PCPs for Arthur-Merlin games and communication protocols

Probabilistically Checkable Proofs (PCPs) are an important class of proof systems that have played a key role in computational complexity theory. In this thesis we study the power of PCPs in two new settings: Arthur-Merlin games and communication protocols. In the first part of the thesis, we give a 'PCP characterization' of AM analogous to the PCP Theorem for NP. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the result for AM might be of particular significance for attempts to derandomnize this class. To test this notion, we pose some 'Randomized Optimization Hypotheses' related to our stochastic CSPs that (in light of our result) would imply collapse results for AM. Unfortunately, the hypotheses appear over-strong, and we present evidence against them. In the process we show that. if some language in NP is hard-on-average against circuits of size 2 [omega](n), en there exist hard-on-average optimization problems of a particularly elegant form. In the second part of the thesis, we study PCPs in the setting of communication protocols. Using techniques inspired by Dinur's proof of the PCP Theorem. we show that functions f (X, y) with nondeterministic circuits of size i have -distributed PCP protocols' of proof length O(poly(m)) in which each verifier looks at a constant number of proof positions. We show a complementary negative result: a distributed PCP protocol using a proof of length f, in which Alice and Bob look at k bits of the proof while exchanging t bits of communication, can be converted into a PCP-free randomized protocol with communication bounded by In both parts of the thesis, our proofs make use of a powerful form of PCPs known as Probabilistically Checkable Proofs of Proximity. and demonstrate their versatility. In our work on Arthur-Merlin games, we also use known results on randomness-efficient soundness- and hardness-amplification. In particular, we make essential use of the Impagliazzo-Wigderson generator; our analysis relies on a recent Chernoff-type theorem for expander walks.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2010

Author dc:creator

• Drucker, Andrew Donald

Advisor dc:contributor.advisor

• Scott Aaronson.

• Electrical Engineering and Computer Science.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/60160

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/60160