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

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 provide a full characterization of this class of languages in terms of a restricted version of the one-way nondeterministic multihead finite automaton model. 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.