suffer the most sacred rights of our Nation and our people to be
- 66. ignored or violated. The wrongs against which we now array ourselves are no common wrongs; they cut to the very roots of human life. With a profound sense of the solemn and even tragical character of the step I am taking and of the grave responsibilities which it involves, but in unhesitating obedience to what I deem my constitutional duty, I advise that the Congress declare the recent course of the Imperial German Government to be in fact nothing less than war against the Government and people of the United States; that it formally accept the status of belligerent which has thus been thrust upon it; and that it take immediate steps not only to put the country in a more thorough state of defense, but also to exert all its power and employ all its resources to bring the Government of the German Empire to terms and end the war. What this will involve is clear. It will involve the utmost practicable co-operation in counsel and action with the governments now at war with Germany, and, as incident to that, the extension to those governments of the most liberal financial credits in order that our resources may, so far as possible, be added to theirs. It will involve the organization and mobilization of all the material resources of the country to supply the materials of war and serve the incidental needs of the Nation in the most abundant and yet the most economical and efficient way possible. It will involve the immediate full equipment of the Navy in all respects, but particularly in supplying it with the best means of dealing with the enemy’s submarines. It will involve the immediate addition to the armed forces of the United States already provided for by law in case of war at least five hundred thousand men, who should, in my opinion, be chosen upon the principle of universal liability to service, and also the authorization of subsequent additional increments of equal force so soon as they may be needed and can be handled in training. It will involve also, of course, the granting of adequate credits to the Government, sustained, I hope, so far as they can equitably be sustained by the present generation, by well- conceived taxation. I say sustained so far as may be equitable by taxation because it seems to me that it would be most unwise to base the credits which will now be necessary entirely on money borrowed. It is our duty, I most respectfully urge, to protect our people so far as we may, against the very serious hardships and evils which would be likely to arise out of the inflation which would be produced by vast loans. In carrying out the measures by which these things are to be accomplished we should keep constantly in mind the wisdom of interfering as little as possible in our own preparation and in the equipment of our own military forces with the duty—for it will be a very practical duty—of supplying the nations already at war with Germany with the materials which they can obtain only from us or by our assistance. They are in the field and we should help them in every way to be effective there. I shall take the liberty of suggesting, through the several executive departments of the Government, for the consideration of your committees, measures for the accomplishment of the several objects I have mentioned. I hope that it will be your pleasure to deal with them as having been framed after very careful thought by the branch of the Government upon which the responsibility of conducting the war and safeguarding the Nation will most directly fall. While we do these things, these deeply momentous things, let us be very clear, and make very clear to all the world what our motives and our objects are. My own thought has not been driven from its habitual and normal course by the unhappy events of the last two
- 67. months, and I do not believe that the thought of the Nation has been altered or clouded by them. I have exactly the same things in mind now that I had in mind when I addressed the Senate on the twenty-second of January last; the same that I had in mind when I addressed the Congress on the third of February and on the twenty-sixth of February. Our object now, as then, is to vindicate the principles of peace and justice in the life of the world as against selfish and autocratic power and to set up amongst the really free and self- governed peoples of the world such a concert of purpose and of action as will henceforth insure the observance of those principles. Neutrality is no longer feasible or desirable where the peace of the world is involved and the freedom of its peoples, and the menace to that peace and freedom lies in the existence of autocratic governments backed by organized force which is controlled wholly by their will, not the will of their people. We have seen the last of neutrality in such circumstances. We are at the beginning of an age in which it will be insisted that the same standards of conduct and of responsibility for wrong done shall be observed among nations and their governments that are observed among the individual citizens of civilized states. We have no quarrel with the German people. We have no feeling toward them but one of sympathy and friendship. It was not upon their impulse that their Government acted in entering this war. It was not with their previous knowledge or approval. It was a war determined upon as wars used to be determined upon in the old, unhappy days when peoples were nowhere consulted by their rulers and wars were provoked and waged in the interest of dynasties or of little groups of ambitious men who were accustomed to use their fellow men as pawns and tools. Self-governed nations do not fill their neighbor states with spies or set the course of intrigue to bring about some critical posture of affairs which will give them an opportunity to strike and make conquest. Such designs can be successfully worked out only under cover and where no one has the right to ask questions. Cunningly contrived plans of deception or aggression, carried, it may be, from generation to generation, can be worked out and kept from the light only within the privacy of courts or behind the carefully guarded confidences of a narrow and privileged class. They are happily impossible where public opinion commands and insists upon full information concerning all the nation’s affairs. A steadfast concert for peace can never be maintained except by a partnership of democratic nations. No autocratic government could be trusted to keep faith within it or observe its covenants. It must be a league of honor, a partnership of opinion. Intrigue would eat its vitals away; the plottings of inner circles who could plan what they would and render account to no one would be a corruption seated at its very heart. Only free peoples can hold their purpose and their honor steady to a common end and prefer the interests of mankind to any narrow interest of their own. Does not every American feel that assurance has been added to our hope for the future peace of the world by the wonderful and heartening things that have been happening within
- 68. the last few weeks in Russia? Russia was known by those who knew it best to have been always in fact democratic at heart, in all the vital habits of her thought, in all the intimate relationships of her people that spoke their natural instinct, their habitual attitude toward life. The autocracy that crowned the summit of her political structure, long as it has stood and terrible as was the reality of its power, was not in fact Russian in origin, character or purpose; and now it has been shaken off and the great, generous Russian people have been added in all their native majesty and might to the forces that are fighting for freedom in the world, for justice, and for peace. Here is a fit partner for a League of Honor. One of the things that has served to convince us that the Prussian autocracy was not and could never be our friend is that from the very outset of the present war it has filled our unsuspecting communities and even our offices of Government with spies and set criminal intrigues everywhere afoot against our national unity of council, our peace within and without, our industries and our commerce. Indeed, it is now evident that its spies were here even before the war began; and it unhappily is not a matter of conjecture, but a fact proved in our courts of justice, that the intrigues which have more than once come perilously near to disturbing the peace and dislocating the industries of the country have been carried on at the instigation, with the support, and even under the personal direction of official agents of the Imperial Government accredited to the Government of the United States. Even in checking these things and trying to extirpate them we have sought to put the most generous interpretation possible upon them because we knew that their source lay, not in any hostile feeling or purpose of the German people toward us (who were, no doubt, as ignorant of them as we ourselves were), but only in the selfish designs of a Government that did what it pleased and told its people nothing. But they have played their part in serving to convince us at last that that Government entertains no real friendship for us and means to act against our peace and security at its convenience. That it means to stir up enemies against us at our very doors, the intercepted note to the German Minister at Mexico City is eloquent evidence. We are accepting this challenge of hostile purpose because we know that in such a Government, following such methods, we can never have a friend; and that in the presence of its organized power, always lying in wait to accomplish we know not what purpose, there can be no assured security for the democratic governments of the world. We are now about to accept gauge of battle with this natural foe to liberty and shall, if necessary, spend the whole force of the Nation to check and nullify its pretensions and its power. We are glad, now that we see the facts with no veil of false pretense about them, to fight for the ultimate peace of the world and for the liberation of its peoples, the German peoples included: for the rights of nations great and small and the privilege of men everywhere to choose their way of life and of obedience. The world must be made safe for democracy. Its peace must be planted upon the tested foundations of political liberty. We have no selfish ends to serve. We desire no conquest, no dominion. We seek no indemnities for ourselves, no material compensation for the sacrifices we shall freely make. We are but one of the champions of the rights of mankind. We shall be satisfied when those rights have been made as secure as the faith and the freedom of nations can make them.
- 69. Just because we fight without rancor, without selfish object, seeking nothing for ourselves but what we shall wish to share with all free peoples, we shall, I feel confident, conduct our operations as belligerents without passion and ourselves observe with proud punctilio the principles of right and of fair play we profess to be fighting for. I have said nothing of the governments allied with the Imperial Government of Germany because they have not made war upon us or challenged us to defend our right and our honor. The Austro-Hungarian Government has, indeed, avowed its unqualified indorsement and acceptance of the reckless and lawless submarine warfare adopted now without disguise by the Imperial German Government, and it has therefore not been possible for this Government to receive Count Tarnowski, the Ambassador recently accredited to this Government by the Imperial and Royal Government of Austria-Hungary; but that Government has not actually engaged in warfare against citizens of the United States on the seas, and I take the liberty, for the present at least, of postponing a discussion of our relations with the authorities at Vienna. We enter this war only where we are clearly forced into it because there are no other means of defending our rights. It will be all the easier for us to conduct ourselves as belligerents in a high spirit of right and fairness because we act without animus, not in enmity toward a people nor with the desire to bring any injury or disadvantage upon them, but only in armed opposition to an irresponsible Government which has thrown aside all considerations of humanity and of right and is running amuck. We are, let me say again, the sincere friends of the German people, and shall desire nothing so much as the early re-establishment of intimate relations of mutual advantage between us —however hard it may be for them, for the time being, to believe that this is spoken from our hearts. We have borne with their present Government through all these bitter months because of that friendship—exercising a patience and forbearance which would otherwise have been impossible. We shall, happily, still have an opportunity to prove that friendship in our daily attitude and actions toward the millions of men and women of German birth and native sympathy who live amongst us and share our life, and we shall be proud to prove it toward all who are in fact loyal to their neighbors and to the Government in the hour of test. They are, most of them, as true and loyal Americans as if they had never known any other fealty or allegiance. They will be prompt to stand with us in rebuking and restraining the few who may be of a different mind and purpose. If there should be disloyalty, it will be dealt with with a firm hand of stern repression; but, if it lifts its head at all, it will lift it only here and there and without countenance except from a lawless and malignant few. It is a distressing and oppressive duty, Gentlemen of the Congress, which I have performed in thus addressing you. There are, it may be, many months of fiery trial and sacrifice ahead of us. It is a fearful thing to lead this great peaceful people into war, into the most terrible and disastrous of all wars, civilization itself seeming to be in the balance. But the right is more precious than peace, and we shall fight for the things which we have always carried nearest our hearts—for democracy, for the right of those who submit to authority to have a voice in their own governments, for the rights and liberties of small nations, for a universal dominion of right by such a concert of free peoples as shall bring peace and safety to all nations and make the world itself at last free.
- 70. To such a task we can dedicate our lives and our fortunes, everything that we are and everything that we have, with the pride of those who know that the day has come when America is privileged to spend her blood and her might for the principles that gave her birth and happiness and the peace which she has treasured. God helping her, she can do no other. Larger Image
- 71. Lanett Corp. Joe F. Adams Company F 167th Infantry Pvt. George Alexander Company E 167th Infantry Pvt. Loyd Allen Company F 167th Infantry Pvt. Will T. Anderson Company C 106th Am. Train Pvt. Clyde Andrews Company B 3d Infantry Pvt. Chas. H. Barnett Battery C 6th Field Artillery Corp. Harry Bachelor Company F 167th Infantry Pvt. Claude Barnett Bakery Co. 357 Sailor George Bankston U.S.S. Rhode Island Pvt. Jesse Berry Company C 106th Am. Train Pvt. Earl Beal Battery F 53d Artillery C.A.C. Pvt. Edgar Blakely Medical Corps
- 72. Sgt. James Blackmon 19th Division Supply Train Corp. Mark B. Blackmon Company C 106th Am. Train Pvt. Willie H. Brewer Company G 2d Training Reg. Pvt. Earnest G. Brewster Company 39 157th Depot Brigade Pvt. Eddie E. Buchannan 1st Company 1st Army Corps School Det. Sgt. Thos. H. Cason Company C 106th Am. Train Pvt. George Caldwell Company B 324th Infantry Pvt. Merritt E. Carlisle Company L 327th Infantry Corp. Henry Carlisle Battery E 21st Field Artillery Sgt. Jno. G. Chapman Quartermaster Corps Pvt. T. G. Clements 2d Provisional Depot Battalion Sgt. Maj. Guy Coffee Hdqtrs. Company 384th Infantry
- 73. Tipton Coffee Y. M. C. A. Wendell Coffee Ph. M.1 U.S.S. Kentucky Sgt. Ewell Coffee Company B 17th Engineers Corp. Harvey R. Collins Company B 6th Repl. Reg. Inf. Pvt. A. Fennimore Cox Company F 167th Infantry Pvt. Jesse W. Coleman Company B 151st Mach. Gun Btn. Pvt. Hoyt Crowder 3d Company Developing Btn. Corp. Lester D. Crowder Company F 167th Infantry Cook O. W. Culpepper Company I M.T.C.R.U. 307 Pvt. Leroy Daniel Hdqtrs. Company 167th Infantry Pvt. Elijah Daniel 6th Company Development Btn. Pvt. Robert Dailey Battery E 117th Field Artillery
- 74. Pvt. Winfred L. Deloach Battery C 7th Field Artillery Pvt. Huburt Denham Battery D 117th Field Artillery Pvt. Radney Dobson Company H 161st Infantry Pvt. Gay Dunn Company B 48th Mach. Gun Btn. Pvt. A. E. Fincher 2d Provisional R.R.C. Pvt. George Fincher Company B 359th Infantry Pvt. Isac Free Mach. Gun Company 167th Infantry Pvt. William E. Freeman Company F 167th Infantry Pvt. Wesley Foster Company F 167th Infantry Pvt. Will H. Gill Company C 321st Infantry Corp. Tolbert H. Gray Company F 167th Infantry Corp. Ben W. Griffeth Company B 34th Engineers
- 75. Pvt. Allie Griffin Company E 123d Infantry Pvt. J. B. Grier Company G 321st Infantry Pvt. Alver Gunn Company E 7th Engineers Pvt. John B. Gunn Battery F 117th Field Artillery Pvt. Richard Hadaway Company E 167th Infantry Pvt. Brinton Hall Company H 161st Infantry Sgt. Will H. Hammock 20th Company 156th Depot Brigade Pvt. Robert Hammock 65th Company 6th Group M.T.D. Pvt. L. Clyde Harmon Bakery Co. 326 Pvt. Grady Harmon Company 7 Infantry Repl. Unit Pvt. Hobson H. Harmon Supply Battery 56th Field Artillery Pvt. Phillip H. Heard Company D 66th Engineers
- 76. Sgt. James Heard Company A 59th Engineers Roland Shaefer Heard Yeoman 3 c. 8 U.S. Navy Yard Charleston, S.C. Corp. Buford Heggood 118th Infantry Band 59th Brigade Pvt. Hobson Heggood Post Military Band Edgewood Arsenal Pvt. F. M. Heggood 118th Infantry Band Pvt. Emmit Henderson Company G 165th Infantry Corp. S. Calloway Herring Company F 167th Infantry Pvt. Charles Frank Hill Battery C 3d Field Artillery Corp. John J. Seymore Company C 106th Am. Train Musc. David Holloway 167th Infantry Band Pvt. Minor Hood Company D 106th Am. Train Pvt. Jack Howard Company 17 5th Reg. U.S. Marine Corps
- 77. Pvt. Jno. M. Howarth S.A.T.C. Auburn, Ala. Pvt. Reuben J. Jennings S.A.T.C. Marion Inst. Pvt. John Johnson Company A 106th Engineers Sgt. Frank P. Jones Company F 167th Infantry Pvt. Oscar King Company C 54th Mach. Gun Btn. Pvt. Belah King 5th Company Coast Artillery Pvt. Marion W. Knight Quartermaster Corps Pvt. Joe W. Knight Marine Guard Naval Radio Station Pvt. John C. Leonard Casual Co. 63 162d Depot Brigade Pvt. Hobson Lewis Company E 3d Infantry Pvt. Evans McGhee Company C 3d Infantry Pvt. Gip. L. McGhee 23d. Infantry
- 78. Corp. James McGlon Company H 167th Infantry Pvt. Jesse McGlon 64th Engineers R.O.T. Pvt. Curtis McNaron Company L 115th U.S.G.N.A. Pvt. Brant F. Maguire 13th Company 5th Platoon Pvt. J. T. Manley Battery D 117th Field Artillery Pvt. Luther Martin 39th Company 10th Training Btn. 157th Depot Brigade Pvt. Earnest R. Mitchell Hdqtrs. Company 152d Depot Brigade Pvt. Lofton Mitchell Company E 106th Am. Train Pvt. Cluster Morgan Company M 70th Infantry Pvt. Edd L. Newby Company F 167th Infantry Pvt. Walter Newsome Company A 168th Infantry Corp. Eugene Oliver Company H 167th Infantry
- 79. Pvt. Calvin Parker Company F 167th Infantry Pvt. Henry M. Parker Quartermaster Corps Sgt. Watson Phillips Quartermaster Corps Sgt. George C. Pryor Medical Dept. 6th Engineers Corp. William C. Raines Headquarters Band 116th Field Artillery Pvt. Willie Rogers Company A 321st Infantry Pvt. Charles E. Sanders Motor Truck Co. 332 Pvt. Charles Sedinger Company D 6th Infantry Pvt. Jimmie Seymour Company A 101st Infantry Pvt. Thomas M. Simms Company E 307th Engineers Pvt. Grady Smith Medical Dept. 157th Depot Brigade Pvt. Joe Smith Company F 167th Infantry
- 80. Pvt. Ollie Smith Company C 321st Mach. Gun Btn. Pvt. John W. Stewart Company H 43d Infantry Sgt. James Stearns Battery C 117th Field Artillery Pvt. Harvey D. Stephens Company C 321st Mach. Gun Btn. Corp. Eugene Stiff Company G 122d Infantry Pvt. Charles Tally Hdqtrs. Troops 314th Cavalry Horseshoer Thomas Tally Battery D 117th Field Artillery Pvt. Lomas Thomaston Company A 1st Infantry Regl. and Trn. Btn. Corp. Thomas Thomaston Company F 167th Infantry Pvt. Hugh Turner Company D 19th Btn. U.S.G.N.A. Pvt. James Ward Company F 167th Infantry Corp. Quincer W. Whittle Company B 116th Supply Train
- 81. Pvt. Ocie T. Wilbanks Company E 20th Engineers Pvt. Colvin Wilbanks 71st Company 6th Group M.T.D. Pvt. Robert Williams Company F 167th Infantry Sgt. Jesse Von Williams Company F 167th Infantry Sailor Charles Winningham U.S.S. Camden Detail League Island Navy Yard Charles H. Yarbrough Ph. M.3 Bay Ridge Rec. Ship Pvt. Dan H. Hart Company H 123d Infantry Pvt. Carl Smith Company H 123d Infantry Corp. William D. Purcell Company A 306 Ammunition Train Pvt. Walter Geter Company 21 R.R.D. Pvt. Chester D. May Company F 167th Infantry Corp. Eugene Herring Company C 106th Am. Train
- 82. Pvt. Robert Hollis Company K 16th Infantry Pvt. James E. Robinson 8th Field Artillery Pvt. Hobson Cummings S.A.T.C. Auburn, Ala. Pvt. Walter Peppers Company 39 New Receiving Camp Pvt. Jim B. Morris Hdqtrs. Company 115th Field Artillery Roll of Honor ‡ Killed in action † Died of disease * Photo *Adams, J. F. Allen, Marshall Alexander, Ben *Alexander, George *Allen, Loyd *Anderson, Will †*Andrews, Clyde Andrews, J. C. Aughtman, John *Hill, Charles Frank Hill, Charlie *Hollis, Robert *Holloway, David *Hood, Minor *Howard, Jack *Howarth, John M. Jenkins, Hamp *Jennings, Rube J. *Johnson, John
- 83. ‡*Bachelor, Harry Baker, William *Bankston, George Barnett, Claude Barnett, Charles H. Barton, Tebe *Beal, Earl *Berry, Jesse *Blackmon, James *Blackmon, Mark *Blakely, Edgar Boggs, James G. Bowling, I. L. *Brewer, Willie H. Brewster, Earnest G. Brown, Jesse Brumaloe, C. C. *Buchannan, Edward E. *Caldwell, George *Carlisle, Henry ‡*Carlisle, Merritt Carmichael, George Carmichael, Jim *Cason, Thomas *Chapman, John *Clements, T. G. *Coffee, Ewell *Coffee, Guy *Coffee, Tipton *Coffee, Wendell *Coleman, J. W. *Collins, Harvey R. ‡*Cox, Fennimore *Crowder, Hoyt ‡*Crowder, Lester D. *Culpepper, Orein W. Cummings, Hobson *Dailey, Robert *Daniel, Elijah *Daniel, Leroy *Deloach, Winfred L. *Denham, Huburt *Dobson, Radney *Dunn, Lonnie G. East, Albert *Jones, Frank P. Kendrick, John *King, Belah *King, Oscar *Knight, Marion *Knight, Joe Knight, Horace Kynard, O. D. *Leonard, John C. *Lewis, Hobson J. Lewis, Edd Manning, E. Martin, Clarence *May, Chester D. *Mitchell, Earnest *Mitchell, Lofton *Morgan, Cluster *Morris, Jim B. *Maguire, Brant F. *Manley, J. T. *Martin, Luther *McGhee, Evans McGhee, Gip L. *McGlon, Jesse *McGlon, James *McNaron, Curtis Neese, Kenny *Newby, Edd L. *Newsome, Walter Norman, Raemon *Oliver, Eugene *Parker, Calvin *Parker, Mose Henry Peppers, Walter *Phillips, Watson *Pryor, George C. *Purcell, William D. *Raines, William C. Robinson, James E. Robinson, Oscar *Rogers, William
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