Constant-Space, Constant-Randomness Verifiers with Arbitrarily Small Error

TL;DR

This paper characterizes the subset of NL languages verifiable by constant-space, constant-randomness probabilistic machines with arbitrarily small error, focusing on linear-time multi-head automata.

Contribution

It introduces a method for low-error verification of certain NL languages using extremely weak, resource-limited machines, and relates their power to complexity classes.

Findings

Verifiers with constant coins and space can achieve arbitrarily low error for languages recognized by linear-time 2nfa(k).

The class of languages verifiable under these constraints is a subset of NL.

The paper discusses the limitations of extending this verification method to all of NL.

Abstract

We study the capabilities of probabilistic finite-state machines that act as verifiers for certificates of language membership for input strings, in the regime where the verifiers are restricted to toss some fixed nonzero number of coins regardless of the input size. Say and Yakary{\i}lmaz showed that the class of languages that could be verified by these machines within an error bound strictly less than $1/2$ is precisely NL, but their construction yields verifiers with error bounds that are very close to $1/2$ for most languages in that class when the definition of "error" is strengthened to include looping forever without giving a response. We characterize a subset of NL for which verification with arbitrarily low error is possible by these extremely weak machines. It turns out that, for any $\varepsilon>0$, one can construct a constant-coin, constant-space verifier operating within error $\varepsilon$ for every language that is recognizable by a linear-time multi-head nondeterministic finite automaton (2nfa($k$)). We discuss why it is difficult to generalize this method to all of NL, and give a reasonably tight way to relate the power of linear-time 2nfa($k$)'s to simultaneous time-space complexity classes defined in terms of Turing machines.