Short Proofs for the Determinant Identities
SIAM journal on computing (Print)
We study arithmetic proof systems P c (F) and P f (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 P c (F) proofs can be balanced: if a polynomial identity of syntactic degree d and depth k has a P c (F) proof of size s, then it also has a P c (F) proof of size poly(s, d) in which every circuit has depth O(k + log 2 d
... + log d • log s). As a corollary, we obtain a quasipolynomial simulation of P c (F) by P f (F). Using these results we obtain the following: consider the identities where X, Y and Z are n×n square matrices and Z is a triangular matrix with z 11 , . . . , z nn 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 P c (F) proofs of polynomial-size using circuits of O(log 2 n) depth. Moreover, there exists an arithmetic formula det of size n O(log n) such that the above identities have P f (F) proofs of size n O(log n) . This yields a solution to a basic open problem in propositional proof complexity, namely, whether there are polynomial-size NC 2 -Frege proofs for the determinant identities 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 NC 2 -Frege proofs, and quasipolynomial-size Frege proofs.