Efficient Interactive Proofs for Non-Deterministic Bounded Space

Abstract

The celebrated IP = PSPACE Theorem gives an efficient interactive proof for any bounded-space algorithm. In this work we study interactive proofs for non-deterministic bounded space computations. While Savitch’s Theorem shows that nondeterministic bounded-space algorithms can be simulated by deterministic bounded-space algorithms, this simulation has a quadratic overhead. We give interactive protocols for nondeterministic algorithms directly to get faster verifiers. More specifically, for any non-deterministic space S algorithm, we construct an interactive proof in which the verifier runs in time ˜ O ( n + S 2 ). This improves on the best previous bound of ˜ O ( n + S 3 ) and matches the result for deterministic space bounded algorithms, up to polylog( S ) factors. We further generalize to alternating bounded space algorithms. For any language L decided by a time T , space S algorithm that uses d alternations, we construct an interactive proof in which the verifier runs in time ˜ O ( n + S log( T ) + Sd ) and the prover runs in time 2 O ( S ) . For d = O (log( T )), this matches the best known interactive proofs for deterministic algorithms, up to polylog( S ) factors, and improves on the previous best verifier time for nondeterministic algorithms by a factor of log( T ). We also improve the best prior verifier time for unbounded alternations by a factor of S . Using known connections of bounded alternation algorithms to bounded depth circuits, we also obtain faster verifiers for bounded depth circuits with unbounded fan-in.