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 ...
MllfflMAL N O N T R I V I A L SPACE C O M P L E X I T Y OF PROBABILISTIC O N E - W A Y T U R I N G M A C H I N E S
J ~ s Ka~eps and R~s~ffFreivalds Institute o f M a t h e m a t i c s and C o m p u t e r Science University of Latvia RaiI~a bulv~ris 29 Riga, L a t v i a
Abstract L a n g u a g e s r e c o g n i z a b l e in o(log log n) space by probabilistic one - w a y Turing machines are p r o v e d to b e regular. This solves an open p r o b l e m in [4].
1.INTRODUCTION 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 _l,r2 by a probabilistic one-way Turing machine which never exceeds a space bound S(n)=o(log log n) whatever random options are taken by the probabilistic Turing machine. The results in [4] may seem trivial since B.Trakhtenbrot [10] and J.Gill [5] proved that determinization of probabilistic Turing machines increases the space complexity no more than exponentially.Unfortunately, precise formulations of these theorems include the propei~ty of space constructibility but no function o(log log n) can be space constructible. It was formulated explicitly in [4] as an open problem, to eliminate the abovementioned restriction (either p > 2/3 or space bound S(n)=o(log log n) for all random options). In spite of many attempts, this problem turned out to be very hard. We solve it only now in this paper by considering a notion of n-similar pairs of words and proving the crucial Lemma 2 which, we believe, may be of some interest itself. It is interesting to note that for two-way machines there is no minimal nontrivial space complexity such that capabilities of probabilistic and deterministic machines differ only starting from this complexity. R.Freivalds [2] proved that there is a nonregular language which can be recognized by probabilistic two-way finite automata with arbitrary probability 1 - e (e>0). We r e m i n d the r e d u c t i o n theorem by M . O . R a b i n [7] . Let X be a finite set, and L c X* be a language. The words w',w" • X* are called equivalent with respect to the language L if ( V w • X*) (w'w • L) ¢=~(w"w • L). By weight (L) we denote the number of the classes of equivalence with respect to the Ianguage L (language L is regular if weight (L)< o% M.O.Rabin [7] proved the following theorem. If a language L is recognized by a finite probabilistic 1 way automaton with k states with probability 1/2 + ~5then weight (L) < (1+ 1/8 )k-1. (In fact, M.O.Rabin formulated his theorem in a slightly more general form. Any case,it follows from this theorem that finite probabilistic automata with isolated cut-point recognize only regular languages). Let X be a finite set, L ~_ X* be a l a n g u a g e and n _> 0 be an integer. The words w',w" • x -g(n) ). For arbitrary triple (q",u",r') e c n we denote by pn(x,(q',u'~r),(q",u",r')) the probability of the associated period ending in moving the head on the input tape, and in this moment M finding itself in the configuration (q",u",l"). Now we consider the associated period for arbitrary configuration (q',u',r) ~ c n where the head on the input tape observes #. In this case we associate a period of work of M until one of the events take place: I) the input word is accepted or rejected; 2) M enters a configuration (q",u",r') where lu" I >g(n). We denote by pn(#,(q',u',r),inf'mity) the probability of the associated period lasting infinitely long time. By pn(#,(q',u',r),fiJll) we d e n o t e the p r o b a b i l i t y of the a s s o c i a t e d period e n d i n g lu"l > g(n). By pn(#,(q',u',l'),qaccept),
in a t r i p l e ( q " , u " , l " )
where
Pn(#,( q',u',r),qreject ) we denote the probabilities of the acceptation and rejection, respectively, at the end of the associated
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period. Additionally for aJrbitrary z e Xu{#} we define pn(z,qaccept,Stop)=pn(z,qreject,Stop)=l, pn(z,infinity,infinity)=pn(z,fifll,full)=Pn(Z,stop,stop)=1 Let n be arbitrary positive integer. Consider finite probabilistic one-way automaton Iglnin alphabet X w {#} with the set of states Sn=C n ~0 {infinity,fifil,qaccept,qreject,Stop}, the initial state (ql,A J ) where A is the empty symbol and qaccept is the only accepting state. The automaton A n works as follows. When A n in the state s' ~ Sn reads from the input an arbitrary symbol z e X w {#}, the automaton ~ moves m the state s" c Sn with the probability pn(z,s',s"). It is easy to see that the automaton An accepts arbitrary word w#, where w e X ~ , with probability hn, w. Every word not of the form w#, where w E X*, is rejected by An. Since (w e X 1f2+5, (w ~ X-
