Witnessing Matrix Identities and Proof Complexity

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(x1, . . . , xn) 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 (fn)n∈N, such that each fn has 2n variables and requires at least Ω(n ) 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 [12]. 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) 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.