theory of - Department of Computer Science - University of Warwick

The University of Warwick

THEORY OF COMPUTATION REPORT N0,7

THE DEPTH OF ALL BOOLEAN FIJ N(:TIOr'IS

l,l,F, lJ|cC0LL |lI, S, PATERSON

DEPARIT'1ENT

IT{I\ERS

COVENTRY ENGI-AI{D

OF COPUTER SCIENCE

ITY OF WARI{ICr.

CV4 7AL

ATJGUST

I975

THE DEPTH OF ALL BOOLEAN FUNCTIONS

by

W"F. McCol

I and M"S.Paterson

of ComPuter Science University of Warwick, CoventrY

Department ENGLAND

Abstract It is shown that every Boolean function of n argumentshasacircuitofdepthn+]overthe basis {fl

f :

{0,.| }2

*

{0'1

}i.

l.

Lntroduction

ipira showed in If] that for any k >

On

there is a r'.uber N(k)

sucli that if n > N(k) then any n argument Boolean function has

a

circuit of denth n + log2lo82 .lo8rn.

LJpper trounds on depth f or

specific values o{ n, g-iven fiy

Prelaratil and Mul.ler [ 2] , are n

for

n\< I

n+l for

n- I and m is odd then in the hypothesis, take

BC

same expansion we mayr

by the incluctive

m-r-1 subfonmulae of depth to be a conjunction of 2

S*-2 .td a smaller subfonmula of depth S*-3. Since 6a is essentially

s*-., m2

conjunction of rn arguments and rr.< 2

a

s-

- ,-m-3, it

may be conjoine.

m=3 and the sequence a ut DJ unction

of

102

3

of the fonm:

x_/ x.,l-o.+r:).c) -x,.x- x^

1 4

I

Ieftmost subformula may be given in X- X^ | / t6

I'TN

2

X^ X.^ IY IIU

X

i

lrzl

t4

detaiL

x,t J-o

dD.

i+

x.

where the base functions associated with certain gates are not

defined if they depend on

D.

Each of the ha(RonRl_oR2) subfonmulae

x,f,

x^ rb

1"

:n6

of the form: x^x^

*2*,

1

tl il 4

I

FIG.

l

4

na(*o

where the leftmost subformula

oR1rR2 )

ID.

and the associated base functions depend on

C.

4.

Main Result

It remains to be

shown how

the approximation g to f over

can be used to conpu'te f. r ,...,R*p-r,

R,

are disjoint subsets of Xn. I'or all f(Xn),

Lemma. Suppose Rrr...r\

such that en(Xn) = f t O

there exist fr(Xn-Rr),...rfk(xn-\) is an approximation to

some

w.r.t.

function

{Rf

r... r\}

Proof. This is by induction on k. The lemma holds trivially Let k>0, and suppose the result is true for k-I.

t-t

for

k=0"

Then,

there exist f.|(Xn-Rr)r... rft -t(Xn-FU-r) such that for all j-, lt = 12r2r4r6

r8

tl-Or...>

so that each f. contains 2 fewer anguments than the previous one. The result is a scheme of dePth n+3.

For b consisting of one ^-type function and one v-type function we can obtain n+4 by taking the unate construction and conplementing subformul-ae as necessary"

Conjecture.

For any b 5 BZ, if ther"e is a scheme over b which covers B, then there is a constant c such that

for all n there is a which covers

scheme

over b of depth ntc

Brr.

Fon b = {p u q, p} we haver at presentn achieved no better

than n + 0(1o8rn).

Wemustdistinguishthenotionsofcompletebasesforforntrr};re and for schemes. For example, b = {NAND} is complete for formulae but obviously no singleton basis can be complete fon the condition on b given in the conjecture'

schenes

'

hence

6.

Conclusion We

have described a unifor"m scheme for expnessing all n-argument

Boofean functions in depth n+l, and have matched this upper bound with

a lower bound of n-l

under^

the nestriction of unifo::mity' For a basis

of unate functibns only, our upper bound is n+3' In OUr construction we used a

Sequence 1212 r3rq 151..

secuence which grows much faste:: could be used instead.

.)r

but

a

The effect

of the choice of sequence on formula size has not been consid'ered but n+l n- -I 2'-'2" and within size to possible the limit arguments easy counting for our: method. Lupanovrs construction [3] yields formufae of sjze about 2n/Log^n, though not of coul?se using schemes' This raises the 1_

iollowlng: Open problem

:

Doesalowenboundof||n-constant||ondepttistji]

hol-d when the restriction

to schemes is

removed?

13

l).ef

i 1l

|

?

I

erences

, t'On the time necessary to compute sw-itciring functionsrr, IEEE Tnans. Computer"s vol"c-20 (Jan.197]), 104-105" l,pira,

P.M.

l'reparata, f.P. and Muller, D.E. ' "0n the delay required lo:'ealise Bc-rofean functionsr', IEEE Trans" Computers vol.C-20 (Apr.I97r), 459-461.

L;l

Lupanov, 0oB., "Complexity of Formula Realisation of

I'unctions of Logical Algebra", Prob. cybernetics Vol.3 (1962), 7

B2-8

tr "