UTCS Colloquia - Algorithms and Computation Theory - Eli Ben-Sasson, Technion and MIT, "Constant rate PCPs for circuit-SAT with sublinear query complexity"

Contact Name: 
Anna Gal
Location: 
GDC 4.516
Date: 
Apr 22, 2013 11:00am - 12:30pm

Talk Audience: UTCS Faculty, Grads, Undergrads, Other Interested Parties

Host:  Anna Gal

Talk Abstract: The PCP theorem says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up incurred by this encoding, i.e., how long is the PCP compared to the original NP-proof. The state-of-the-art work of Ben-Sasson and Sudan and Dinur shows that one can encode proofs of length n by PCPs of length n poly log n that can be verified using a constant number of queries. In this work, we show that if the query complexity is relaxed to n^epsilon, then one can construct PCPs of length O(n) for circuit-SAT, and PCPs of length O(n log n) for any language in NTIME(n).

More specifically, for any epsilon>0 we present (non-uniform) probabilistically checkable proofs (PCPs) of length O(n) that use n^epsilon queries for circuit-SAT instances of size n. Our PCPs have perfect completeness and constant soundness. This is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity (o(n)).

Our proof replaces the low-degree polynomials in algebraic PCP constructions with tensors of transitive algebraic geometry (AG) codes. We show that the automorphisms of an AG code can be used to simulate the role of affine transformations which are crucial in earlier high-rate algebraic PCP constructions. Using this observation we conclude that any asymptotically good family of transitive AG codes over a constant-sized alphabet leads to a family of probabilistically checkable proofs (PCPs) of length O(n) that use n^epsilon queries for circuit-SAT instances of size n. Our PCPs have perfect completeness and constant soundness. This is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity (o(n)).

Our proof replaces the low-degree polynomials in algebraic PCP constructions with tensors of transitive algebraic geometry (AG) codes. We show that the automorphisms of an AG code can be used to simulate the role of affine transformations which are crucial in earlier high-rate algebraic PCP constructions. Using this observation we conclude that any asymptotically good family of transitive AG codes over a constant-sized alphabet leads to a family of constant-rate PCPs with polynomially small query complexity. Such codes are constructed by Stichtenoth in the appendix to this work for the first time for every message length, after they have been constructed for infinitely many message lengths by him. 

Joint work with Yohay Kaplan, Swastik Kopparty and Or Meir, with an appendix by Henning Stichtenoth.

Tags: