Some results on replacement rules in monotone Boolean networks

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Original citation:

Dunne, P. E. (1984) Some results on replacement rules in monotone Boolean networks. University of Warwick. Department of Computer Science. (Department of Computer Science Research Report). (Unpublished) CS-RR-064

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The

Univer-sity

of

Warwick

THEORY

OF

COMPUTATION

REPORT

Some Resulrs

0{

RepLqc$.1EMt-

Ruus

rr,t

lhlororc

Boor-enr'r NerwoRr<s

No

64

January 1984

Paul

E Dunne

University

of

Warwick

Dep artmeDt

of

Conputer Science

Some

Results

On

Replacement

Rules

In

ltronotone

Boolean

Networks

Panl, E. Dutt'n'e

Department Of Computer Science

University Of Warwick

Coventry CV4'lAL

Great Britain

1.

Introduction

ReplaceraenL n:les were introcluced by Paterson _[4] and Mehlhorn _[3] and used to prove Light Iower bound.s on the monotone network

complefty

of

boolean matrix multiplieation. Th.e results applied prove that in networks

com-puting boolean matrix product, gates computing certain functions may be

replaced by the eonstants 0 or 1 or by an input of the network. Applications of

replacement rules to otber functions have been made by TTeiss _16] for boolean

convolution anil Dunne _{[2] for} threshold functions.

In this paper we investigate the f ollowing problem:

(P1) Glven a pair of boolean functions/:[0,1J"->[0,1J, ir:[0,11D->[0,1l what are the

monotone boolean functions

h

such that for any monotone network S

com-puLi::g

_/,

containing a gate u which computes the function 9 , 5lPrs(-t);=b

t11

computes

_/?

We prove the folloning results:

RZ) For any pair of funclioBs

_t,

g we determ-ine closed lorm expressions for:

(i) min s such tbat

g

is replaceable by s in a network comFuting _.f

(ii) max s sucb that

g

is replaceable by

s

in a network compuling

_/

R3) For any pair of functions

_t,g

we determine closed form expressions for:

(i) min s such tbat s is replaceable by

g

in a Detwork computir:g

_/

(ii) max

s

sucb that s is replaceable by

g

in a network computing

/

Using (RZ) we obtain a complete solution for (P1).

The remainder of this paper is organised as follows. Below we give basic

definitions and the notation used. In SecUon(Z) we derive (R1), (RZ) and (R3) above. ln Section(S) we extend our results to multiple-output functions and illus-trate some applications. ln Section(4) we prove the existence of

"pseuclo-complemenLs" for aII single-valued monotone boolean functions and

character-i-se lhese.

Definition 1

A rrLottotone netunrk S is a directed acyclic graph with 2 distinguished sets

ol nodes: X(the inpr.its of S) is the set of norles with in-degree 0. These nodes are

labelled ruith members of the set _{[t1,...,r,, J.} G is the remsinihg set of nodes,

wbich have in-degree 2 (the gdtes of S). Gates are labeUed by

n

or w (Boolean

conjunction and disjunction).

-o-Detrnition ?

If S is a monotone network and u is a node of S ,?.AS(u) is the boolean

func-Uon recursively defineil by:

q

iJ u is input

u;

of S

Rf,s(u) =

_|

.Rrs(uJ

^.RES(uz) iJ

uis

Br

^

gare

I

r?n',S(u) v-RE,S(ue) if u is an

v

gate

where ur and u9 are the inputs of uiJ

uis

a

gate.

tl

cl

Definition 3

,f(!

<

g(x)

<=>

v

_{a € [0,11"t(d)=} 1 _{=1 9} (a)=1

I)efiniUon 4

A monom m is a function of the f orm:

IIr ₌ zl

_{r^ltte/\.../\,,ri{}

tij

€X

A monom m,is an *npl:i,cotzt of the monotoDe boolean function./' if and only

if

m<t.

m is a prime intgtti,cutt Lt:

1)

m is aa implic ant of

t

2)

w

n

such that m' < m',

B

is not an

inplicant

of

_/

Ilefnition

5

A clouse c is a function of the f orm;

C

_=:rit

w

tiz

w... v

q{

zi1

€X

A clause

cis

an itr:.pliog;nd, of tl1e honotone boolean function

_/

if and only

if

.f

.

c. c is

aprime

clanlsa X:

1)

c is an implicand of

/

2)

w c- such that

e

< c,

s

is not an i-rrplieanct of

/,

Notation

PI(Jf)

_{= [m}

_I

mis

a prime implicant of

I

J

PC(Jf)

_{= {c}

_{lcisaprimeclauseolJ}

_}

-5-2.

Results

Def.uition 6

Let/:[0,1]L>[0,11. g :[0,1JD->[0,1J and h:[0,1JL>{o,1J be any monotone

boolean functions.

g

Ls h+efl.o.ceahle ui.threryect to ₇

k \

n)

n:

Sliss{ ):=h still computes

/

for any monotone network S computing

_/

which contains a node u with

nf,s(u)=s tr

The following result is the replacement rule due to Mehlhorn _[3].

Icmma

1

s

{

oiff:

v

m€PI(g)

-3

d

suchthat

L.J

'We _{sball now cbaracterise the largest}

O-replaceable and smallest 1-replaceable functions

vith

respect to any

functiont.

Although the results are

implieil by Theorem(3) below, we shall present these cases separately as the

analysis is clearer, The characterisations use tJre representation of

/

is

a

con-junctioD of prime clauses (Coqjunctive Normal Form) or as a ilisjuDction of

prime implicants (Disjunctive Normal Form).

DefihitioD 7

LetJ

be a monotone boolean function over

_Irr,...,rDl.

Tlne

dtnl

of

/ f)

is

the monotone function:

7

₌

-(j

(

-"r,...,-r"))

where "

-"

deDotes negatioD.. r't

Ilefinition

I

Let m be a monom, c be a clause and

t

be a monotone boolean function.

lben:

1)

_X(m)

₌

v

_{[: In$zl}

2)

ocLt)

₌

-.t4ax(-)

-?-4)

ICU|)

₌

.€Xo,')rr(c)

tr

lbeorem

1(

O,l-replacements

₎

(A)

OC(t) is the unique largest Dreplaceable function with respect to

_/.

i.e

OCA) <

S

_=>

g

is not O-replaceable

w.r.t/.

(B)

IC(J) is the unique smallest 1-replaceable function'with respect

tojl'.

i.e.

ICV)

>

g

_=>

g is not 1-replaceable w.r.t

_t.

.koof

Let:

f

Dross

(/)

₌

wls

₁₉₁₀₁

lYe shall show that:

oc(J)

₌

-Doss(/)

(D

ol(f)

<Dross(/)

Suppose m,

<

OC(J), but that m _$ Dross

(/).

Then

tlere

is some monom m'

such that m

n

E- €

PI(t)

and so

mis

a sbottening of some prime implieant

r

of

/.

But:

By monot onicity

But m $

y(r)

if m is a shortening of r. CoDtraaliction.

(n) Dross

(1)

-

oc(J)

Let m € PI ( Dross

(/)

_), by Lern'na(1) there does not exist

d

such that

m.

n rn

e

PI(/).

Thus:

v

r€

PI(/)

m<X(r)

-=n.t{,Dx(n) <

ocv)

Thus;

OCA)

₌

Dross(/)

and (A) follows as Dross

(/)

is by defnition the largest Freplaceable function witb respect to

l.

(B)

It is easy to see tbat:

IC(J)

₌

Thus, by duality, JC(t) ts tJre unique inihirnal l-replaceable fi:ncUon with

respect to .f ,

'YIe _shall

_nor

_{consider non-eonstant replacements of the form.f :=s.}

-

_{and <Ietermine tninimurn anil lndaimum sotu[oos for} these,

oc6

-9-Definition

I

l,et

I[=[6r,...,rnrl

be a set ol monoms, and

let/

be a monotone boolean

function- TJ:e Prim.e- Irnpl:inwt Eztensinz of I[ with respect to

/

(mr(U)) is

.

deffneil to be:

my(II)

₌

_[pePI("f)

_{l=m,e llwithp<}

q

J

Th.e Pritne-d.anse Extensi.on ol a set of clauses C _{= [ cr,...,cl i} with respect

to/

(c&(c))

_{is eiven by:}

cE;(c)

_{= [p€Pc(/)}

_l=qe

cwitbq<

pl

A(J'g)

₌

-.nyo{ot)-BU

_,s)

₌

".

q,f;"q";;"

tr

lbeorem 2

Let

_/,

gr be monotone booleau functions. Then:

t) l|,il

is the minimal function

s

such that

₉

_{

s

2)

.B(/,9) is the maximal

s

such tJmt

₉

{

s

koof

1)

Certainly

g

is ,4 (f _{,9 )-replaceable} when computing

l,

as by deffnition

IFy( pt(C) _{) is} tJre set of all prime implicants of

/

towhichg

could be

extended. So suppose some turction

s.

exists, also satisf5ring these

require-ments and

that,4(/,9)

_{$ s} There must be some prime implicant of

l(f

,g), p

say, which is not an implicant of s . Now:

p €

PI(t)

atr.l

!

m € PI(g) such that

p<

m

So; PI(g

^

p) n

PI(f)

=

lpJ

But; PI(s

n

ps)

n

ntf) =

_$

as

p{

s

CoatradicUon, as g is not replaceable by s wheD computilg

_/,

Thus

A(/,g)

is a minimal function. lf e now establisb uniqueDess. Suppose sr, sB are ilistinct i.e incomparable minirnal funetions. 'Ihen:

Pl(gnh)nPl(/) c

PI(s

ns1

n h)

n

PI(/)

c

PI(g n

st

n

sz n

h)

n

PI(/)

asg

n

sr

^h

<,

Thus s1

n

ss (< s1,s2) is also a suitable, but smaller funetion. Contrad.iction

l'1

Codfary

2.1)

C

!nr"ndonlyif:

D,efinition 1O

PI-*(J,c)

₌ PI(.f)-IE;( PI(g) )

Pc".-f

,c) =

Pc(/)-trr'(

Pc(s) )

E(f ,g) =

_{_.4ro/(d}

DV,s)

₌

_*rq:rr,!r(.)

tr

-I

1)

EA'c)

I

c

By Cor(z.1) we need only show:

eA,eA,cD<s

<EAg)

(i)

_s

<

E(J,c)

l,etp<g.Thenby

dennition ot

PI,.-(f

,g ):

-3n

such tbat p

n rL

€

PI--(.f

,S ₎

Ttrus:

w m e

Pl--(/,g

)

p<

_X(-)

By the definition of -O(/,9) this implies the result.

(n)

AU,EU:,s))-s

let: pePI(,a(J,rf

,s)) ). Now:

rer(n@VgD

) n

Iry( pl-"-(.f,s)

)

=

{l

Ttru6 either p e PI(/), in which case p is a lengthening of some

pribe

tmplicant m of

g

or p=0, In both cases

p<

g.

e)

.6'(/,9 ₎ is maximal

f

Suppose

u*

E(.f ,c)

anttu!

g.'Iben

3 p e PI(u) sucb that

p$

.8(f ,g) thus: E(f ,9) < X(P) _".ct therefore,

-13-It'\r

_{= r}

€

PI(t)

gr\t*r(Asr€IL-(f,g))

.

Ilrus

uis

not g-replaceable rrith respect

tof;

Contratliction-

El

Corotlary A1)

n

_I

_s

ilaad only if:

o{nr,lo(c)

=

n =

_-o4;o,o,*{-)

tr

Beynoa _[1] has considerecl a concept of "computational equivaleDce" within

a difierent framework. g is said to be equivaleDt to h when computinC

t

(

l.

,.

t.

C H

h)

if and only

Ilg

_lh

anat

h

_=l

g.

In this context Cor(z.f) and Cor(3.1)

yield;

lheorem 4

Obvious

-15-3. Ilultiple

Ou[rut F]rnctirrn*

Let F ₌

lfr,...l-

J be any set of m monotone boolean functions over

X

.

theoren

S

oc(F)

_=,4

oc(/t

rc(n

=

_J,

t"tfrl

l(F,e)

₌

wrrr

(rrk))

B(F,g)

₌

ncr*

(Pc(g))

E(F,s)

₌

_L

eA',s)

D(F,s)

₌

_P,

a(J',s)

where;

As an illustration we reprove the replacement rule due to Paterson _{[4] for}

81P

_{Io,1lhE->[o,1j"?}

where each output

ct'

is defined by:

'

cv

₌

,-Y=, (z* ^Uri ) l<a

I.ernrna

g

(Paterson _[+])

Let BMP _{= I}

crr,.,,,cr-

_l, where cV is as defined above. Let 1<

i,i<

n (i

*

i)

andl<j,j<D(j#il.

BP

3.1)

cp

w

zr1

₌

_|

1

BAP 3.2) yX w

yo'

₌

_|

t

BITP 3.3)

zn

w

yr{

₌

_|

r

koof

.IC(cir)

₌

OC("ir-)

=

OC(

_,4*

(a1 w _{ya) )}

IC( BMP _{) =}

_,;"

(

n

z*y,"

cr, ₎

-11

-It is easy to see that for each of the function s in (3.1)-(3.3):

IC(BMP)

vs

₌

s =>

IC(BMP)

<

s

TI

A k-slice function of

/

is a function of the form:

. t

t\TPvTP.r

The following is due to ltegener[5].

Lemma4

If g is a k-slice firnction then:

vrr€X

_{&t -} I,_| z5

n

Tf($

hoof

Easily deriveil from Theorem(Z)

above,

tr

Gorollary

If

g

is a k-slice function then:

E.

koof

tr)

-

rv-4,

Replacement Rules lc Pseuilo.f.omplements

Defnition

11

Let;f

be a monotone boolean firnction over

X

L pseudo-com4tlernzr* for

zlis

a monotone boolean tunction hi, such that in any (,^,v, -,)-network T computing

/,

in which negation is applied only to the iDputs, any instance of

-zi,

can be

replaced by hi and the resulting network will still compuLe

/.

theorem.6

w monotone jf , th is a pseudo" complement for zi if anil only if:

.fl"r'=e

₌

h,

<

fl"t='

Proof

Let/s

denote the function computed by T after some instance, z say, of

-ti

is replacect

by/le'o.

si-il*ly

let;f1 denote the function computed by T after

tlris instaoce, z, is replaced by

_tla=t.

No.o

"io" e

f

o

<

f

t

it

is sufficient to prove

that:

fr<I<Io

_"

.:

In terms of the irxtance z

of

-zi, T compules:

where the functions gopT are such that:

1)

vm€PI(

gdF?):

("t

_)"

(

-"i

_)P (z)7

n m

is a moDom computeil at T.

2) n

does not depend onzi

_,

-rl-

or z.

wbere:

CIe arly:

.f

=

gooo

v

5i9 _roo

w

-zr ( goro

v

_9or

v

gorr ₎

Now let z :=

_t

la=o so that

_t

'= _J6. lYe must prove

_/

..f0.

IYe need only show: -trtgoor

\/

-zigorr

1

Jo

-...--_---...

However;;la=o

^goor

I

goor and

-"r.f

la=o _gorr

=

-tr

gorl thus

/

< _/s. Now consider the replaceEent

,,=

flt'=',lYe must show tJmt

_/1

<

t

Simi-l,arly we need. only prove:

tlq=tgoo,

v

-rrtln=lgo' v

titlq=lg

lor

<

.r But:

tfr=t

r.,g.r

=

g _oot

<

|

-zr

nJll=t

n gorr

€

gorr

<.f

I

t

uo=o

(')o

₌

_I

-zt-z1

nTlt=t

^gm

<

f

Thus .f

r

<

J,

anil the theorem

follows,

tr)

Corollary 6.1)

tet

F _{= Lfr,...,.f-J} be a set of m monotone boolean functions. Then h; is a

pseuilo-complement for .zi if antl only if:

,Y/,'**<tli<*th=r

L'

'We _note

_tlat

_{for sets of monotone boolean functions, in general the interval}

of Corollary(6.1) is not well-defined. However for special cases, such as slice

functions, pseuilo-complements exist. For slice fu-nctions Theorem(6) and

5.

Relerences

[1]

Beynon, M

Replaceability anil computational equivalence in ffnite distributive lattices: Iheory Of Computation Report No.61, Univ. Of lYarwick, Mareh 1984.

[2]

Durne, P.E.

A 2.5n tower Bounil OnThe Monotone Network Complexity Of T$, Theory Ot Computation Report No.62, Univ Of Warwick, March 1984

[3]

Mehlhorn, K. & Galit, Z.

Monotone Switching Networks and Boolean Matrix Product, CompuUng (f6),

99-111, 19?6

[4]

Paterson" M.S.

Complexity Of Monotone Networks r.or Boolean Matrix Product, Theoretical Computer Science (1), 13-20, 19?5

[5]

1{egener, I

On The Complexity Of Slice-Functioos, Internal Report, Univ. Of Frankfurt, Jtrly 1983

[6]

Yieiss. J

A nve lower Bor:ncl On Ttre Boolean Convolution, Univ. Of Bielefeltl, West