It is shown that every Boolean function of n arguments has a circuit of depth n+1 over the basis {f|f:{0,1}^2 -> {0,1}}

McColl, William Finlay

Paterson, Michael S.

English

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http://wrap.warwick.ac.uk/            Original citation: McColl, W. F. and Paterson, Michael S. (1975) The depth of all Boolean functions. Coventry, UK: Department of Computer Science. (Theory of Computation Report).  CS-RR-007  Permanent WRAP url: http://wrap.warwick.ac.uk/46304       Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions.  Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners.  To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.  Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge.  Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.  A note on versions: The version presented in WRAP is the published version or, version of record, and may be cited as it appears here.  For more information, please contact the WRAP Team at: publications@warwick.ac.uk  The University of WarwickTHEORY OFCOMPUTATIONREPORTN0,7THE DEPTH OF ALL BOOLEAN FIJ N(:TIOr'ISl,l,F, lJ|cC0LL|lI, S, PATERSONDEPARIT'1ENT OF COPUTER SCIENCEIT{I\ERS ITY OF WARI{ICr.COVENTRY CV4 7ALENGI-AI{DATJGUST I975THE DEPTH OF ALL BOOLEAN FUNCTIONSbyW"F. McCol I and M"S.PatersonDepartment of ComPuter ScienceUniversity of Warwick, CoventrYENGLANDAbstract It is shown that every Boolean function of nargumentshasacircuitofdepthn+]overthebasis {fl f : {0,.| }2 * {0'1 }i.l. Lntroductionipira 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 acircuit of denth n + log2lo82 .lo8rn.LJpper trounds on depth f or specific values o{ n, g-iven fiyPrelaratil and Mul.ler [ 2] , aren for n\< In+l for n-<28+B=264n+2 for ,t -a 2254 + 264er c.Whereas Knuth has showno by computer analysis, that therear:,e 4 .intiument Boolean fr.rnctions requiring depth 4.Iri this papere we describe a construction which yieids anupper bound of n+I for all values of n.2. Schemes' Our present constnuctions, and all pr:evious ones for minimizingdepth that we know of, have the property of being "uniform" for aj-'lfunctions of n arguments. The same directed graph with 1-he same:assignment of arguments to inputs is used for all the furrctionst"th::necessary variation being only in the assignment of base functionsto the nodes. Lupdnovts construction for minirnizing for:'mi.ifa size [ 3lis notable for escaPing this form.wc iorm.rlize this restriction ill our definition of "circuitschemet' and show that for Schenes our construction achieves theoptirlal, depth to within an addjtive constant.^_:dlluLiS ELets = {rlfnXn = (x .X-...,o' I'in forrnul-ae andLet C.. c B.. andrr- nbasis b if for each fto the gates of S suchshows a formula schemefrom the expansionf(xorx,ox,){o,r}}the set of fonmal anguments we shallr {Ort1t *,x-. ,) bert-acircuits.Definition. A qircui! sch.e.m-e is a connected acyclic directed g,raph inwhjch nodes have either in-deglee 2 (gates) in which case the pair ofincoming arcs are ondered, or else in-degnee 0 (inpr$ nodel) in whichcase an argument x. is assigned to the node. A fonmu_la @ is acircuit scheme in which all gates have out-degree at most one.b 3 Br. A circuit scheme S c.ov?:s Cn over. cr, ti"oe is an assignment of functions from bthat the resulting circuit computes f. Figure 1which covers B^ over basis B^. This fol-Iows3t= (*o ,r fr(xr rxr)) $ fo(xrrxr)whene 0 denotes sum modulo 2, fo(*lrx2) = f(Or"IrX2) andfa(xrrxr) = f (l '*Lrx2; o f(Ooxrrxr). We have ver:ified that this isthe unique formula scheme (to within obvious symmet::ies) with fewerthan five gates that covers B". Its depth of 3 is therefore optinai.We prove a lower bouncl on the clepth of schemes by a simple couilt-ingargunent.Tireorent l. Any circuit scheme which covers Br., over any basis h c ll.,has tlepttr .tt least n-l.T)Prool'. A scheme of depth D has at most 2"-l gates, and so by var'/irigthe assignment to gatcs trom b can cover a set of at most lllzD-rdifferent functions,Since ler., | = 22n we haveI62D-1 u, 22nwhich yields D >z n-lfiot-e th;it this arzument produces no better bognd even when lbl - 2,In the next two ssglions we describe the main result of tnepaper, a scheme of depth n-t-I to cover Bn over basis Br. The firststage is to produce anrrapproximation[ to such a scheme in depth n.il . J'iie approxirnation of depth n.Il C : (crr...r"kt. {uoLik and Y - <y'r...rY/, i';' slialiwrite Y : c for l. Yi = "i and o for (o,... rD.r-< f( kAnother'.rbbr:eviation will be to write f(Y) for^ f(ytr.."rYp),g(Y,Z ) for g(ytr. .. rYk ,zLr... ), etc-Definition. Given t = {Rtr... r\} where R, : Xn for all j 'and any f(Xn), we define g(Xn), the apProximation to f with respe-c.ttoSbyg(Xn) : O if lR. e S such that oj - 0= f(Xn) otherwisenVCe{0,}may be expnessed as a gigjjSgliyg. expansion-6c(z) n f(Y,c)]"^- 0 otherwiseDuallyr the co.n-i unctive eI]:_ansion about Z isnf(YrZ) = / \ 6^(2) v f(Y,C)- tct uwhene 5a denotes the complementIt is evident that 6a and AUrequine fonmulae ofAny function f(YrZ) u Bk+*about Z byF(V 7\ -r \ r to / -where 6a(Z) : 1I'or eactr n>3,whrch satisfywe rnay definethe followingqanrrAn^6q nf:conditions,(negation) of 6ar^^+]. ^*1.- r. ^_^m '-l .usy LII vlll_v I J QY.Iintescr"s (n -. -....r' .l-'-..'-i,m: ) r. fonallm:evenodd^^^-'+.'-t^yvJr La vswhere Sm(i)( rr /(afa/(iv IWe choose the sequence defined by-a- -r - 's' "m-2{2< 2^-t _ 2 *-JI'or e.rctr rrrc.:lt n, let pr be maxima.L suctr that,p(p+l)+t<n2form>o and mform>1 and mrobprmrmro].tlp-Z: i+l- for= n-Sn-lp> 1>O['or example if n:12 we get 12o21314r]>. This sequence satisfj,-s(i)-(iv). The fastest-growing sequence satisfying (i)-(iv) begins,1or 1.arge n, with <212)4,].212561220-?56, 2276r..c>Given (ror...r"pt =12r2r314n...) l-et RorRlo...rRo be acomesponding partition of Xn with lRi l = r. fon all i'We shall- describe oun construction in ter"ms of forrnulae patherthan in d more abstract way as schemes. It wiII be cl-ear throughouthowever that the formulae are uniform.Theorem 2. for all f e B-, (n>4), there is a formula of depth nfor the appr:oximation to f w.r.t. S = {Rar.."rRn-1}.Proof. Since n > 4, then p > J-. T,"e express f as an exDansion aboutR^ which is disjunctive if p is odd and conjunctive if p is even.trIiach of ihe 2 P t""*s in this expansion is expnessed in depth S--: --- p_1by using the nesults and constructions of the following lemma {'lI-or an inductive proof we must incorporate a more detajled-^^^.i€"'^-+-i^- ^€ the fOr"mulae at eaCh Stage.JPEcJffUdLrvrr vrLemila. Let R .R.,... R--(m>o) be disjoint sets of arguments wlth- o, I- mtheir cardinal-ities ro!o"rr* satisfying conditions (i)' (iii) ancl (iv)above. Then fon any function f(Ror...r*r), there is a forr'ruia for:+- .im:{-.'^n q w.r.t. {R"....,R } consisting of :-ILS ql)PruAirrrqLlv * I_ _ mrCase(a): if m is odd, a disjunction of 2 m-l subformulae each ofdePth Sr_arCase(b): if m is €v€rir a conjunction of 2 m-l- subformulae each ofdepth S*-l utd another subfonmula of depth Sr-Z'Proof. we proceed by induction on m using two alternative expansions.6In case (a)r\/g(Ro,,..,o*) = ya(6c(Rm) n ec(Ro,...oRm-t))and in case (b ) o Ag(Ro0,..,**) = 6o{R*) ^{),o.tn*) v gc(Ro,...,R *r))where in each case gC is the approximation to f (Ro,"' rRm-lrc)over {Rl ,...rRrn_r}.The validity of these expansions is easily verified.If m = I, then the first e><pansion is of the r:equired forrn since both 6aand ga have de;;th I and so we have a conjunction of 3 formulae of depth 2'If m > I and m is odd then in the same expansion we mayr by the incluctivehypothesis, take BC to be a conjunction of 2 m-r-1 subfonmulae of depthS*-2 .td a smaller subfonmula of depth S*-3. Since 6a is essentially as*-., s-conjunction of rn arguments and rr.< 2 m 2 - ,-m-3, it may be conjoine<iwith the smaller subformufa to pnoduce a fo::mula of depth Sr-2. Therresulting conjunction of 2 m-J fonmulae of depth S*-2 ".. be writtenin depth "m-l t S*-2 = Sm-I. The requirements of case (a) ane thereby]f m is even then the second expansion is usedo the 6a are thernselves ofdepth S*-2 .td case (b ) is easily satisfied trThe lemma mayThe resultingformulae eachbe illustratedapp:roximationof the fonm:with n=17, m=3 andn(P R R R ) is a5r,.or,'fr,.2r-'32 -.1sequence {2 ,2 ,3, tc>.unction of 102 3theut DJ14Ix_ x. -x,.x- x-^ / ,l-o.+r:).c)Ieftmost subformula may be given in detaiL dD.X- X^| / t6 X^ X.^ XIY IIU i lrzl t4 x,-t J-o i+ x.I'TN 2where the base functions associated with certain gates are notdefined if they depend on D.Each of the ha(RonRl_oR2) subfonmulae :n6 of the form:1" 1 *2*, x^x^x- x^,f, rbFIG. 4na(*o oR1rR2 )where the leftmost subformula ID.tlil4Iland the associated base functions depend on C.4. Main ResultIt remains to be shown how the approximation g to f overR, ,...,R* , can be used to conpu'te f.r p-rLemma. Suppose Rrr...r\ are disjoint subsets of Xn. I'or all f(Xn),there exist fr(Xn-Rr),...rfk(xn-\) such that en(Xn) = f t O t-tis an approximation to some function w.r.t. {Rf r... r\}Proof. This is by induction on k. The lemma holds trivially for k=0"Let k>0, and suppose the result is true for k-I. Then,there exist f.|(Xn-Rr)r... rft -t(Xn-FU-r) such that for all j-, lt<i<k,' k_]R_. = O * E"_-,,(Xn) = f O @ f- - O-L N-J .. -rWe define fU(Xn-RO) = 0 if li' l<i<k' *i = o= g._rlu =o othenwiseIr(rnd n:n rronifrr +hat o has the yaenrrined nnonertv t]ull\f eolr vsr rrJ LrruL 5k Iruu vs rt "r*- -JMain TheoremFor al_l n, n)1, there is a formula scheme with depth n+l whictrcovers Bn over B,r.Proof . Schemes for" BrrB, a::e obvious, while for BrrBu expans'ionscan be made about I and 2 arguments respectively to yielC schemesof deoth 3 and 4. By the previous lemmar anY function f(Xn) may beo-l-computed as g .(xn) o ls fj(Xn-R.) where g^ ,(xn) is an-""'r-- "p_l"---' H 1 r -p-rapproximation to some function w.n.t. {RIr...rRo-l}.I0By Theorem 2, any function f(xn) may be approximated w.r.t.{Rtr.." rRn-1}..-'+1-'-i n zlant-h nw! Lrrrlr uvy Lrr rr.1ls5 for n>4, there is an approximation fo with depth n and weneed to "add inil appnopriate functions frr...rfp-I where f' hasn-.:n - i- 16r^guments. WheneverDi( 4rafor"mul-aforf is1rconstructed directly, otherwise the present constnuction is usedrecursively to yield a formula of depth n. + I = n - i'Thus f is expressed asr-r- ff^(Xn) o\17 f-(Xn-R.)1=lnn :f+an naaqsrlgiatiOn, aSvr t ur Lvrto@ (t, @ (t, * .., o tp_r))...)Since f. has depth n-i for i=or...oP-fo this represents a formulal--of depth n+I.Again it is clear that the construction is unifonm and thusrrr'a'l rl< 2 q.hamo n5. Restricted BasesThe formufae considered so far have used alt of B, as the basis.Provided that the basis b permits a scheme to cover B, and contains atleast one function from each of the following three types:zl Jt^-type p ^ qv-type p"' u q'!@typ. p"' o qwhere a starred vaniable represents either the variable or its complement,the constnuction can be followed more or less as befone' complementingsubformulae as necessary to achieve depth n+2.l1An interesting basis is the set which excludes the two o -ti'pefunctions. In using this unate basis we may ::eplace @ byp @ q" = p ^ [ .v. p n qIn orde:r to fit in the coz'recting functions efficientlv wechoose a new sequence("o0o1r"2r...> = 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-typefunction we can obtain n+4 by taking the unate constructionand conplementing subformul-ae as necessary"Conjecture. For any b 5 BZ, if ther"e is a scheme over b whichcovers B, then there is a constant c such thatfor all n there is a scheme over b of depth ntcwhich covers Brr.Fon b = {p u q, p} we haver at presentn achieved no betterthan n + 0(1o8rn).Wemustdistinguishthenotionsofcompletebasesforforntrr};reand for schemes. For example, b = {NAND} is complete for formulaebut obviously no singleton basis can be complete fon schenes ' hencethe condition on b given in the conjecture'6. ConclusionWe have described a unifor"m scheme for expnessing all n-argumentBoofean functions in depth n+l, and have matched this upper bound witha lower bound of n-l under^ the nestriction of unifo::mity' For a basisof unate functibns only, our upper bound is n+3'In OUr construction we used a Sequence 1212 r3rq 151.. .)r but asecuence which grows much faste:: could be used instead. The effectof the choice of sequence on formula size has not been consid'ered butn- I n+leasy counting arguments limit the possible size to within 2" - and 2'-'-for our: method. Lupanovrs construction [3] yields formufae of sjzeabout 2n/Log^n, though not of coul?se using schemes' This raises the1_iollowlng:Open problem :hol-d whenDoesalowenboundof||n-constant||ondepttistji]the restriction to schemes is removed?13l).ef erencesi 1l l,pira, P.M. , t'On the time necessary to compute sw-itciringfunctionsrr, IEEE Tnans. Computer"s vol"c-20 (Jan.197]), 104-105"| ? I l'reparata, f.P. and Muller, D.E. ' "0n the delay requiredlo:'ealise Bc-rofean functionsr', IEEE Trans" Computersvol.C-20 (Apr.I97r), 459-461.L;l Lupanov, 0oB., "Complexity of Formula Realisation ofI'unctions of Logical Algebra", Prob. cybernetics Vol.3 (1962),7 B2-8 tr "

The depth of all Boolean functions

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The depth of all Boolean functions

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