Windable Heads & Recognizing NL with Constant Randomness

TL;DR

This paper introduces the concept of windable heads in 2nfa automata and improves the error bounds for constant-randomness verification algorithms, enabling more efficient verification of certain NL languages.

Contribution

It defines windable heads and presents a modified verification algorithm that reduces error, expanding the class of NL languages verifiable with constant resources.

Findings

Error bound improved to (k_W^2 - 1)/(2k_W^2)

Languages with at most one windable head can be verified with arbitrarily low error

Verification uses constant space and randomness

Abstract

Every language in NL has a $k$-head two-way nondeterministic finite automaton (2nfa($k$)) recognizing it. It is known how to build a constant-space verifier algorithm from a 2nfa($k$) for the same language with constant-randomness, but with error probability $\tfrac{k^2-1}{2k^2}$ that can not be reduced further by repetition. We have defined the unpleasant characteristic of the heads that causes the high error as the property of being "windable". With a tweak on the previous verification algorithm, the error is improved to $\tfrac{k_{\textrm{W}}^2-1}{2k_{\textrm{W}}^2}$, where $k_{\textrm{W}} \le k$ is the number of windable heads. Using this new algorithm, a subset of languages in NL that have a 2nfa($k$) recognizer with $k_{\textrm{W}} \le 1$ can be verified with arbitrarily reducible error using constant space and randomness.