Witnessing matrix identities and proof complexity
International journal of algebra and computation
We use results from the theory of algebras with polynomial identities (PI algebras) to study the witness complexity of matrix identities. A matrix identity of d × d matrices over a field F is a non-commutative polynomial f (x 1 , . . . , x n ) over F, such that f vanishes on every d × d matrix assignment to its variables. For any field F of characteristic 0, any d > 2 and any finite basis of d × d matrix identities over F, we show there exists a family of matrix identities (f n ) n∈N , such
... each f n has 2n variables and requires at least Ω(n 2d ) many generators to generate, where the generators are substitution instances of elements from the basis. The lower bound argument uses fundamental results from PI algebras together with a generalization of the arguments in  . We apply this result in algebraic proof complexity, focusing on proof systems for polynomial identities (PI proofs) which operate with algebraic circuits and whose axioms are the polynomialring axioms [13, 14] , and their subsystems. We identify a decreasing in strength hierarchy of subsystems of PI 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 an Ω(n 2d ) lower bound on the number of proof-steps needed to prove certain identities. Finally, we present several concrete open problems about non-commutative algebraic circuits and speed-ups in proof complexity, whose solution would establish stronger size lower bounds on PI proofs of matrix identities, and beyond.