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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 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 k 2 W − 1 /2k 2 W , where k W ≤ 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 W ≤ 1 can be verified with arbitrarily reducible error using constant space and randomness.

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