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- W2034804004 abstract "We study arithmetic proof systems ${mathbb P}_c({mathbb F})$ and $ {mathbb P}_f({mathbb F})$ operating with arithmetic circuits and arithmetic formulas, respectively, and that prove polynomial identities over a field ${mathbb F}$. We establish a series of structural theorems about these proof systems, the main one stating that ${mathbb P}_c({mathbb F})$ proofs can be balanced: if a polynomial identity of syntactic degree $ d $ and depth $k$ has a ${mathbb P}_c({mathbb F})$ proof of size $s$, then it also has a ${mathbb P}_c({mathbb F})$ proof of size $ {rm poly}(s,d) $ in which every circuit has depth $ O(k+log^2 d + log dcdot log s) $. As a corollary, we obtain a quasi-polynomial simulation of ${mathbb P}_c({mathbb F})$ by ${mathbb P}_f({mathbb F})$. Using these results we obtain the following: consider the identities $det(XY) = det(X)cdotdet(Y) mbox{ and } det(Z)= z_{11}cdots z_{nn},$ where $X,Y$, and $ Z$ are $ntimes n$ square matrices and $Z$ is a triangular matrix with $z_{11},dots, z_{nn}$ 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 ${mathbb P}_c({mathbb F})$ 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 ${mathbb P}_f({mathbb F})$ 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 $mathbf{NC}^2$-Frege proofs for the determinant identities and the hard matrix identities, as considered, e.g., in Soltys and Cook [Ann. Pure Appl. Logic, 130 (2004), pp. 277--323] (cf. Beame and Pitassi [Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 65 (1998), pp. 66--89]). We show that matrix identities like $ AB=I rightarrow BA=I $ (for matrices over the two element field) as well as basic properties of the determinant have polynomial-size $mathbf{NC}^2$-Frege proofs and quasi-polynomial-size Frege proofs." @default.
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- W2034804004 date "2015-01-01" @default.
- W2034804004 modified "2023-10-14" @default.
- W2034804004 title "Short Proofs for the Determinant Identities" @default.
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- W2034804004 doi "https://doi.org/10.1137/130917788" @default.
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