Regular frequency computations

Abstract

An ( m , n ) -computation of a function f is given by a deterministic Turing machine which on n pairwise different inputs produces n output values where at least m of the n values are in accordance with f. In such a case, we say that the Turing machine computes f with frequency ⩾ m / n . The most prominent result for frequency computations is due to Trakhtenbrot: The class of ( m , n ) -computable functions equals the class of computable functions if and only if 2 m > n . Via characteristic functions the definition of ( m , n ) -computability carries over to sets. Here Trakhtenbrot's result reads as: The class of ( m , n ) -computable sets equals the class of recursive sets if and only if 2 m > n . The notion of frequency computation can be extended to other models of computation. For resource bounded computations, the behavior is completely different: for e.g., whenever n ′ - m ′ > n - m , it is known that under any reasonable resource bound there are sets ( m ′ , n ′ ) -computable, but not ( m , n ) -computable. However, scaling down to finite automata, the analogue of Trakhtenbrot's result holds again: We show here that the class of languages ( m , n ) -recognizable by deterministic finite automata equals the class of regular languages if and only if 2 m > n . This was originally stated by Kinber, but his proof has a flaw, as pointed out by Tantau. Conversely, for 2 m ⩽ n , the class of languages ( m , n ) -recognizable by deterministic finite automata is uncountable for a two-letter alphabet. When restricted to a one-letter alphabet, then every ( m , n ) -recognizable language is regular. This was also shown by Kinber. We give a new and more direct proof for this result.