Generating Matrix Identities and Proof Complexity Lower Bounds

Motivated by the fundamental lower bounds questions in proof complexity, we investigate the complexity of generating identities of matrix rings, and related problems. Specifically, for a field F let A be a non-commutative (associative) F-algebra (e.g., the algebra Matd(F) of d × d matrices over F). We say that a non-commutative polynomial f(x1, . . . , xn) over F is an identity of A, if for all c ∈ A, f(c) = 0. Let B be a set of non-commutative polynomials that forms a basis for the identities of A, in the following sense: for every identity f of A there exist noncommutative polynomials g1, . . . , gk, for some k, that are substitution instances of polynomials from B, such that f is in the (two-sided) ideal 〈g1, . . . , gk〉. We study the following question: Given A,B and f as above, what is the minimal number k of such generators g1, ..., gk for which f ∈ 〈g1, . . . , gk〉? In particular, we focus on the case where the algebra A is Matd(F), and F has characteristic 0. Our main technical contribution is a generalization of the lower bound presented in Hrubeš [6] (for the case d = 1) to any d > 2: For every natural number d > 2 and every finite basis B for the identities of Matd(F), where F is of characteristic 0, there exists an identity fn with n variables, that requires Ω(n) generators (i.e., substitution instances from B) to generate. Note that for any d > 2, it is an open problem to find a basis for the identities of Matd(F) (while the existence of a finite basis was proved by Kemer [11]). Nevertheless, using results from the theory of algebras with polynomial identities (PI-algebras) together with a generalization of the arguments in [6], we conclude the above lower bound for every finite basis B. We then explore connections to lower bounds in proof complexity. We consider arithmetic proofs of polynomial identities that operate with algebraic circuits and whose axioms are the polynomial-ring axioms (which can be considered as an algebraic analogue of the Extended Frege propositional proof system). We raise the following basic question: is it true that using the generators of the (non-commutative) polynomial identities over Matd(F) as axiom (schemes) is an optimal way to prove such identities, with respect to proof size? Namely, is it true that proving matrix identities by reasoning with polynomials whose variables X1, . . . , Xn range over matrices is as efficient as proving matrix identities using polynomials whose variables range over the entries of the matrices X1, . . . , Xn? We show that a positive answer to this question may lead, under further assumptions (which are generalization of the assumptions presented in [6]), up to exponential-size lower bounds on arithmetic proofs. Institute for Theoretical Computer Science, The Institute for Interdisciplinary Information Sciences (IIIS), Tsinghua University, Beijing. Supported in part by NSFC grant 61373002 Institute for Theoretical Computer Science, The Institute for Interdisciplinary Information Sciences (IIIS), Tsinghua University, Beijing [email protected] Supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of P. R. China; Grants 61033001, 61061130540, 61073174, 61373002. 1 ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 185 (2013)