Efficient Probabilistically Checkable Debates

نویسنده

Andrew Drucker

چکیده

Probabilistically checkable debate systems (PCDSs) are debates between two competing provers, in which a polynomial-time verifier inspects a constant number of bits of the debate. It was shown by Condon, Feigenbaum, Lund, and Shor that every language in PSPACE has a PCDS in which the debate length is polynomially bounded. Using this result, they showed that the approximation versions of some natural PSPACE-complete problems are also PSPACE-complete. We give an improved construction of these debates: for any language L that has an ordinary debate system definable by uniform circuits of size s = s(n), we give a PCDS for L whose debate is of total bitlength Õ(s), with a verifier that uses only log2 s + log2(polylog(s)) bits of randomness. This yields a much tighter connection between the time complexity of natural PSPACE-complete problems and the time complexity of their approximation versions. Our key ingredient is a novel application of error-resilient communication protocols, as developed by Schulman; we use the more recent protocol of Braverman and Rao. We show that by requiring ordinary debates to be encoded in an error-resilient fashion, we can endow them with a useful “stability” property. Stable debates can then be transformed into PCDSs, by applying efficient PCPPs (as given by Dinur). Our main technical challenge in building stable debates is to enforce error-resilient encoding by the debaters. To achieve this, we show that there is a constant-round debate system, with a very efficient verifier, to debate whether a communication transcript follows the Braverman-Rao protocol.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Probabilistically Checkable Proofs: A Primer

Probabilistically checkable proofs are proofs that can checked probabilistically by reading very few bits of the proof. Roughly ten years back it was shown that proofs could be made probabilistically checkable with a modest increase in their size. While the initial proofs were a little too complex, a recent proof due to Irit Dinur gives a dramatically simple (and radically new) construction of ...

متن کامل

On the role of algebra in the efficient verification of proofs

This article extracts the elements of algebra that play a central role in the design of efficient probabilistic verifiers or “probabilistically checkable proof systems (PCPs)”. The main algebraic elements are low-degree polynomials over finite fields. Their role can be broken up into three essential elements: 1. Their classical role in the design of error-correcting codes. 2. Their recently dis...

متن کامل

Talk given at workshop on Algebraic Methods in Complexity Theory On the role of algebra in the efficient verification of proofs

متن کامل

On Dinur’s Proof of the Pcp Theorem

Probabilistically checkable proofs are proofs that can be checked probabilistically by reading very few bits of the proof. In the early 1990s, it was shown that proofs could be transformed into probabilistically checkable ones with only a modest increase in their length. The initial transformations, though elementary, were a little too complex. A recent work due to Irit Dinur gives a dramatical...

متن کامل