Complexity and Probability of Some Boolean Formulas

TLDR: The construction of a probabilistic distribution on formulas in the basis of∧, [oplus ] 1 in some given set of n variables and of size at most [lscr ](k)=4k is a step towards developing a model appropriate for investigation of the properties of a typical (random) Boolean function of some given complexity.

Abstract: For any Boolean function f, let L(f) be its formula size complexity in the basis {∧, ⊕ 1}. For every n and every k≤n/2, we describe a probabilistic distribution on formulas in the basis {∧, ⊕ 1} in some given set of n variables and of size at most l(k)=4k. Let pn,k(f) be the probability that the formula chosen from the distribution computes the function f. For every function f with L(f)≤l(k)α, where α=log4(3/2), we have pn,k(f)>0. Moreover, for every function f, if pn,k(f)>0, then***** Insert equation here *****where c>1 is an absolute constant. Although the upper and lower bounds are exponentially small in l(k), they are quasi-polynomially related whenever l(k)≥lnΩ(1)n. The construction is a step towards developing a model appropriate for investigation of the properties of a typical (random) Boolean function of some given complexity.

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Complexity and Probability of Some Boolean Formulas
Savick
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1997
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INSTITUTE OF COMPUTER SCIENCE
ACADEMY OF SCIENCES OF THE CZECH REPUBLIC
Complexity and Probability of some Bo olean
Formulas
Petr Savicky
Technical rep ort No. 679
October 1997
Institute of Computer Science, Academy of Sciences of the Czech Republic
Pod vo drenskou v 2, 182 07 Prague 8, Czech Republic
phone: (+4202) 66 05 3690 fax: (+4202) 85 85 789
e-mail: savicky@uivt.cas.cz

INSTITUTE OF COMPUTER SCIENCE
ACADEMY OF SCIENCES OF THE CZECH REPUBLIC
Complexity and Probability of some Bo olean
Formulas
Petr Savicky
1
Technical rep ort No. 679
October 1997
Abstract
For any Boolean function
f
let
L
(
f
) b e its formula size complexity in the basis
f^



1
g
.
For every
n
and every
k

n=
2, we describ e a probabilistic distribution on formulas in
the basis
f^



1
g
in some given set of
n
variables and of the size at most
`
(
k
)=4
k
.
Let
p
nk
(
f
) be the probability that the formula chosen from the distribution computes
the function
f
.For every function
f
with
L
(
f
)

`
(
k
)

, where

=log
4
(3
=
2), wehave
p
nk
(
f
)
>
0. Moreover, for every function
f
,if
p
nk
(
f
)
>
0, then
(4
n
)
;
`
(
k
)

p
nk
(
f
)

c
;
`
(
k
)
1
=
4

where
c>
1 is an absolute constant. Although the upp er and lower b ounds are ex-
ponentially small in
`
(
k
), they are quasipolynomially related whenever
`
(
k
)

ln
(1)
n
.
The construction is a step towards developping a mo del appropriate for investigation
of the prop erties of a typical (random) Bo olean function of some given complexity.
Keywords
Complexity, probability, Bo olean formulas
1
This researchwas supported by GA CR, Grant No. 201/95/0976, and by Heinrich-Hertz-Stiftung
while visiting Universitat Dortmund, FB Informatik, LS2.

1 Intro duction
Probabilistic methods app ear to b e very powerful in combinatorics and computer sci-
ence. A natural p oint of view on these metho ds is that weinvestigate the prop erties
ofatypical ob ject chosen from a set. One of the very rst facts proven on Bo olean
functions is that a typical Bo olean function chosen from the set of all functions has
exponential complexityin any reasonable computation mo del. In particular, for the
Boolean formulas the result may be found in 5]. Hence, the prop erties of functions
chosen from the set of all functions cannot saymuch ab out functions of mo derate
complexity.
In this situation, it is natural to ask what are the typical prop erties of functions
chosen among the functions of some given complexity rather than among all functions.
One p ossibility to construct a probabilistic distribution on functions of limited com-
plexity is to describe the distribution in terms of their representations. In this case, it
is easy to guarantee the complexity b ound just using only representations of an appro-
priate size. However, if the distribution is dened only in terms of syntactic prop erties
of the representations, it may easily b e the case that the distribution is concentrated
on a small set of functions, e.g. on the twoconstant functions.
In the presentpaper a syntactically dened probabilistic model of Bo olean formu-
las is describ ed. The mo del is constructed by iterating the 4-ary Bo olean operation
x
1
x
2

x
3

x
4
starting from a simple distribution on variables, their negations and
the constants. After
k
iterations, the mo del generates a distribution on functions of
the formula size complexity in the basis
f^



1
g
bounded by
`
(
k
)=4
k
. The set of
functions having nonzero probabilitycontains all functions of complexityat most
`
(
k
)

,
where

=log
4
(3
=
2). An upp er b ound on the probabilityofeach of the functions with
a positive probability is given. The upp er b ound is quasip olynomially related to a
trivial lower b ound on this probability (Theorem 3.2). It follows that the distribution
is not concentrated on any small set of functions.
The result is proved for a particular case of the model investigated in 6] and 7]. For
this particular case, comparing to the b ounds from 6] and 7], much stronger b ounds
on the probability of single functions are obtained.
A similar mo del based on balanced formulas build up from the NAND op eration
(or equivalently from alternating levels of ANDs and ORs) and with randomly cho-
sen literals was suggested byFriedmann 2] in order to get information on Boolean
complexity.Friedman suggested to study the distributions using their moments and
presented an application of this metho d to iterated AND, namely to random 1-SAT
and random 2-SAT.
Formulas with a xed tree of connectives and with the leaves assigned to variables
or some other simple functions at random were used also for some other more specic
purposes. Let us mention the construction of a monotone formula of size
O
(
n
5
:
3
)
presented in 10 ] and the proof of existence of e.g. Ramsey graphs on 2
n
vertices,
whose adjacency matrix is representable by a Bo olean formula of p olynomial size in
n
,
see 4], 8].
A dierent mo del of random Bo olean formulas based on the uniform distribution
on all AND/OR formulas of size tending to innitywas investigated in 3]. It is proved
1

that the distributions on functions obtained in this wayconverge to a limit distribution,
in which the probabilityof every function
f
is positive and related to the complexityof
f
as follows. If
L
0
(
f
)

(
n
3
), then the probability
p
(
f
)of
f
in the limit distribution
satises
(8
n
)
;
L
0
(
f
)
;
2

p
(
f
)

c
;
L
0
(
f
)
=n
3
1

where
c
1
>
1 is an absolute constantand
L
0
(
f
)istheformula size complexityof
f
in
the basis
f^

_

:g
. The existence of a limit distribution with all probabilities p ositive
was investigated also for a more general mo del of random trees, see 12].
The number
B
(
n `
) of distinct Bo olean functions of
n
variables expressible byan
AND/OR form
ula of size at most
`
is estimated in 9]. In a wide range of the values
of
`
, matching lower and upp er b ound on
B
(
n `
)isproved. Namely,if both

(
n
) and

(
n
) tend to innitywith
n
and

(
n
)

`

2
n
=n

(
n
)
, then
B
(
n `
)=((
c
2
;
o
(1))
n
)
`
,
where
c
2
=2
=
(ln 4
;
1).
2 The probabilitymodel
Let
n

2 b e a xed natural numb er throughout the pap er. The Bo olean functions of
n
variables are the functions
f
0

1
g
n
!f
0

1
g
.Since
n
is xed, we call them simply
Boolean functions. The pro jection functions are denoted
x
i
for
i
=1
:::n
as usual.
The negation of
x
i
is denoted as 
x
i
. The conjunction is denoted likethemultiplication,
i.e. without any op eration symbol. Recall that

is the addition mo d 2.
For any Bo olean function
u
let
u
;
1
(1) be the set of
a
2f
0

1
g
n
for which
u
(
a
)=1.
Moreover, let
j
u
j
=
j
u
;
1
(1)
j
.For arbitrary Bo olean functions
u v
let
h
u v
i
=
M
a
2f
0

1
g
n
u
(
a
)
v
(
a
)
:
If
A
f
0

1
g
n
and
g
is a Bo olean function, we denote as
g
j
A
the restriction of
g
to the
set
A
.Let
X
A
be the characteristic function of
A
.
For an
y nonconstant function
f
,let
L
(
f
) b e the formula size complexityof
f
in the
basis
f^



1
g
, i.e. the minimum number of o ccurrences of the variables in a formula
expressing
f
in the given basis. Moreover, let
L
(
f
)=1, if
f
is a constant function.
The probability distributions studied in the present pap er are dened as follows.
Denition 2.1
Let ~
g
n
0
2f
0

1
x
1
:::x
n


x
1


x
2
:::

x
n
g
be a random Bo olean function
such that Pr( ~
g
n
0
= 0) = Pr( ~
g
n
0
=1)=1
=
4 and each of the literals occurrs as ~
g
n
0
with
probability1
=
(4
n
). For every
k

0let ~
g
nk
+1
=~
g
nk
1
~
g
nk
2

~
g
nk
3

~
g
nk
4
, where ~
g
nkj
are independent realizations of ~
g
nk
. Finally,for every
k

0, let
p
nk
(
f
)=Pr(
f
=~
g
nk
).
For the purp ose of the present paper, ~
g
nk
is dened to be a Bo olean function.
Clearly, the denition of this function implicitly describ es a Boolean formula expressing
~
g
nk
, which contains
`
(
k
)=4
k
occurrences of variables, their negations and constants.
Hence,
p
nk
(
f
)
>
0 implies
L
(
f
)

`
(
k
).
The distribution is chosen so that if
a
and 
a
are complementary points in
f
0

1
g
n
, i.e.
they have the Hamming distance
n
,then ~
g
n
0
(
a
)and ~
g
n
0
(
a
) are indep endent random
2

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Complexity and probability of some boolean formulas" ?

For every n and every k n the authors describe a probabilistic distribution on formulas in the basis f g in some given set of n variables and of the size at most k k Let pn k f be the probability that the formula chosen from the distribution computes the function f 

Q2. what is the rst fact on boolean functions?

One of the very rst facts proven on Boolean functions is that a typical Boolean function chosen from the set of all functions has exponential complexity in any reasonable computation model 

Q3. what is the constant of the Boolean cube?

Then every point of the Boolean cube is within Hamming distance at most n from A HencejA j bn cX j n jnBy using the estimatedX j n jnd d n n d n d ne d dfor d bn c and using the fact that the estimate is even larger with d n the authors obtainjA j n e n n Let K log K where K is the constant from Theorem Clearly for every nonempty A A the authors have K log n A K log n A K 

Q4. What is the induction hypothesis for v v w?

The induction hypothesis says that for all nonzero v v w and every s log n w m the authors have j s v j qjvj mIf jvj m then and the induction hypothesis with s r imply j r v j j s v j j s v j q jvj m 

Q5. How is the probability of each function of a given set of variables estimated?

The existence of a limit distribution with all probabilities positive was investigated also for a more general model of random trees seeThe number B n of distinct Boolean functions of n variables expressible by an AND OR formula of size at most is estimated in 

Q6. What is the probability of gn f?

Let gn f x xn x x xng be a random Boolean function such that Pr gn Pr gn and each of the literals occurrs as gn with probability n 

Q7. what is the probability of a function appearing as a realization of gn?

There exists a constant K such that for every n every nonempty subset A f gn every f A f g and every k k A log jAj log n A K the authors have Pr gkjA f jAj jAj k k ANote that the number of di erent functions which may appear as a realization of gn k does not exceed nk Hence for k log jAj log log n not every function f A f g has a positive probability 

Q8. How is the distribution of functions related to the complexity of f?

It is provedthat the distributions on functions obtained in this way converge to a limit distribution in which the probability of every function f is positive and related to the complexity of f as follows 

Q9. how is the distribution of gk determined?

It is very natural to present the proof with such a general value of this parameter although the theorem is nally proved by setting pTheorem is proved at the end of this section as a consequence of an upper bound on the Fourier coe cients of the distribution of gk The upper bound will have the form j gk w j qjwj k rm for all w w XA and all k r where a real number q and integers r and m are appropriately chosen Extending an estimate in this form from any k r to k instead of k is guaranteed by Lemma on the assumption that m is large enough 

Q10. what is the upper bound on d f?

This bound implies the theorem since L f k log is equivalent to log L f kIf L f then f is a constant a variable or negation of a variable Hence D f Let L f and let the upper bound on D f be true for all functions of complexity less than L f The authors will nd functions f f and f such that f f f f and L fj L f for j 

Q11. what is the simplest way to guarantee the complexity bound?

In this case it is easy to guarantee the complexity bound just using only representations of an appro priate size However if the distribution is de ned only in terms of syntactic properties of the representations it may easily be the case that the distribution is concentrated on a small set of functions e g on the two constant functions