Stochasticity of the languages acceptable by two-way finite probabilistic automata

It is proved that two-way finite probabilistic automata can accept only stochastic languages, i. e. the capabilities of one-way and two-way finite probabilistic automata to accept languages are the same. The increase in the number of states when we replace a two-way automaton by a one-way automaton accepting the same language is estimated. The capabilities of automata of various types in language recognition have been widely investigated since the seventies. Usually the research concerns the recognition with probability p > 1/2 which is a special case of acceptance with isolated cut-point. Freivald published a series of papers proving the advantages of probabilistic automata of numerous types over their deterministic counterparts. Two-way finite probabilistic automata (2-FPA) were first considered by Kuklin [1]. In [2] Freivald proved that such automata with probability p > 1/2 can recognize a non-regular language {Ol | n > 1}. The result was somewhat unexpected since deterministic and non-deterministic two-way finite automata accept only regular languages [3]. The algorithm of Freivald has exponential expected time. Greenberg and Weiss [4] proved that the language { O n l n | n > l } cannot be recognized by 2-FPA with probability p > 1/2 in less than exponential expected time. Recall that there is a different situation with one-way finite probabilistic automata (1-FPA). Rabin (see [12], Theorem 4.1.1) proved that 1-FPA with isolated cut-point can accept only regular languages, i. e. their capabilities in this sense equal the capabilities of deterministic and non-deterministic one-way finite automata. For 1-FPA in contrast to more complicated types of probabilistic automata, the extensive research on language acceptance capabilities in a broader sense, without the cut-point isolation requirement, began in the sixties. The languages acceptable by 1-FPA with non-isolated cut-point are called stochastic. There are non-regular stochastic languages. Elements of the theory of stochastic languages are presented in [5], [13]. 2-FPA is the type of probabilistic automata which is the most close to 1-FPA. Hence, it is interesting to compare the class of languages acceptable by 2-FPA with the class of stochastic languages. This problem was posed in [1]. The main result of the present paper shows that every language acceptable by 2-FPA is stochastic. Hence in a broader sense (without the restriction of the cut-point to be isolated) capabilities of 2-FPA and 1-FPA in language acceptance are equal. 1. THE BASIC NOTATION Bellow we will use the notation £,0 for the unit and zero matrices; R for the set of *UDC 519.71. Originally published in Diskretnaya Matematika (1989) 1, No. 4, 63-77 (in Russian). Translated by the author.