Finite state verifiers with constant randomness

TL;DR

This paper characterizes NL as the class of languages verifiable by finite state machines with constant randomness, showing that interaction and additional randomness do not extend this class.

Contribution

It provides a new characterization of NL using finite state verifiers with constant randomness and interaction, contrasting with traditional logarithmic-space verifiers.

Findings

Finite state verifiers with constant randomness characterize NL.

Two-way interaction does not expand verifiable languages.

Limited randomness is sufficient for constant-memory language recognition.

Abstract

We give a new characterization of $\mathsf{NL}$ as the class of languages whose members have certificates that can be verified with small error in polynomial time by finite state machines that use a constant number of random bits, as opposed to its conventional description in terms of deterministic logarithmic-space verifiers. It turns out that allowing two-way interaction with the prover does not change the class of verifiable languages, and that no polynomially bounded amount of randomness is useful for constant-memory computers when used as language recognizers, or public-coin verifiers. A corollary of our main result is that the class of outcome problems corresponding to O(log n)-space bounded games of incomplete information where the universal player is allowed a constant number of moves equals NL.