Windable Heads & Recognizing NL with Constant 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 $\frac{k^2-1}{2k^2}$ 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 $\frac{k_{\text{W}}^2-1}{2k_{\text{W}}^2}$, where $k_{\text{W}}\le k$ is the number of windable heads. Using this new algorithm, a subset of languages in NL that have a 2nfa($k$) recognizer with $k_{\text{W}}\le 1$ can be verified with arbitrarily reducible error using constant space and randomness.

12 pages, accepted and to be published in Language and Automata Theory and Applications - LATA 2020 Alberto Leporati, Carlos Martín-Vide, Dana Shapira, Claudio Zandron (Eds.)