Abstract

This paper studies a generalization of multi-prover interactive proofs in which a verifier interacts with two competing teams of provers: one team attempts to convince the verifier to accept while the other attempts to convince the verifier to reject. Each team consists of two provers who jointly implement a no-signaling strategy. No-signaling strategies are a curious class of joint strategy that cannot in general be implemented without communication between the provers, yet cannot be used as a black box to establish communication between them. Attention is restricted in this paper to two-turn interactions in which the verifier asks questions of each of the four provers and decides whether to accept or reject based on their responses. We prove that the complexity class of decision problems that admit two-turn interactive proofs with competing teams of no-signaling provers is a subset of PSPACE. This upper bound matches existing PSPACE lower bounds on the following two disparate and weaker classes of interactive proof: 1. Two-turn multi-prover interactive proofs with only one team of no-signaling provers. 2. Two-turn competing-prover interactive proofs with only one prover per team. Our result implies that the complexity of these two models is unchanged by the addition of a second competing team of no-signaling provers in the first case and by the addition of a second no-signaling prover to each team in the second case. Moreover, our result unifies and subsumes prior PSPACE upper bounds on these classes.

Highlights

  • Interactive proofs were introduced in the mid-1980’s as a generalization of the concept of efficient proof verification and the complexity class NP [2, 4, 18]

  • We prove that the complexity class of decision problems that admit two-turn interactive proofs with competing teams of no-signaling provers is a subset of PSPACE

  • This small distinction is enough to increase the power of the model from PSPACE all the way up to NEXP [3, 17], even when the interaction is restricted to only two turns with only two provers [13]

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Summary

Introduction

Interactive proofs were introduced in the mid-1980’s as a generalization of the concept of efficient proof verification and the complexity class NP [2, 4, 18]. The polynomial-space algorithm of Feige and Kilian for two-turn interactive proofs with competing provers [12] is a complicated and highly specialized precursor to the MWUM that, like our algorithm, optimizes over stochastic matrices that represent strategies for the provers Their algorithm works by nondeterministically guessing the entries of the matrix and scanning them in a read-once fashion. The parallel algorithm of Ito for two-turn, two-prover interactive proofs with no-signaling provers [22] is essentially a reduction to the mixed packing and covering problem, which is a special type of linear program that is known to admit an efficient parallel algorithm [37] This approach, too, cannot be extended to competing teams of no-signaling provers, as any linear programming formulation of the protocol is unlikely to be a mixed packing and covering problem.

Definition of two-turn interactive proofs with competing teams of provers
Min-max formalism for interactive proofs with competing provers
Notation for marginal distributions
Characterization of no-signaling strategies
A relaxed min-max problem with penalties
Bounds on two-turn verifiers
Definition of the relaxed min-max problem
Equivalence of the two min-max problems
Lemmas used in the rounding theorem
A parallel multiplicative weights algorithm
The parallel algorithm
Compute
Implementations of the best-response oracle for Team Bob
Containment in PSPACE
Full Text
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