Generating Matrix Identities and Proof Complexity

Motivated by the fundamental lower bounds questions in proof complexity, we initiate the study of matrix identities as hard instances for strong proof systems. A matrix identity of d × d matrices over a field F, is a non-commutative polynomial f(x1, . . . , xn) over F such that f vanishes on every d × d matrix assignment to its variables. We focus on arithmetic proofs, which are proofs of polynomial identities operating with arithmetic circuits and whose axioms are the polynomial-ring axioms (these proofs serve as an algebraic analogue of the Extended Frege propositional proof system; and over GF (2) they constitute formally a sub-system of Extended Frege [9]). We introduce a decreasing in strength hierarchy of proof systems within arithmetic proofs, in which the dth level is a sound and complete proof system for proving d× d matrix identities (over a given field). For each level d > 2 in the hierarchy, we establish a proof-size lower bound in terms of the number of variables in the matrix identity proved: we show the existence of a family of matrix identities fn with n variables, such that any proof of fn = 0 requires Ω(n 2d) number of lines. The lower bound argument uses fundamental results from the theory of algebras with polynomial identities together with a generalization of the arguments in [7]. Specifically, we establish an unconditional lower bound on the minimal number of generators needed to generate a matrix identity, where the generators are substitution instances of elements from any given finite basis of the matrix identities; a result that might be of independent interest. We then set out to study matrix identities as hard instances for (full) arithmetic proofs. We present two conjectures, one about non-commutative arithmetic circuit complexity and the other about proof complexity, under which up to exponentialsize lower bounds on arithmetic proofs (in terms of the arithmetic circuit size of the identities proved) hold. Finally, we discuss the applicability of our approach to strong propositional proof systems such as Extended Frege. Institute for Theoretical Computer Science, The Institute for Interdisciplinary Information Sciences (IIIS), Tsinghua University, Beijing. Supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61361136003 and NSFC grant 61373002. Department of Computer Science, Royal Holloway, University of London. Email: [email protected] Supported in part by the NSFC Grant 61373002. 1 Background Proving super-polynomial size lower bounds on strong propositional proof systems, like the Extended Frege system, is a major open problem in proof complexity, and in general is among a handful of fundamental hardness questions in computational complexity theory. An Extended Frege proof is simply a textbook logical proof system for establishing Boolean tautologies, in which one starts from basic tautological axioms written as Boolean formulas, and derives, step by step, new tautological formulas from previous ones by using a finite set of logical sound derivation rules; including the so-called extension axiom enabling one to denote a possibly big formula by a single new variable (where the variable is used neither before in the proof nor in the last line of the proof). It is not hard to show (see [11]) that Extended Frege can equivalently be defined as a logical proof system operating with Boolean circuits (and without the extension axiom). Lower bounds on Extended Frege proofs can be viewed as lower bounds on certain nondeterministic algorithms for establishing the unsatisfiability of Boolean formulas (and thus as a progress towards separating NP from coNP). It is also usually considered (somewhat informally) as related to establishing (explicit) Boolean circuit size lower bounds. In fact, it has also another highly significant consequence, that places such a lower bound as a step towards separating P from NP: showing any super-polynomial lower bound on the size of Extended Frege proofs implies that, at least with respect to “polynomial-time reasoning” (namely, reasoning in the formal theory of arithmetic denoted S 2), it is not possible to prove that P = NP; or in other words, it is consistent with S 2 that P 6=NP (cf. [15]). Accordingly, proving Extended Frege lower bounds is considered a very hard problem. In fact, even conditional lower bounds on strong proof systems, including Extended Frege, are not known and are considered very interesting; here, we mean a condition that is different from NP 6= coNP (see [18]; the latter condition immediately implies that any propositional proof system admits a family of tautologies with no polynomial-size proofs [4]). The only size lower bound on Extended Frege proofs that is known to date is linear Ω(n) (where n is the size of the tautological formula proved; see [14] for a proof). Establishing super-linear size lower bounds on Extended Frege proofs is thus a highly interesting open problem. That said, although proving Extended Frege lower bounds is a fundamental open problem in complexity, it is quite unclear whether such lower bounds are indeed far from reach or beyond current techniques (in contrast to other fundamental hardness problems in complexity, such as strong explicit Boolean circuit lower bounds, for which formal so-called barriers are known). Another feature of proof complexity is that, in contrast to circuit complexity, even the An additional simple technical axiom is needed to formally define this proof system ([11]). Informally, we call a proof system strong if there are no known (non-trivial) size lower bounds on proofs in the system and further such lower bounds are believed to be outside the realm of current techniques.