4 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
Short Propositional Refutations for Dense Random 3CNF Formulas
Random 3CNF formulas constitute an important distribution for measuring the
average-case behavior of propositional proof systems. Lower bounds for random
3CNF refutations in many propositional proof systems are known. Most notably
are the exponential-size resolution refutation lower bounds for random 3CNF
formulas with clauses [Chvatal and Szemeredi
(1988), Ben-Sasson and Wigderson (2001)]. On the other hand, the only known
non-trivial upper bound on the size of random 3CNF refutations in a
non-abstract propositional proof system is for resolution with
clauses, shown by Beame et al. (2002). In this paper we
show that already standard propositional proof systems, within the hierarchy of
Frege proofs, admit short refutations for random 3CNF formulas, for
sufficiently large clause-to-variable ratio. Specifically, we demonstrate
polynomial-size propositional refutations whose lines are formulas
(i.e., -Frege proofs) for random 3CNF formulas with variables and clauses.
The idea is based on demonstrating efficient propositional correctness proofs
of the random 3CNF unsatisfiability witnesses given by Feige, Kim and Ofek
(2006). Since the soundness of these witnesses is verified using spectral
techniques, we develop an appropriate way to reason about eigenvectors in
propositional systems. To carry out the full argument we work inside weak
formal systems of arithmetic and use a general translation scheme to
propositional proofs.Comment: 62 pages; improved introduction and abstract, and a changed title.
Fixed some typo