Abstract
We study interactive proofs with sublinear-time verifiers. These proof systems can be used to ensure approximate correctness for the results of computations delegated to an untrusted server. Following the literature on property testing, we seek proof systems where with high probability the verifier accepts every input in the language, and rejects every input that is far from the language. The verifier's query complexity (and computation complexity), as well as the communication, should all be sublinear. We call such a proof system an Interactive Proof of Proximity (IPP). On the positive side, our main result is that all languages in NC have Interactive Proofs of Proximity with roughly √n query and communication and complexities, and polylog(n) communication rounds. This is achieved by identifying a natural language, membership in an affine subspace (for a structured class of subspaces), that is complete for constructing interactive proofs of proximity, and providing efficient protocols for it. In building an IPP for this complete language, we show a tradeoff between the query and communication complexity and the number of rounds. For example, we give a 2-round protocol with roughly n3/4 queries and communication. On the negative side, we show that there exist natural languages in NC1, for which the sum of queries and communication in any constant-round interactive proof of proximity must be polynomially related to n. In particular, for any 2-round protocol, the sum of queries and communication must be at least ~Ω(√n). Finally, we construct much better IPPs for specific functions, such as bipartiteness on random or well-mixing graphs, and the majority function. The query complexities of these protocols are provably better (by exponential or polynomial factors) than what is possible in the standard property testing model, i.e. without a prover.
Highlights
The power of efficiently verifiable proof systems is a central question in the study of computation
In building an Interactive Proof of Proximity (IPP) for this complete language, we show a tradeoff between the query and communication complexity and the number of rounds
On the negative side, we show that there exist natural languages in N C1, for which the sum of queries and communication in any constant-round interactive proof of proximity must be polynomially related to n
Summary
The power of efficiently verifiable proof systems is a central question in the study of computation. We study the power, and the limitations, of interactive proof systems with very efficient sublinear time verifiers. As in the standard interactive proof model, the verifier’s input is reliable and fixed throughout the protocol, whereas the prover is unreliable and untrusted, and might deviate from the protocol in an adaptive and arbitrary manner. We find this to be the most natural model for sublinear-time proof verification, but we provide some motivating applications below. We define and study a new complexity class: IPP, the class of languages that have interactive proofs of proximity with sublinear-time verification. We refer loosely to the complexity class IPP as the class of languages that have Interactive Proofs of ε-Proximity, with sublinear query and communication complexities, for some non-trivial ε
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