On Interactivity in Arthur-Merlin Communication and Stream Computation

We introduce online interactive proofs (OIP), which are a hierarchy of communication complexity models that involve both randomness and nondeterminism (thus, they belong to the Arthur–Merlin family), but are online in the sense that the basic communication flows from Alice to Bob alone. The complexity classes defined by these OIP models form a natural hierarchy based on the number of rounds of interaction between verifier and prover. We give upper and lower bounds that (1) characterize every finite level of the OIP hierarchy in terms of previously-studied communication complexity classes, and (2) separate the first four levels of the hierarchy. These results show marked contrasts and some parallels with the classic Turing Machine theory of interactive proofs. Our motivation for studying OIP is to address computational complexity questions arising from the growing body of work on data stream computation aided by a powerful but untrusted helper. By carefully defining our complexity classes, we identify implicit assumptions in earlier lower bound proofs. This in turn indicates how we can break the mold of existing protocols, thereby achieving dramatic improvements. In particular, we present two-round stream protocols with logarithmic complexity for several query problems, including the fundamental INDEX problem. This was thought to be impossible based on previous work. We also present a near-optimal, one-round stream protocol for counting triangles in a dynamic graph. Our protocols leverage classic algebraic techniques, including multilinear extensions, sum check, and arithmetization of Boolean formulas. ∗This work was supported in part by the Simons Institute for the Theory of Computing, and was performed while the authors were visiting the Institute. †Department of Computer Science, Dartmouth College. Supported in part by NSF grant CCF-1217375. ‡Department of Computer Science, University of Warwick. §Department of Computer Science, UMass Amherst. Supported in part by NSF grant IIS-1251110. ¶Simons Institute for the Theory of Computing, UC Berkeley. Supported by a Research Fellowship from the Simons Institute for the Theory of Computing. ‖School of Computing, University of Utah. Supported in part by NSF grant IIS-1251049 and the Simons Institute for the Theory of Computing. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 180 (2013)