Circuit Complexity, Proof Complexity and Polynomial Identity Testing (Extended abstract) Joshua A. Grochow Toniann Pitassi Santa Fe Institute Depts. of Computer Science and Mathematics Santa Fe, NM, USA University of Toronto Email: [email protected] Toronto, Canada Email: [email protected] Abstract—We introduce a new and natural algebraic proof prior to our work, essentially all implications went the op- system, which has tight connections to (algebraic) circuit posite direction: a circuit complexity lower bound implying complexity. In particular, we show that any super-polynomial a proof complexity lower bound. We use this result to begin lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size to explain why several long-open lower bound questions algebraic circuits (VNP 6= VP). in proof complexity—lower bounds on Extended Frege, 0 As a corollary, super-polynomial lower bounds on the on AC [p]-Frege, and on number-of-lines in Polynomial number of lines in Polynomial Calculus proofs (as opposed Calculus-style proofs—have been so apparently difficult. to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. A. Background and Motivation Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any Algebraic Circuit Complexity.: The most natural way to computational lower bound. compute a polynomial function f(x1; : : : ; xn) is with a se- Our proof system helps clarify the relationships between quence of instructions g1; : : : ; gm = f—called an algebraic previous algebraic proof systems, and begins to shed light circuit or a straight-line program—starting from the inputs on why proof complexity lower bounds for various proof x ; : : : ; x , and where each instruction g is of the form g ◦ systems have been so much harder than lower bounds on the 1 n i j corresponding circuit classes. In doing so, we highlight the im- gk for some j; k < i, where ◦ is either a linear combination portance of polynomial identity testing (PIT) for understanding or multiplication. The goal of algebraic complexity is to un- proof complexity. derstand the optimal asymptotic complexity of computing a 1 Keywords-AC0[p]-Frege; algebraic circuit complexity; Grob- given polynomial family (fn(x1; : : : ; xpoly(n))n=1, typically ner bases; lower bounds; polynomial identity testing; proof in terms of size and depth of algebraic circuits. In addition complexity; syzygies to the intrinsic interest in these questions, since Valiant’s All proofs are present in the freely available draft of the work [2] algebraic complexity has become more and more full version [1]. important for Boolean computational complexity. Valiant argued that understanding algebraic complexity could give new intuitions that may lead to better understanding of other I. INTRODUCTION models of computation; several direct connections have been NP versus coNP is the very natural question of whether, found between algebraic and Boolean complexity [3], [4], for every graph that doesn’t have a Hamiltonian path, there is [5]; and the Geometric Complexity Theory Program (see, a short proof of this fact. One of the arguments for the utility e. g., the overview [6] and references therein) suggests how of proof complexity is that by proving lower bounds against algebraic techniques might be used to resolve major Boolean stronger and stronger proof systems, we “make progress” complexity conjectures. towards proving NP 6= coNP. However, until now this Two central functions in this area are the determinant argument has been more the expression of a philosophy or and permanent polynomials, which are fundamental both be- hope, as there is no known proof system for which lower cause of their prominent role in many areas of mathematics bounds imply computational complexity lower bounds of and because they are complete for various natural complex- any kind, let alone NP 6= coNP. ity classes. In particular, the permanent of f0; 1g-matrices We remedy this situation by introducing a very natural is #P-complete, and the permanent of arbitrary matrices algebraic proof system, which has tight connections to is VNP-complete. Valiant’s Permanent versus Determinant (algebraic) circuit complexity. We show that any super- Conjecture [2] states that the permanent of an n × n matrix, polynomial lower bound on any Boolean tautology in our as a polynomial in n2 variables, cannot be written as the proof system implies that the permanent does not have determinant of any polynomially larger matrix all of whose polynomial-size algebraic circuits (VNP 6= VP). Note that, entries are variables or constants. Unlike in Boolean circuit complexity, (slightly) non-trivial Definition 1 (Ideal Proof System). An IPS certificate that a lower bounds for the size of algebraic circuits are known system of F-polynomial equations F1(~x) = F2(~x) = ··· = [7], [8]. However, their techniques only give lower bounds Fm(~x) = 0 is unsatisfiable over F is a polynomial C(~x;~y) up to Ω(n log n). Moreover, their methods do not give in the variables x1; : : : ; xn and y1; : : : ; ym such that lower bounds for polynomials of constant degree. Recent 1) C(x1; : : : ; xn;~0) = 0, and exciting work [9], [10], [11] has shown that polynomial- 2) C(x1; : : : ; xn;F1(~x);:::;Fm(~x)) = 1. size algebraic circuits computing functions of polynomial The first condition is equivalent to C being in the ideal degree can in fact be computed by subexponential-size depth generated by y ; : : : ; y , and the two conditions together 4 algebraic circuits. Thus, strong enough lower bounds for 1 m therefore imply that 1 is in the ideal generated by the F , depth 4 algebraic circuits for the permanent would already i and hence that F (~x) = ··· = F (~x) = 0 is unsatisfiable. prove VP 6= VNP. 1 m An IPS proof of the unsatisfiability of the polynomials Proof Complexity.: Despite considerable progress ob- Fi is an F-algebraic circuit on inputs x1; : : : ; xn; y1; : : : ; ym taining super-polynomial lower bounds for many weak proof computing some IPS certificate of unsatisfiability. systems (resolution, cutting planes, bounded-depth Frege systems), there has been essentially no progress in the last For any class C of polynomial families, we may speak 25 years for stronger proof systems such as Extended Frege of C-IPS proofs of a family of systems of equations (Fn) systems or Frege systems. More surprisingly, no nontrivial where Fn is Fn;1(~x) = ··· = Fn;poly(n)(~x) = 0. When we lower bounds are known for the seemingly weak AC0[p]- refer to IPS without further qualification, we mean VP-IPS, Frege system. In contrast, the analogous result in circuit i. e., the family of IPS proofs should be computed by circuits complexity was resolved by Smolensky over 25 years ago of polynomial size and polynomial degree. [12]. The Ideal Proof System is easily shown to be sound; To date, there has been no satisfactory explanation for its completeness (without size bounds) follows from the this state of affairs. In proof complexity, there are no Nullstellensatz. known formal barriers such as relativization [13], Razborov– We typically consider IPS as a propositional proof system Rudich natural proofs [14], or algebrization [15] that exist in by translating a CNF tautology ' into a system of equations Boolean function complexity. Moreover, there has not even in the following standard way. We translate a clause κ of been progress by way of conditional lower bounds. That is, ' into a single algebraic equation F (~x) using x 7! 1 − x, trivially NP 6= coNP implies superpolynomial lower bounds x_y 7! xy.A f0; 1g-assignment thus satisfies κ if and only for AC0[p]-Frege, but we know of no weaker complexity if it satisfies the equation F = 0. We add to this system 2 assumption that implies such lower bounds. The only formal of equations the equations xi − xi = 0, which forces any implication in this direction shows that certain circuit lower solutions over F to be f0; 1g-valued. bounds imply lower bounds for proof systems that admit Like previously defined algebraic systems [18], [19], [20], feasible interpolation, but unfortunately even very weak [21], proofs in our system can be checked in randomized proof systems such as Frege and AC0-Frege don’t have this polynomial time. The key difference between our system and property, under standard complexity-theoretic assumptions previously studied ones is that those systems are axiomatic [16], [17]. In the converse direction, there are essentially no in the sense that they require that every sub-computation implications at all. For example, we do not know if AC0[p]- (derived polynomial) be in the ideal generated by the original Frege lower bounds—nor even Frege nor Extended Frege polynomial equations Fi, and thus be a sound consequence lower bounds—imply any nontrivial circuit lower bounds. of the equations F1 = ··· = Fm = 0. In contrast our system has no such requirement; an IPS B. Our Results proof can compute potentially unsound sub-computations In this paper, we define a simple and natural proof system (whose vanishing does not follow from F1 = ··· = Fm = that we call the Ideal Proof System (IPS) based on Hilbert’s 0), as long as the final polynomial is in the ideal generated Nullstellensatz. Our system is similar in spirit to related by the equations. This key difference allows IPS proofs to algebraic proof systems that have been studied previously, be ordinary algebraic circuits, and thus nearly all results in but is different in a crucial way that we explain below. algebraic circuit complexity apply directly to the Ideal Proof Given a set of polynomials F1;:::;Fm in n variables System. To quote the tagline of a common US food chain, x1; : : : ; xn over a field F without a common zero over the IPS is a “No rules, just right” proof system.