Windable heads and 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 (k2-1)/2k2 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 (k2W-1)/2k2W, where kW≤ k is the number of windable heads. Using this new algorithm, a subset of languages in NL that have a 2nfa(k) recognizer with kW≤ 1 can be verified with arbitrarily reducible error using constant space and randomness.