Relations between communication complexity classes

The study of property testing is concerned with algorithms that solve approximate decision problems, while only probing a small fraction of their inputs. More specifically, a tester for a property Pi receives query access to an object x and is required to determine whether x is in Pi or x is far from Pi, using as little as possible queries to x.A fundamental question that arises in any computational model is to understand the power of proof systems within the model. Indeed, the P neq NP conjecture, which deals with the power of proofs in polynomial time computation, is arguably the most important open problem in the theory of computation. The focus of this thesis is on understanding the power and limitations of proof systems within the framework of property testing.We study locally verifiable proofs of proximity (LVPP), which are probabilistic proofs systems wherein the verifier queries a sublinear number of bits of a statement and is only required to reject statements that are far from valid. In their most basic form, the verifier receives, in addition to query access to the statement, also free access to a proof of sublinear length; such proof systems are called Merlin-Arthur proofs of proximity (MAP) and can be viewed as the MA (i.e., ``randomized NP'') analogue of property testing.Other notable forms of LVPPs include interactive proofs of proximity (IPP), in which the verifier is allowed to communicate with an omniscient prover (rather than obtain a static proof), and probabilistically checkable proofs of proximity (PCPP), in which the verifier is only allowed to make a small number of queries to both statement and proof (which is typically longer than the statement, in the case of PCPPs). These proofs systems can be viewed as the IP and PCP analogues of property testing.In this thesis, we initiate the study of some types of LVPPs and continue the study of others. Our main contributions include:Introducing the notion of non-interactive (Merlin-Arthur) proofs of proximity (MAP) and initiating its systematic study.Exponential separations between the power of property testers, MAPs, and IPPs. In particular, denoting by PT, MAP, and IPP the classes of properties that admit testers and verifiers of polylogarithmic query and communication complexity, we show that PT subsetneq MAP subsetneq IPP, which can be interpreted as separating BPP, MA, and IP in the settings of property testing.A hierarchy theorem for IPPs, which shows that the power of IPPs gradually increases with the number of rounds of communication allowed between the prover and the verifier.Constructions of MAPs and IPPs for several complexity classes, including constraint satisfaction problems (such as 3SAT formulas), properties of graphs, languages accepted by small branching programs, and context-free languages; as well as a strong form of PCPPs for affine subspaces.Several constructions of error-correcting codes admitting local features (such as a strong form of local testability, a relaxed form of local decodability, and testability of numerous subcodes) that are useful for applications to LVPPs.

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