On the Size Complexity of Deterministic Frequency Automata

Austinat, Diekert, Hertrampf, and Petersen [2] proved that every language L that is (m, n)-recognizable by a deterministic frequency automaton such that m > n/2 can be recognized by a deterministic finite automaton as well. First, the size of deterministic frequency automata and of deterministic finite automata recognizing the same language is compared. Then approximations of a language are considered, where a language L′ is called an approximation of a language L if L′ differs from L in only a finite number of strings. We prove that if a deterministic frequency automaton has k states and (m, n)-recognizes a language L, where m > n/2, then there is a language L′ approximating L such that L′ can be recognized by a deterministic finite automaton with no more than k states. Austinat et al. [2] also proved that every language L over a single-letter alphabet that is (1, n)-recognizable by a deterministic frequency automaton can be recognized by a deterministic finite automaton. For languages over a single-letter alphabet we show that if a deterministic frequency automaton has k states and (1, n)-recognizes a language L then there is a language L′ approximating L such that L′ can be recognized by a deterministic finite automaton with no more that k states. However, there are approximations such that our bound is much higher, i.e., k!.