Probabilistically checkable proofs and the testing of hadamard-like codes

Recent results concerning probabilistically checkable proofs (PCPs) enable the encoding of mathematical proofs so as to allow very efficient probabilistic verification. Probabilistic verification consists of a simple randomized test that looks at a few bits of the proof and decides to accept or reject the proof's validity by performing a simple computation on those bits. Valid proofs are always accepted. Incorrect proofs are rejected with a non-negligible probability. The interest in PCPs stems from their complexity theoretic implications. Especially interesting are the surprising way in which they characterize NP and the hardness of approximation results derived through them. This thesis contributes to further our understanding of complexity theoretic issues through the study of PCPs. We address a basic problem that arises in the construction of PCPs and the derivation of hardness of approximation results: linearity testing over the two element field. We give a thorough analysis of this problem. Some of our results are obtained by showing a new connection between testing over the field of size two and discrete Fourier analysis. This relationship is further illustrated by studying a pair of tests that are related to the Hadamard code test. In addition, we establish a connection between testing and the theory of weight distributions of dual codes. This connection allows us to formulate a new way of testing for linearity over finite fields. We then show how to analyze, through the MacWilliams Theorems, the tests that fall into our framework. The results we obtain imply nontrivial facts about the functions being tested even when these functions fail the test with relatively large probability. We discuss why these types of results are desirable in the PCP context. Finally, we propose two models of interactive proof systems. In the first of these proof systems, a computationally limited verifier interacts with a ranked sequence of oracles. In the second proof system, the verifier now interacts with a ranked sequence of provers. The distinguishing characteristic of these proof systems is the ordered relationship amongst the provers and amongst the oracles. We show that, with suitable restrictions on the randomness and query complexity, both proof systems exactly characterize each level of the polynomial hierarchy. The characterization via oracle proof systems extends that of NP via PCPs. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)