Short proofs for the determinant identities

We study arithmetic proof systems Pc(F) and Pf (F) operating with arithmetic circuits and arithmetic formulas, respectively, and that prove polynomial identities over a field F. We establish a series of structural theorems about these proof systems, the main one stating that Pc(F) proofs can be balanced: if a polynomial identity of syntactic degree d and depth k has a Pc(F) proof of size s, then it also has a Pc(F) proof of size poly(s, d) in which every circuit has depth O(k + log2 d + log d ∙ log s). As a corollary, we obtain a quasipolynomial simulation of Pc(F) by Pf (F). Using these results we obtain the following: consider the identities det(XY) = det(X) ∙ det(Y) and det(Z) = z11 ∙ ∙ ∙ znn, where X,Y and Z are n×n square matrices and Z is a triangular matrix with z11,..., znn on the diagonal (and det is the determinant polynomial). Then we can construct a polynomial-size arithmetic circuit det such that the above identities have Pc(F) proofs of polynomial-size using circuits of O(log2 n) depth. Moreover, there exists an arithmetic formula det of size nO(log n) such that the above identities have Pf (F) proofs of size nO(log n). This yields a solution to a basic open problem in propositional proof complexity, namely, whether there are polynomial-size NC2-Frege proofs for the determinant iden-tities and the hard matrix identities, as considered, e.g. in Soltys and Cook [SC04] (cf., Beame and Pitassi [BP98]). We show that matrix identities like AB = I → BA = I (for matrices over the two element field) as well as basic properties of the determinant have polynomial-size NC2-Frege proofs, and quasipolynomial-size Frege proofs.