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The University of Wanvi ck |HEORY OF COMPUTATION REPORT NO. 16 NEltl BOIINI]S FOR FORMULA SIZE II. Department of University of Computer Scjence l^lan'.rick COVENTRY CV4 7AL ENGLAi'ID. S. PATERSON December 1976. New bounds on formula size* 11. S. Paterson University of Abs tract. A variety of War:wick theorems bounding the formula size of rather simple Boolean functions are described here for the first time. The principal results are irnproved lower syme tri c functions and uPper bounds for . 1. Introduction. In preparing tny presentation to this in nind the higir 1evel of 1977 GI-Conference knowledge and expertise I have borne in the audience at such a neeting. I resolved that it would be most appropriate to talk about some of the most recent research upon whiqh I results ro be described have been engaged. The have been obtained by myself usually in collaboration wiEh others, notably Mike Fischer, Albert Meyer and Bill McColl, during the past year or so. appeared One was completed in published BooLean form. function conplexity is a key area of theoretical conputer science. Questions of actual appear here for and potential effieiency in computation in their ultimately refined form. It is the sticking-place many problems automata only a few days ago and none has yet theory. arising from circuit design, algorithmic analysis The principal measures of complexity for and Boolean functions are circuit size, formula size and depth. Wtrile the first must be regarded as the most fundament.al measure, it is an unhappy historical fact that no lovrer bound non-linear in the nurnber of arguments has yet, been proved the circuit size of an explicitly described function. This can be juxtaposed with the classical result that all but a vanishing fraction * Invited lecture at 2nd GI Conference, Darnstadt 1977. for of Boolean functions have exponential circuit size. In contradistinction several lower bound theorems have liberated formula size from this ttl-inear strait-jacket", disclosing a richer structure of function complexities. It is to be hoped that the present exploration into formula size will overtaken within the next decade be by similar developrnents with respect to circuit size. The technical nature to assume that the interested reader has some previous definitions explicitly, of the results presented here makes it and knovrledge reasonable of the basic results, so for example I sha11 not define "formula" For a brief survey of the field may I suggest [Pat 76]; for a very ful1 accotrrt, [Sav 76] is to be recomxended. 2, Definitions. Let B- {h : iort}n * [or1]] be the set of n-argument Boolean funcrions. n We consider formulae for such functions conposed of argument variables x1r...rXr,8rd arbitrary binary connectives from the full set 82. We find it convenient to account the size of a formul-a as the nurnber of occurrences of variables in it. (The alternative eount of the ntmber of connectives result in a size of exactly one Less.) The formula size F(h) would of a function h e Bis defined to be the size of the shortest fornula representing t. n Our choice of the whole of B, as the set of connectives is important in that two kinds of ones and for distinguished. Str.o.Jrg. contain either rOr (not-equivalence, sum modulo 2) or r=t (equivalence) complete basis glls ones contain neither. are {Or+} and {NAND} Bn can be Examples of minimal strong and weak bases respectively. Pratt [Pra 75] has demonstraEed that, to within a constant factor in formuLa size, there are just two equivalence classes of complete basesr the strong and the weak, and has quantified the 2 r rnaximum as well disparity betvJeen them. For the results presented here, we could choose any strong complete basis. The set Sn of syryletriq ftmctions contains those functions in J ----- just B u which depend only on the nrsrber of lfs among their arguments. Equivalently, h e Bo is syuunetric if and only if there is a function q : {0r...1n} +'{011}1 the char:cteristic function of h, such'that for all. x, h(x) = xn (xxr). Hence we have L 2t*1, as compared wirh lr-l = z2n. ltol n' = ' ll' To express lower and upper conplexity botnds trore succinctly the following notation is convenient. Let f(n), g(n) be nonlregative-valued functions from the natural nurnbers. We f(n) - O(g(n)) if write 3a>0 such that f(n)sa.g(n) for alI sufficiently large n f(n) - n(s(n)) if g(n) = o(f(n)) f(n) - O(g(n)) if f(n) = o(f(n)) 3._S giporuk' s theo lsm and f(n) = n(s(n)). and_co r.o.{a_ries . A sinple yer powerful theorem yielding Lol'er bornds of up to g(nZ on formula size is irnplicit in [Nec 66]. Suppose the argrnen,ts of h e Br, to be partitioned into bloslg R1r...rRp. arguments Wtren for some in all the block" *j, j f irare fixed to o or I in result is a lej.qri.c-rion of h to Rrr a. i, some /tog n) some the way, the function hr depending only on R.. Let mt--be the nurnber of different such hr for all possible fixations of the other variables. In this notation the theorem is concisely expressed. -r.h."olg (Neciporuk) F(h) = ,(n, rog \i=1 r-). 7 This result establishes lower bounds for a variety of functions of interest. Most applications have been to functions with a rather combinatorial nature such as rdeterrninantr [r1o 651, tmarriage problemt [HaS 72] addressing schemes [Nec 66], [Pau 75J. 1\^ro or indirect, 'corollaries of a sonrewhat different character have emerged recently. rf we idenrify rhe n = rr cetts of a f:= with the arguments array in r dirnensions xlr...rxrrr then any predicate on patterns.of black and white colourings of the cells corresponds with a function in 8.,. 0f particular appeaI are.those predicates of a geometric or topologicaL natue. 'The predicate CONN!") e B- which rn is true if and only if the black cells of the r-dimensional array are connected has been considered by Hodes [ttod 7O]. Thegrem (Paterson & The Eischer) f CCO,oU(l) = Q(n log n) fot r >.2. proof uses Neciporukrs theorem applied with blocks of size 2s = O(.,l&t|. By embedding arbitrary pernutation connections 6elween s of these ce1ls and the other s, we can show that at least s! restrictions are possible for each block. In the case r = 1, it is an easy observation that r(coNNt:'r= tCtCtl")), where T, is the threshold fr:nction defined in the next section. With a sirnilar correspondence lre can identify 3 set of binary strings * of lengthnwitha function of Br,. For anyLS[, where X = t0r1], we aefir,e e{n) e B' to be the function corresponding to the set L n In. The lower bounds on context-free language recognition proved by Hotz [t{ot 75] and Mehlhorn [Meh 76] can be improved slightly as follows. theorem There is a context-free language L such that rCsf")) = n(nLllog n). The l.anguage used in the proof is defined L = {o* 1w oo x* rR x* | w e l*1. by The stated bound results from Neciporukts theorem when we choose the arguments for each block and o* 1b,1b^... LZ where set the other arguments to impose a form Lb 100x brr...rb* are the block variables, and m e2 logrn. Each block in turn is thus forced into the r6le of rwt in the definition of L. ? f-\ An upper bowrd of- O(n- 1og n) for F(t;"'), encouragingly close to .lower bound proved, O(n) formulae, each is with a formula which is a disjunction of corresponding to rfoldingt the string about some position. demonstrable string of L is detected by a formula where the fold is at the of symmetry of w, w Each 4._ Boglds. for. Since -sy.nu'negr_ic ls-l tm' - when applied f ,m+1, Neciporul<ts theorem produces only to symnetric functions. The triwial bor.mds firsc non-linear lower bounds are corollaries of the following general result. for ", )4|, l. and sirnilarly for (E Theorem (Hodes & Specker) such cencre ulrsriots. for any function in S' hrere proved by Hodes and Specker [HoS 68]. !' the There that for all crm, if n >t We Ihey write . is a (very rapidly growing) function G(rnrc) and @ G is any formula of size less than el with argtrment set X = tx1 ,...rxrr), then there is a subset Y g x with lVl = m and constants bo, blr b, such that the restricted formula QY of m argtrnents got by setting the variables of x\Y to 0 is equivalenr t.o bo.@.bl n-.!v.o.b2^e t This theorem has been applied for [Hod 701 some geometric and topoLogical also for some symrnetric functions. to prove non-linear Lower bounds predicates as considered earlier The statement of the theorem is and somewhat comPlicated, but for application Eo S' orrly: a considerable simplificarion is possible. The following corollary appears not to have been stated i previous 1y. l Lheorep (Meyer & Paterson) For aLl c ) or for n sufficiently large, only 16 functions of S. have formula size less than cn. fhe sixteen ftrnctions are characterized by the property of the characteristic -frnction that X(r) = X(r+2) for 1 r< r < n-3. The threshold ftmctions rf") "r" defined by xT(r)=l<..+r>k -k so r{rf")) is cerrainly nonlinear in n for any fixed k >.2. It ls t(tl")) shown by Ktrasin [IQa 69] and Pippenger [Pip 75] thar = O(n 1og n) for all fixed k, but only existence proofs are given. Elernentary recursion using dichotomy generates explicit forrnulae with sizes which are g(n6fog ,,)k-1) for fixed k. McCoLl and Paterson have deferred this growth rate initially {heo-rgm For 2{k( Our nethod by using increasingly tortuous identities[t"tcC 76]. 6, tCtrl")) =O(n.logn (loglogo)k-2). is to partition the n arguments ir,to rrl blocks of si"e ol. The predicace that ar Least I Tr(Ts). for example, our recursi.ve construction for Ta uses the Then blocks satisfy Ts(ttl) i" abbreviated as identity. r (n) - ,r, rr ^rtr1) ^3 - v T1(T3) v (Tr(Tr) n rr(Tr)). With the aim of raising the very s1-ow1y increasing lor'rer bounds of llodesf and Speckerrs theorem; Fischer, l,Ieyer and Paterson weakened 5 somewhat the conclusion of the theorem and proved lower bounds of Q(n.1og n/log1og n) tfMP 751. Recent improvements in both the scope of functions covered and the bound attained provide a theorem analogous to Hodesr and Speckerrs, with the follorving corollary for syrrnetric functions. ThugtgL For any h e Srrr if Xnft) * Xn(t+Z) for f(h) = 0(n.J.og k). some k = k(n), rhen For example, Lovrer bounds of Q(n.log n) are established for threshold. functions T^(t) rhure 0 is constant, o < 0 < l, and for the congruence v ll /n\ functions Q"', k fixed, k > 2, where Cn is defined by X,. (r) = 1 if and only if r is a multiple of k. t, Our especial favourite t (al") The upper bound .f") = is )is CO since we now have e(n. log n) shown . using the identity . Df') ,u -oj") where Dor DL, DZ, . . . represent successive digits (lowest first) of the binary representation of the sum of the arguments. The Drs are e:gressed recursively by constructing the Dts.for each of two equal halves of the argunent set and perforrning a binary addition on the results. p(l)r*l -r \"' ' o(zn1 f-\ r(DJ ) = n, An if r = o =0ifr>0 =x (5, 1) = oj") (1) o oj") t1l of'") (r, Since Thus D = of") (1) o of") c1l o (oj"'(3) n oj") cr) we have lt-\ t{ol"") = 0(n.1og n). arbitrary function of Sr, is representable as some function of Do, Dlr...rDm where m = [.og(n+11'l -1, and.may be expressed in a forraula using just one occurrence of Dr, two of Drnl, four of Dm-2, et cetera. T Sma1l formulae for the Dts yield therefore Using approximately this F(Sn) .=- r"* def nrethod Pippenger {FCh) sma11 forrnulae tPip for all S.r. 741 has determined that ltr . srr} = o(na) where a=Lo8r(6+4'fZ1 * 3.54. rn his binary adder Pippenger uses "fu11 adders' to digic d of the sumurands carry digit c from two digits dt, sum and new and the previous compute each new carry ct. dr r, of the t{e employs the formula pair d = dr O drt O ct c = (ct ^ (dt v drr)) v (dt A dtt). A small refinement in the construction is to replace the latter formula by I = dr and spl-it the @ argument ((ct O dt) ,r (dti e df )) set into two unequal parts at each stage. , Optinization of the ratio of this split (with the help of Colin lJhitby-Strevens and the warwick university Burroughs 67oo cornputer) decreases the exponent. O(nB) where B < 3.41866 705572. !Lqo.1e+. r(S-) n = 5. Conclusion. A number of new bounds for symmetric, or otherwise fairly sirnple, functions are sketched here in a prelirninary form. f hope that by the time of the Gr neeting at least some rnay be available in a more polished and complete version. I should like to ackncrwledge the cooperation and encouragement derived from many other people than those. named explicitly, and the valuable stimulation provided by such international meetings as this. 8 References IFUP 75] M.J. Fischer, A.R. Msysl and M.S. paterson. t'Lower bounds on the size of Boolean formulas: preliminary reportr', !roc. Ttlr Ann. AW S/ne. or.r-Jh. of .CoJnp-qting (L975), Lffi. 7'i-vt* [HaS 72] L.H. Harper and J.E. Savage. "on the gomplexity of the marriage problem'r, Advjr,nge: ijl.Matlr.egarics 9, 3 (L972), Zgg-3LZ. [uoa ZO] [I{oS 68] L. Hodes. "The logical complexity of geometric properties in the plane"r. !_3s 17, 2 (1970) , 339-347, L. Hodes and E. Specker. "Lengths of formulas and elimination gi quantif iers I", in Contributions- to }{athsp-a-tigal L-ogig, K. Schutte, ed., North Holland Pub1. Co., (1968), 175-188. [ttot 75] G. ]lotz. ttUntere Schranken fiir das AnalyseprobLem kontext-freier Sprachen", Techn. Berichtr Univ. des Saarlandes, L9j6. [xtra 0g] L.S. Khasin. "Complexity bounds for the realization of monotone symmetrical functions by means of formulas in the basis vr &, -.tt, Eng. trans. in Soviet_P.hysics D_ok1. r 14 LZ (1970),1149-1151; orig. 9o51, .Akad. Nauk_SSSR, 19, 4 (t969), 752-755 B.M. Kloss. "Estimates of the complexity of solutions of systems of linear equations", Eng. trans. in sgviet.Math Dpkf, Z,6 (1966), 1537-1540; orig. D_ok1.. Aka-d..-Nguk SSS3., !L, 4 (t96b), 781-783. [rto 66] [UcC 76] W.F. McCol1. t'Some resuLts on circuit depth", Ph.D. dissertation, Computer Science Dept., Wanuick University, 1976. [t'tetr 76] K. Mehlhorn. "An improved bound on the formula complexity of context-free recognition". Unpublished report, I976. [Nec 66] E.I. Neciporuk. "A Boolean function"r sov-iet Ygrh..]>kJ. ]r 4 (1966), 999-1000, orig. pokl. Aka.d. Nqujc SS.SB 1_69, 4 (t966), lPat 76] [Pau 75] M.S. Paterson. rrAn introduction to Boolean furction complexity". Stanford Computer Science Report STAN-CS-76-557 Stanford University, 1976; to appear in Ast6risque. W. Paul. nA 2.5 N lower bound for the combinational complexity of Boolean functionsrr, 3ro,c._7-th Ann. Albuquerque (1975), 27-36. tPip 74i 765-766. .A-C4 Symp....on Th_. of._g_ojng. N. Pippenger. rrshort formulae for syrmrstric functionsrr, Research Report RC-5143, Yorktciwn 9 Hts., L974. IBM tPip 75J N. Pippenger. "Short IBII Research [Pra 75] for thieshold functionstt. Report RC 54.05, Yorktown Hts.r 1975. monotone formulae V.R. Pratt. 'rThe effect of basis on size of Boolean expressionstt. Proc. 16tI, l\rua.1 rEEE_s.y*po:iy* on l.g-rndari.o.n-s qI cpmqq,r_er gqience, 119- l_21. [Sav 76 ] J.E. Savage. New ISha 49] . York, T.Lre .Co,Ip]Srxi.ty of gomp-y-tgrg, Wiley-Inrers cience, 1976 c.E. Shannon. "T'l-re synthesis of two-terrninal switching circuits", Bell 9Jjten Le.ch€c.al Jgu.rnal Ag (1949), 59-98. {fhis is the full text of an invited l-ecture at the 3rd GI Conference on Theoretical Conrputsr Science, March I977, Darmstadt, Germany. 1 Decernber ,Department 19 76 of University of COVENTRY ENGI-A,ND" 10 Computer Science Warwick CV4 7AL