Real-Time, Constant-Space, Constant-Randomness Verifiers

TL;DR

This paper explores the capabilities of real-time finite-state machines with constant space and randomness to verify membership proofs for various languages, including some beyond regular languages.

Contribution

It establishes that all languages recognized by two-head deterministic finite automata have such verifiable proofs, and identifies languages outside this class that are still verifiable under these constraints.

Findings

All two-head deterministic finite automaton languages are verifiable.

Some non-regular languages are verifiable despite limitations.

Nonpalindromes are verifiable within these constraints.

Abstract

We study the class of languages that have membership proofs which can be verified by real-time finite-state machines using only a constant number of random bits, regardless of the size of their inputs. Since any further restriction on the verifiers would preclude the verification of nonregular languages, this is the tightest computational budget which allows the checking of externally provided proofs to have meaningful use. We show that all languages that can be recognized by two-head one-way deterministic finite automata have such membership proofs. For any $k>0$, there exist languages that cannot be recognized by any $k$-head one-way nondeterministic finite automaton, but that are nonetheless real-time verifiable in this sense. The set of nonpalindromes, which cannot be recognized by any one-way multihead deterministic finite automaton, is also demonstrated to be verifiable within these restrictions.