267,582 research outputs found
Circuit complexity, proof complexity, and polynomial identity testing
We introduce a new algebraic proof system, which has tight connections to
(algebraic) circuit complexity. In particular, we show that any
super-polynomial lower bound on any Boolean tautology in our proof system
implies that the permanent does not have polynomial-size algebraic circuits
(VNP is not equal to VP). As a corollary to the proof, we also show that
super-polynomial lower bounds on the number of lines in Polynomial Calculus
proofs (as opposed to the usual measure of number of monomials) imply the
Permanent versus Determinant Conjecture. Note that, prior to our work, there
was no proof system for which lower bounds on an arbitrary tautology implied
any computational lower bound.
Our proof system helps clarify the relationships between previous algebraic
proof systems, and begins to shed light on why proof complexity lower bounds
for various proof systems have been so much harder than lower bounds on the
corresponding circuit classes. In doing so, we highlight the importance of
polynomial identity testing (PIT) for understanding proof complexity.
More specifically, we introduce certain propositional axioms satisfied by any
Boolean circuit computing PIT. We use these PIT axioms to shed light on
AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no
satisfactory explanation as to their apparent difficulty. We show that either:
a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not
have polynomial-size circuits of depth d - a notoriously open question for d at
least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or
b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we
have a lower bound on AC^0[p]-Frege.
Using the algebraic structure of our proof system, we propose a novel way to
extend techniques from algebraic circuit complexity to prove lower bounds in
proof complexity
A sub-determinant approach for pseudo-orbit expansions of spectral determinants in quantum maps and quantum graphs
We study implications of unitarity for pseudo-orbit expansions of the
spectral determinants of quantum maps and quantum graphs. In particular, we
advocate to group pseudo-orbits into sub-determinants. We show explicitly that
the cancellation of long orbits is elegantly described on this level and that
unitarity can be built in using a simple sub-determinant identity which has a
non-trivial interpretation in terms of pseudo-orbits. This identity yields much
more detailed relations between pseudo orbits of different length than known
previously. We reformulate Newton identities and the spectral density in terms
of sub-determinant expansions and point out the implications of the
sub-determinant identity for these expressions. We analyse furthermore the
effect of the identity on spectral correlation functions such as the
auto-correlation and parametric cross correlation functions of the spectral
determinant and the spectral form factor.Comment: 25 pages, one figur
Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I. Generalizations of the Capelli and Turnbull identities
We prove, by simple manipulation of commutators, two noncommutative
generalizations of the Cauchy-Binet formula for the determinant of a product.
As special cases we obtain elementary proofs of the Capelli identity from
classical invariant theory and of Turnbull's Capelli-type identities for
symmetric and antisymmetric matrices.Comment: LaTeX2e, 43 pages. Version 2 corrects an error in the statements of
Propositions 1.4 and 1.5 (see new Remarks in Section 4) and includes a Note
Added at the end of Section 1 comparing our work with that of Chervov et al
(arXiv:0901.0235
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