Why is Bo olean Complexity Theory Dicult? L.G. Valiant Aiken Computation Lab oratory Harvard University Cambridge, MA 02138 1 Intro duction In the last decade substantial progress has b een made in our understanding of re- stricted classes of Bo olean circuits, in particular those restricted to have constant depth (Furst, Sipser, Saxe [FSS81], Ajtai [Ajt83], Yao [Yao85], Hastad [Has86 ], Razb orov [Raz87], Smolensky [Smo87]) or to b e monotone (Razb orov [Raz85], An- dreev [And85], Alon and Boppana [AB87], Tardos [Tar88 ], Karchmer and Wigderson [KW88]). The question arises, p erhaps more urgently than b efore, as to what ap- proaches could b e pursued that mightcontribute to progress on the unrestricted mo del. In this note we rst argue that if P 6= NP then any circuit-theoretic pro of of this would have to b e preceded by analogous results for the more constrained arithmetic mo del. This is b ecause, as we shall observe, there are proven implications showing that if, for example, the Hamiltonian cycle problem (HC) requires exp onential circuit size, then so do es the analogous problem on arithmetic circuits. Since the set of valid algebraic identities in the latter mo del form a prop er subset of those in the former, a lower b ound pro of for it should b e strictly easier. In spite of the ab ove relationship the algebraic mo del is often regarded as an alternative, rather than a restriction of the Bo olean mo del. One reason for this is that sp eci c computations are usually understandable in one of these mo dels, and not in b oth. In particular, the main p ower of the algebraic mo del derives from the p ossibility of cancellations, and it is usually dicult to express explicitly how these help in computing combinatorial problems. Our second aim in this note is to give an example of an algorithm, namely the Samuelson-Berkowitz metho d for computing Research supp orted by the National Science Foundation NSF-CCR-89-02500, the Oce for Naval Research ONR-N0014-85-K-0445, the Center for IntelligentControl ARODAAL 03-86-K-0171 and byDARPAAFOSR 89-0506. 1 the determinant, where the intermediate terms that are computed but ultimately cancelled by the arithmetic circuit can b e exhibited explicitly in combinatorial terms. The ease of computing the determinant can b e attributed to the existence of such an auxiliary set of monomials with certain computational prop erties. A pro of of an exp onential lower b ound on the complexity of a p olynomial that is b elieved to b e hard, such as the p ermanent or the Hamiltonian circuit p olynomial, would involve establishing the nonexistence of such an auxiliary set. It is dicult to imagine how such a pro of might go. Finally,we observethatinlow-level complexity, the arguments giving precedence to studying the algebraic mo del no longer hold. A ma jor op en problem area is that of proving for some explicit problem that it cannot b e computed by unrestricted Bo olean circuits simultaneously in size O (n) and depth O (log n). We describ e, via some conjectures, one candidate approachtowards proving suchalower b ound for problems such as sorting. Analogous conjectures exist for the algebraic mo del, but resolution of those would not imply the same for the Bo olean case. 2 Algebraic Structures Let x ; ;x b e a set of indeterminates and S a set of constants. Let and be 1 n two binary op erators. We de ne a circuit or a straight line program syntactically as a nite sequence of instructions of the form. f := g op h i 2f1; ;Cg i i i i where, for each i; op 2f ; g,andg ;h 2fx ; ;x g[ S [ff ; ;f g. In i i i 1 n 1 i1 other words a circuit is a sequence of C binary instructions, where each argumentof each one is either an indeterminate, a constant, or the result of the execution of an instruction earlier in the sequence. The complexity of a circuit is C , the number of instructions. Such a circuit can b e interpreted in several ways. In this pap er we shall assume that S is a commutative ring with identity. Then each instruction f canbeidenti ed i with the p olynomial that is computed at f ,if and are interpreted as the ring i op erations in the p olynomial ring S [x ; ;x ]. 1 n Among natural multivariate p olynomials whose complexity in this mo del is of interest are Hamiltonian circuits (HC), the p ermanent (PERM) and the determinant (DET). These are de ned over a matrix X of indeterminates fx ; ;x g where x 11 nn ij can b e thought of as representing edge (i; j ) of the complete directed graph G on n no des f1; ;ng. They are de ned as follows: 2 n X Y PERM(X ) = x ; i; (i) " i=1 n X Y HC(X ) = x i; (i); i=1 " has 1 cycle n Y X #( ) x DE T (X ) = (1) i; (i); i=1 " where is the set of p ermutations : f1; ;ng!f1; ;ng and #( )isthe number of even length cycles in . Clearly the monomials of PERM and DET corresp ond to cycle covers in G ,while n these of HC corresp ond to Hamiltonian circuits. These p olynomials can b e interpreted in anyringS . An imp ortantcaseis S = GF[2], the nite eld having two elements, in which case PERM = DET since then 1=1. Eachsuch problem is actually a p olynomial family indexed bythenumber of indeterminates. When we talk ab out the p ermanentwe mean the family PERM 1 PERM PERM ; where the subscript denotes the number of variables. The com- 4 9 putational complexityofsuch a family is determined by the family of circuits, one for each p olynomial, that are the smallest circuits for eachmember. Thus PERM is p olynomial time computable if and only if there is a family of circuits, of size growing p olynomially in n, that computes PERM. Di erentchoices of S allow for di erent circuits.. For example PERM is p olyno- mial time computable for S =GF[2] but is not known to b e so for any ring S whose characteristic is not a p ower of 2 (see [Val79]). We shall de ne three complexity classes of families of p olynomials in which all co ecients are integral multiples of unity: ARP is the class computable by p olynomial size circuits in all rings, GFP is that computable by p olynomial size circuits for S = GF[2], and GFB is that computable by p olynomial size circuits for S=GF[2], where 2 the algebra is assumed to ob ey the extra axiom x = x. By de nition, clearly, ARP GF P GF B : Now GFB is equivalent to the class of Bo olean functions of p olynomial circuit 2 size, since the p olynomial ring over GF[2] with x = x gives Bo olean algebra over f0; 1g and the op erations and in GF[2] give a complete Bo olean basis. Our observations here are rst that some natural p olynomials in this algebra have natural combinatorial interpretations, and, second, that their complexity can b e related to NP-completeness. 3 In particular, a GFB circuit for HC computes nothing other than the parityofthe numb er of Hamiltonian circuits in a graph. In other words if the indeterminates x of ij such a circuit are set to 1 or 0 according to whether edge (i; j ) is present in the graph then the circuit will compute 1 or 0 according to whether the numb er of Hamiltonian circuits is o dd or even. Furthermore, the following is implicit in [VV86] since the randomized reductions used there can b e simulated by small circuits [Adl78]: Theorem 1 HC 2 GF B ) NP has polynomial size circuits. Denoting the class of Bo olean functions of p olynomial Bo olean circuit size bypC [Val79], we conclude, therefore, that since NP j pC ) HC 2j GFB, any e orts at proving the former should b e retargeted at rst proving the latter or indeed anyof its even more restricted versions: NP j pC ) HC 2j GF B ) HC 2j GF P ) HC 2j ARP We note that NP j pC is equivalenttothe P 6= NP question but with Turing uniformity removed from the de nitions. It is argued in [Val79] that the nonuniform versions of such complexity questions as P 6= NP are at least as natural as the uniform versions. For example, the nonuniform version of NP, there denoted by pD (for p olynomial de nable,) has a completeness class that includes the original Hamiltonian circuits problem, just as NP has. Our conclusion, therefore, is that unless the explanation of the p ossible intractability of NP is to do with uniformity, one should seek it rst in the more restricted structure of ARP or GFP. 3 Cancellations in the Samuelson-Berkowitz Al- gorithm Any Bo olean circuit can b e reexpressed eciently as a Bo olean circuit over and in GF[2]. In almost any ecient such circuit the computation and cancellation of unwanted terms by the op erations plays a central role. Unfortunately,thewayin which ecient computations exploit this facility is little understo o d. It is dicult even to nd interesting examples where the structure of the cancelled terms can b e exhibited explicitly. In this section we study the one example wehave found in a nontrivial computation where the terms that are computed but ultimately cancelled have a natural charac- terization.