CIRCUIT COMPLEXITY, PROOF COMPLEXITY, AND POLYNOMIAL IDENTITY TESTING

Speaker: Toniann Pitassi , University of Toronto

Date: Monday, May 12, 2014

Time: 4:00 PM to 5:00 PM Note: all times are in the Eastern Time Zone

Refreshments: 3:45 PM

Public: Yes

Location: 32-G449

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Host: Ankur Moitra, TOC, CSAIL, MIT

Contact: Holly A Jones, hjones01@csail.mit.edu

Relevant URL: http://toc.csail.mit.edu/node/494

Speaker URL: None

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Reminders to: seminars@csail.mit.edu, toc@csail.mit.edu, theory-seminars@csail.mit.edu

Reminder Subject: TALK: CIRCUIT COMPLEXITY, PROOF COMPLEXITY, AND POLYNOMIAL IDENTITY TESTING

Special Day and Time!
Abstract: 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.

This is joint work with Joshua A. Grochow.

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See other events that are part of the Theory of Computation Colloquium - 2014.

Created by Holly A Jones Email at Tuesday, May 06, 2014 at 1:52 PM.