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Witnessing matrix identities and proof complexity

<span title="">2018</span>
<i title="World Scientific Pub Co Pte Lt">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/467zefb3vfbmdmbauxdzukpug4" style="color: black;">International journal of algebra and computation</a>
</i>

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

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... 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 [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 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.
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