The Univer-sity of
Warwick
THEORY OF COMPUTATION
No 64
REPORT
Some Resulrs
0{
RepLqc$.1EMt-
Ruus
rr,t
lhlororc
Boor-enr'r NerwoRr<s
Paul E Dunne
University of
Warwick
artmeDt of Conputer Science
Coveniry CV4 7AI
Dep
England
January
1984
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 compuLi::g
/,
computes
We
containing a gate u which computes the function 9 , 5lPrs(-t);=b t11
/?
prove the folloning results:
R1) For any function
/
we derive closed form exPressions for the maximal (F
replaceable and rninimal 1-replaceable functions with respect
tot.
-2RZ) 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 characteri-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).
tl
-oDetrnition ?
If
S is a
monotone network and u is a node of S ,?.AS(u) is the boolean func-
Uon recursively defineil by:
q
u is input u; of S
Rf,s(u) = | .Rrs(uJ
iJ uis Br
^.RES(uz)
I r?n',S(u) v-RE,S(ue) if u is an
iJ
where ur and u9 are the inputs of uiJ uis a gate.
^v
gare
gate
tl
Definition 3
,f(!
< g(x) <=>
v
a € [0,11"t(d)=
1
=1 9 (a)=1
cl
I)efiniUon 4
A monom m is a
function of the f orm:
IIr = zl r^ltte/\.../\,,ri{
A monom
tij €X
m,is an *npl:i,cotzt of the monotoDe boolean function./' if and only if
m<t. m is a prime intgtti,cutt
1)
m is aa implic ant
2)
w
T'l
n
of
Lt:
t
such that m' < m',
B is not an inplicant
of
/
-4Ilefnition 5
A clouse c is a function of the f orm;
C
=:rit
w
A clause
.f
.
tiz w... v q{
cis an
zi1
€X
itr:.pliog;nd, of tl1e
honotone boolean function
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 /,
cl
Notation
= [m I mis a prime implicant of I
PC(Jf) = {c lcisaprimeclauseolJ }
PI(Jf)
J
/
if and only if
-52. 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
wh g^h<l
=> h<l
-6L.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
where "
-"
= -(j ( -"r,...,-r"))
deDotes negatioD..
r't
Ilefinition
I
Let m be a monom, c be a clause and
lben:
1) X(m) = v [: In$zl
2)
ocLt) =
-.t4ax(-)
3) p(c) = n[c lz$cf
t
be a monotone boolean function.
-?-
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
(B) IC(J)
ICV)
w.r.t/.
is the unique smallest 1-replaceable function'with respect
>
g => g is not 1-replaceable
w.r.t
tojl'.
i.e.
t.
.koof
Let:
f
Dross
lYe shall show
(/) = wls 19101
that:
oc(J) = -Doss(/)
(D ol(f) <Dross(/)
Suppose m,
such that m
/.
<
OC(J), but that m $ Dross (/). Then
n E- € PI(t) and so mis
tlere is some monom m'
a sbottening of some prime implieant
But:
m<
By monot onicity
OCU|) =
n.ftU,,,X{f) =>
m<x(r)
r of
-8But m $ y(r) if m is a shortening of r. CoDtraaliction.
(n)
Dross (1)
-
oc(J)
Let m € PI ( Dross
m.
n rn e PI(/).
(/)
), by Lern'na(1) there does not exist
d
such that
Thus:
v r€
PI(/) m<X(r)
-=n.t{,Dx(n) < ocv)
Thus;
OCA)
and (A) follows as Dross
(/)
=
Dross(/)
is by defnition the largest Freplaceable function
witb respect to l.
(B) It is easy to see tbat:
IC(J)
=
oc6
Thus, by duality, JC(t) ts tJre unique inihirnal l-replaceable fi:ncUon with
respect to .f ,
I]
'YIe
-
shall nor consider non-eonstant replacements of the form.f :=s.
and <Ietermine tninimurn anil lndaimum sotu[oos for these,
-9Definition
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<
Th.e
to/
q
Pritne-d.anse Extensi.on ol a set of clauses C = [ cr,...,cl i with respect
(c&(c)) is eiven by:
cE;(c) = [p€Pc(/)
A(J'g) =
BU
,s) =
l=qe cwitbq< pl
-.nyo{ot)". q,f;"q";;"
tr
lbeorem 2
Let
J
/,
gr
be monotone booleau functions. Then:
t)
l|,il
2)
.B(/,9) is the maximal s such tJmt 9
is the minimal function s such that
{
9{ s
s
Note: Conventionally the empty monom ( clause ) is
r
( o ),
- IU.
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 !
So;
PI(g
^
p) n
But; PI(s n
m € PI(g) such that p< m
PI(f) =
ps)
n
lpJ
ntf) =
$
as
p{
s
CoatradicUon, as g is not replaceable by s wheD computilg
Thus
sB
A(/,g) is a minimal function.
/,
lf e now establisb uniqueDess. Suppose sr,
are ilistinct i.e incomparable minirnal funetions. 'Ihen:
Pl(gnh)nPl(/) c
c
PI(s ns1 n h) n PI(/)
PI(g n st n sz n h) n PI(/)
asg n sr
Thus s1
Z)
^h
<,
n ss (< s1,s2) is also a suitable, but smaller funetion. Contrad.iction
Duality
-
Codfary 2.1)
C
!nr"ndonlyif:
l'1
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
r_t
-
'12I
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
-3n
ot
PI,.-(f
,g
such tbat p
):
n rL € PI--(.f
,S
)
Ttrus:
wme
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
m of g or p=0, In both cases p< g.
e)
.6'(/,9 ) is maximal
Suppose
u*
f
E(.f ,c)
anttu! g.'Iben
3 p e PI(u) sucb
thus: E(f ,9) < X(P) ".ct therefore,
=
r e PI,'-(/,g ) such that X (r)<X (p)
So
r<
p. Tllrs:
that p$ .8(f ,g)
pribe tmplicant
-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)
nI 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
l.
CH
h) if and only Ilg
yield;
lheorem 4
c Hhtfr
,.
lh
anat
t
(
t.
h =l g. In this context Cor(z.f) and Cor(3.1)
koof
Obvious
t3
-153. 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
Boolean Matrix Multiplic ation. Let:
rule due to Paterson [4] for
- ro -
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<
andl<j,j<D(j#il.
BP
3.1) cp w
zr1 = |
3.2) yX w
yo' = | t
BAP
BITP
3.3)
1
zn w yr{ =
|
r
koof
.IC(cir)
=
OC("ir-)
=
OC(
,4* (a1 w ya) )
Thus:
IC( BMP ) =
( n z*y," cr, )
,;"
l<'i!
i,i<
n (i
* i)
-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
I,
&t - |
z5
n Tf($
hoof
Easily deriveil from Theorem(Z)
above, tr
Gorollary
If g is
a
k-slice function then:
z1
E.
2l
ziw
Tf.r(!
-18-
koof
Duality.
tr
-
4, Replacement Rules
Defnition
rv-
lc Pseuilo.f.omplements
11
Let;f be a monotone boolean firnction over X
a
/,
L pseudo-com4tlernzr*
for
zlis
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
/.
tr)
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
is replacect
by/le'o. si-il*ly
some instance, z say, of
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:
9om V :-i9 rO
v
-ti
-,Bt9Oro
V
Z$ggy
V
'ErW
Orr
w uizgtot
-20where the functions gopT are such that:
1) vm€PI( gdF?):
("t )"
is a moDom computeil at
2) n
(
-"i
)P (z)7
nm
T.
, -rl- or z.
does not depend onzi
wbere:
I t uo=o
(')o = I
Iz
CIe
if
6=r
arly:
.f = gooo v 5i9 roo w -zr ( goro v 9or v gorr )
Now
let z := t
la=o
so that
t
'= J6. lYe must prove / ..f0. IYe need only show:
-trtgoor
\/
-zigorr
1
Jo
-.....--_---...
However;;la=o
I
^goor
goor
and -"r.f
Now consider the replaceEent
la=o
gorr
= -tr gorl thus /
,,= flt'=',lYe must show
tJmt
l,arly we need. only prove:
tlq=tgoo,
v -rrtln=lgo' v titlq=lg lor < .r
But:
tfr=t r.,g.r =
-zr
g
oot
<
nJll=t n gorr €
|
gorr
<.f
<
/s.
/1 < t
Simi-
-ztz1
nTlt=t
Thus .f r
^gm < f
< J, anil the theorem follows,
tr)
Corollary 6.1)
tet
F = Lfr,...,.f-J be a set of m monotone boolean functions. Then h; is a
pseuilo-complement for .zi if antl only if:
,Y/,'**<tli<*th=r
L'
'We
note
tlat for
sets of monotone boolean functions, in general the interval
of Corollary(6.1) is not well-defined. However for special cases, such as slice
functions, pseuilo-complements exist. For slice fu-nctions Theorem(6) and Corollary(6.
t) combined yield the results of Berkoitritz (citeil in [5]) for transforning
combinational networks to monotone networks efiiciently,
-225. Relerences
[1]
Beynon, M
Replaceability anil computational equivalence in ffnite distributive lattices:
Iheory
[2]
Of
Computation Report No.61, Univ.
Of
lYarwick, Mareh 1984.
Durne, P.E.
A 2.5n tower Bounil OnThe Monotone Network Complexity Of T$, 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.
Germany, November 1982
Of
Bielefeltl, West