Finite state verifiers with constant randomness

Abstract

Abstract. We give a new characterization of NL as the class of languages whose members have certificates that can be verified with small error in polynomial time by finite state machines that use a constant number of random bits, as opposed to its conventional description in terms of deterministic logarithmic-space verifiers. It turns out that allowing two-way interaction with the prover does not change the class of verifiable languages, and that no polynomially bounded amount of randomness is useful for constant-memory computers when used as language recognizers, or public-coin verifiers. A corollary of our main result is that the class of outcome problems corresponding to O(log n)-space bounded games of incomplete information where the universal player is allowed a constant number of moves equals NL.

Windable Heads and Recognizing NL with Constant Randomness

“…This work improves on the findings of [6]. For their pertinence, an outline of the algorithms attaining the errors in eqs.…”

“…In this paper, we introduce a characteristic for the languages in NL. Based on this characteristic, we lower the error threshold established in [6] for almost all languages in NL. Finally, we delineate a subset of NL which are verifiable by the constant-randomness 2pfa with arbitrarily low error.…”

“…Presented first in [6], verifier V ′ 1 with the following algorithm manages to achieve ε strong (V ′ 1 ) < 1 /2, outlined as follows: 1. Randomly reject with k − 1 /2k probability by flipping ⌈log k⌉ + 1 coins.…”

“…This last definition is the exact analogue of the verifiers' perspective in the algorithms proposed by [6]. It can be used directly in our next definition, that will lead us to a characterization of the 2nfa(k).…”

“…The language class verifiable by the constant-randomness two-way probabilistic finite automata (2pfa) is the class NL. Curiously, however, the error of these verifiers in recognizing languages of this class seems to be irreducible beyond a certain threshold [6].…”