Minimal Nontrivial Space Complexity of Probabilistic One-Way Turing Machines

Languages recognizab le in o( log log n) space by probabi l i s t ic one w a y Turing machines are proved to be regular . This solves an open p rob lem in [4]. 1 . I N T R O D U C T I O N Deterministic one-way Turing machine has a read-only input tape (the head on it never moves from right to the left) and one work-tape with no restrictions for the head on this tape. (Since we study the space complexity several work-tapes can be easily simulated by one). We denote the input alphabet by X. On the input tape immediately after the last symbol of the input word there is written a special symbol. The head on the input tape initially observes the first symbol of the input word. The set of states is denoted by Q, and it includes states q accept and q rejectThe probabilistic one way Turing machine differs from the deterministic one only in the additional ability to perform random options. Technically this is performed by introduction into every instruction of the program of the machine a special symbol being the output of a random number generator producing symbols from a finite set in such a way that every random option is independent of all the other options. (Since the main result of our paper is negative we allow as general type of probabilistic automata as possible. We allow the probabilities of the random options be arbitrary rational or irrational numbers 0 < p _< 1.) Thus every step of the work by the machine may depend not only of the internal state of the machine and the symbols observed by the heads but also of the output of the random number generator. We say that the probabilistic machine M recognizes language L~_X* in g(n) space with probability p if for arbitrary n ~ N and w c X <_n when M works on the word w the probability of the following event is no less than p: the machine stops after a finite number of steps, and 1) no more than g(n) squares are used on the worktape, and 2) the machine stops in the state {~ accept, if w c L reject, if w ~ L. We say that the probabilistic machine M recognizes the language L in g(n) space with isolated cut-point if there is a p>l/2 such that M recognizes L in g(n) space with probability p.