R. Freivalds. Probabilistic two way machines. In Proc. International Symposium on Mathemarical Foundations of Computer Science, volume 118, pages 33-45. Springer-Verlag, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... by Paz [99] We make use of the undecidability proof given by Lipton and Condon [24] This latter work is developed in the context of research on the power of randomization in various resource bounded computational models, and in particular it is based on a technique developed by Freivalds [42], who shows that nite automata with randomized computations unlike their deterministic counterparts have a semi counting ability. This ability, further developed and adapted, is at the heart of the reduction of Lipton and Condon [24] A contribution of the thesis in this part, re ected in the ....

....problems for nondeterministic and deterministic nite automata are simple reachability computations. Such hardness results point to an added power randomized computations bring to space bounded machines such as nite automata. As we describe in this chapter and was rst developed by Freivalds [42], unlike deterministic automata, probabilistic nite automata have a certain weak ability to count that lies at the core of all the reductions we use. We use the proof in [24] instead of [99] as it was more useful to us in establishing other undecidability results, and in particular for showing ....

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R. Freivalds. Probabilistic two way machines. In Proc. International Symposium on Mathemarical Foundations of Computer Science, volume 118, pages 33-45. Springer-Verlag, 1981.

.... the counter contents are represented in unary, this problem reduces to checking whether two strings have the same length: given a string a , does i = j Although this question cannot be answered exactly by any PFA, a weak equality test developed in [15] and inspired by the work of Freivalds [19] can answer it in a strict and limited sense which is nonetheless sucient to allow the reduction. The weak equality test, shown in Fig. 2, works as follows. The PFA scans its input string a , and with high probability enters an Indecision state (equivalently we say the outcome of the test is ....

....planning problems under average reward criteria, and establishes undecidability results for the in nite horizon criteria based on our results. Early work on probabilistic nite automata include [43,41] Our proof techniques build on the techniques developed by Freivalds and Condon and Lipton [19,15] and are di erent from previous techniques of Paz and Bertoni et al. 41,5] which were based on reductions from Post s correspondence problem. Recently, Blondel and Canterini [7] have shown that several PFA problems such as the emptiness and threshold isolation are undecidable even for a PFA ....

[Article contains additional citation context not shown here]

R. Freivalds. Probabilistic two way machines. In Proc. International Symposium on Mathemarical Foundations of Computer Science, volume 118, pages 33-45. Springer-Verlag, 1981.

.... the counter contents are represented in unary, this problem reduces to checking whether two strings have the same length: given a string a , does i = j Although this question cannot be answered exactly by any PFA, a weak equality test developed in [9] and inspired by the work of Freivalds [13] can answer it in a strict and limited sense which is nonetheless sufficient to allow the reduction. The weak equality test, shown in Fig. 2, works as follows. The PFA scans its input string a , and with high probability enters an Indecision state (equivalently we say the outcome of the test ....

R. Freivalds. Probabilistic two way machines. In Proc. International Symposium on Mathemarical Foundations of Computer Science, volume 118, pages 33--45. Springer-Verlag, 1981.

....it has a log space deterministic simulation [37] In this survey, we focus on language membership problems and on complexity classes corresponding to space functions s(n) that are at least log n (thus omitting the notable body of work on very small space classes and probabilistic automata, e. g, [39, 13, 17, 11, 24, 23]) Within these restrictions, the aim is to be reasonably comprehensive, and apologies are o#ered in advance for the inevitable omissions. Section 2 presents definitions of the relevant models and complexity classes and their connection with Markov chains and matrix computations. Section 3 ....

R. Freivalds. Probabilistic two-way machines. Proceedings of the International Symposium on Mathematical Foundations of Computer Science, Springer-Verlag Lecture Notes in Computer Science, 188:33--45, 1981. 20

....0 initial 1 2 Figure 1.2 Coin ipping pfa that is equivalent to that in Figure 1. 5 3. 2PFA CONTAINS NON REGULAR LANGUAGES Before describing results on the limitations of pfa s, it is useful to rst see a pfa that does something surprising. We describe here a coin ipping 2pfa of Freivalds [Freivalds, 1981] that accepts the language fa n b n j n 0g with bounded error. Freivalds 2pfa does the following. First, deterministically) check that the input is of the form a n b m for some n 0; m 0 and that n = m mod (k 1) for some constant k and if not, reject. Here, k controls the error ....

....is accepted by P and by D, it is possible to determine whether L(D) L(P ) 5.1 UNDECIDABILITY OF THE EMPTINESS PROBLEM FOR 2PFA S Since 2pfa s accept nonregular languages, it is perhaps not surprising that the emptiness problem for 2pfa s is undecidable. This result was proved by Freivalds [Freivalds, 1981]. Theorem 4 The following problem is undecidable. Given a natural number k 2 and a 2pfa P recognizing a language L with acceptance threshold 1=2 1=k, determine whether the language L is empty. The proof of Theorem 4 is via a reduction from the halting problem for Turing machines on empty ....

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Freivalds, R. (1981). Probabilistic two-way machines. In Proc. of the International Symposium on Mathematical Foundations of Computer Science Springer-Verlag Lecture Notes in Computer Science, 188, pages 33-45.

....to move o# the left end of the input. The event defined by A is PA :# # # [0, 1] where PA (w)is the probability that the automaton in a computation on w # # # moves o# the right end entering a final state. The first results on probabilistic 2 way machines were presented by Freivalds [12]. In the oppposite to Rabin s result, Freivalds showed that 2PFA s can recognize nonregular languages also when the cut point is restricted to be isolated. The language 0 n 1 n n # N is an example. Dwork and Stockmeyer [10] completed this result showing that if a 2PFA recognizes a ....

....a corollary. Corollary 4.4 It is decidable whether two 2PFA s define the same event and whether a string is accepted by a given 2PFA with probability greater than a given value. The problem of the membership of a word to the language accepted by a 2PFA was already proved decidable by Freivalds [12]. In [1] it is also proved that the problem is in NC 2 . 4.2 Classes of languages Let state the following notation : L(1PFA) is the set of languages recognized by 1PFA s with cut point; L is (1PFA) is the set of languages recognized by 1PFA s with isolated cut point; L(2PFA) is the set ....

R. Freivalds, Probabilistic Two-way Machines, in Proc. Int. Symp. on Math. Found. of Comp. Sc.,LNCS118 (1981) 333--345.

....control and a two way read head. 2PFA with isolated cut point are referred to as 2BPFA [21] It is shown in [17] that 1PFA and 2PFA, with all real or all rational tran sition probabilities, accept the same classes of languages. As the language fa n b n jn 0g is accepted by a 2BPFA [13], the class of languages accepted by 2BPFA is larger than that of the 1PFA with isolated cut point. In [9] it is further shown that a 2PFA recognizing a non regular language (with bounded error probability) must use exponential expected time (for in nitely many inputs) Bounded error, polynomial ....

R. Freivalds, \Probabilistic two-way machines ", Proc. MFCS, Lecture Notes in Computer Science, Springer-Verlag, New York, Vol. 118, 33-45, 1981.

.... for several additional open problems, including the threshold isolation problem also raised in (Paz 1971) The work in (Condon Lipton 1989) in turn is based on an investigation of Interactive Proof Systems introduced in (Goldwasser, Micali, Rackoff 1985) and an elegant technique developed in (Freivalds 1981) to show the power of randomization in two way PFAs. The paper is organized as follows. The next section defines PFAs and the string existence problem, and sketches the reduction of (Condon Lipton 1989) highlighting aspects used in subsequent proofs. The remainder of the section establishes ....

.... the counter contents are represented in unary, this problem reduces to checking whether two strings have the same length: given a string a n b m , does n = m Although this question cannot be answered exactly by any PFA, a weak equality test developed in (Condon Lipton 1989) and inspired by (Freivalds 1981) can answer it in a strict and limited sense which is nonetheless sufficient to allow the reduction. The weak equality test works as follows. The PFA scans its input string a n b m , and with high probability enters an indecision state (or equivalently we say the outcome of the test is ....

Freivalds, R. 1981. Probabilistic two way machines.

....machines seem to be quite different. This even holds for the subclass of probabilistic machines without any nondeterminism. Freivalds has shown that probabilistic machines with constant space (probabilistic finite automata) are quite powerful already. They can accept nonregular languages (Freivalds 1981) . An example is the language COUNT : f1 n 01 m jn = mg which can be accepted by a probabilistic TM in constant space with an arbitrarily small error probability, that means COUNT 2 AM 1 Space(CON) BPSpace(CON) where BP stands for bounded error probabilistic machines (Monte Carlo) ....

R. Freivalds, Probabilistic two--way machines. In Proc. 10. Int. Symp. on Math. Found. of Comp. Science, LNCS, 1981, 33--45.

....Motivated by these reductions, we establish a connection between the number of heads of a multihead probabilistic finite automaton and the bandwidth of its configuration transition matrix for an input string. Partially inspired by this relation and using the competition method of Freivalds [Fr81] in the setting of unbounded error computation, we improve a result of Jung [Ju84] and find an apparently easier log space complete problem for PL. In the final, we discuss possibilities for deterministic simulation of probabilistic automata in small space. 2 Preliminaries In this section, we ....

....marks 2(c 3) k log m cells on its work tape. If its computation tries to use more, B rejects. Next, B checks whether u is of the form xw, by counting first the blocks of b s in w. Next, in a probabilistic way, it verifies whether w has the right form using a competition technique. Freivalds [Fr81] showed that for any fixed fl 0, there is a probabilistic finite automaton that can recognize a language containing strings like w with error below fl. The basic idea is to simulate (in finite control) two counters of magnitude c(fl) and d(fl) with the properties: 14 2( 1 2 ) d(fl) fl ....

Freivalds, R. Probabilistic two-way machines. Proceedings, Int. Sympos. Math. Found. of Comput. Sci. 1981, LNCS 118, 1981, pp. 33-45.

....finite automata with rational transition probabilities and rational cutpoints, can be recognized by O(n) space deterministic Turing machines. In 1973, Kuklin [Ku 1973] introduced the two way probabilistic finite automaton (2U PFA) as a generalization of PFA. Some time later, Freivalds [Fre 1981] proved that the language fa n b n j n 2 Ng can be recognized by a 2 way probabilistic finite automaton with isolated cutpoint (2 PFA) Jung [Jung 1984] found a O(log n log log n) space deterministic simulation for 2U PFA s and later Wang [Wang 1992] provided a much weaker O(n) space ....

....bound f(n) 2 o(log log log n) can be simulated deterministically in O(c f(n) log n) space, where c ia a constant depending on the size of the worktape alphabet of the simulated machine. The existence of languages with probabilistic space complexities at this level was pointed out by Freivalds [Fre 1981] 3 . In the nondeterministic case, on the other hand, there are no such languages since there is a space complexity gap below log log n. ffl Probabilistic Turing machines with space bound f(n) between Theta(log log log n) and Theta(log n) can be simulated deterministically in O(log n(f(n) ....

Freivalds, R. Probabilistic two-way machines. Proceedings, Int. Sympos. Math. Found. of Comput. Sci., Lecture Notes in Computer Science Vol. 118, Springer, 1981, pp. 33-45.

.... for several additional open problems, including the threshold isolation problem also raised in (Paz 1971) The work in (Condon Lipton 1989) in turn is based on an investigation of Interactive Proof Systems introduced in (Goldwasser, Micali, Rackoff 1985) and an elegant technique developed in (Freivalds 1981) to show the power of randomization in two way PFAs. The paper is organized as follows. The next section defines PFAs and the string existence problem, and sketches the reduction of (Condon Lipton 1989) highlighting aspects used in subsequent proofs. The remainder of the section establishes the ....

.... the counter contents are represented in unary, this problem reduces to checking whether two strings have the same length: given a string a n b m , does n = m Although this question cannot be answered exactly by any PFA, a weak equality test developed in (Condon Lipton 1989) and inspired by (Freivalds 1981) can answer it in a strict and limited sense which is nonetheless sufficient to allow the reduction. The weak equality test works as follows. The PFA scans its input string a n b m , and with high probability enters an indecision state (or equivalently we say the outcome of the test is ....

Freivalds, R. 1981. Probabilistic two way machines.

.... on 2qfa s, which are the quantum analogue of deterministic, nondeterministic and probabilistic 2 way finite state automata (2dfa s, 2nfa s and 2pfa s) While 2dfa s and 2nfa s are known to be equivalent in power to ordinary (1 way) deterministic automata [13, 20, 22] it was shown by Freivalds in [11] that the nonregular language fa m b m jm 1g could be recognized by a 2pfa with arbitrarily small error. However, the 2pfa s for fa m b m j m 1g defined by Freivalds require exponential expected time, and it was subsequently shown by Dwork and Stockmeyer [9, 10] that any 2pfa recognizing ....

R. Freivalds. Probabilistic two-way machines. In Proceedings of the International Symposium on Mathematical Foundations of Computer Science, volume 188 of Lecture Notes in Computer Science, pages 33--45. Springer-Verlag, 1981.

....ffl ) is an intermediate problem between Band Mat Inv(n) for which we do not know o(log 2 n) space deterministic algorithms to solve it, and Band Mat Inv(o(log n) for which we do know. For proving our result, we use the competitions method (and a padding technique) of Freivalds [Fr81]. Although so far it has been used only in the setting of bounded error computation and it seems to be natural only in this setting, when combined with some additional ideas the competition method can work for unbounded error computation as well. 2 Log space completeness for PL We use the standard ....

....its right neighbor with probability of error less than fl 1=2 n c 2 . To do this, on its work tape, B simulates two counters whose maximal values c 1 (fl) and c 2 (fl) satisfy the following conditions: 2( 1 2 ) c 2 (fl) fl and (1 Gamma 1 2 c 1 (fl) 1 ) c 2 (fl) 1 Gamma fl [Fr81]. Note that Freivalds implementation of the two counters is done in the finite control, a fact that guarantees only a constant value for fl. Our main observation is that the value of fl can be made smaller than 1=2 n c 2 using only negligible additional work space. Before we establish the ....

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Freivalds, R. Probabilistic two-way machines. Proceedings, MFCS 1981, Lecture Notes on Computer Science 118, pp. 33-45.

....probability 1 2 if it reaches a rejecting configuration. This gives (N)CFL j (P)CFL. In order to prove the properness, recall that L = fa n b n c n j n 2 Ng 62 (N)CFL, that L 0 = fa n b n j n 2 Ng is recognized by a one head, two way, bounded error probabilistic finite state automaton [Fre81], and that one head, two way, probabilistic finite state automata can be simulated by onehead, one way, probabilistic finite state automtat [Kan89] Thus, L 0 is in (P)CFL. Consider a probabilistic pda that, on input u = a k b l c m , checks probabilistically whether a k b l is in L ....

R. Freivalds. Probabilistic two-way machines. In Proceedings of the 6th Symposium on Mathematical Foundations of Computer Science, pages 33--45. SpringerVerlag Lecture Notes in Computer Science #118, 1981.

....an input x is the probability of eventually reaching the accepting configuration when processing x. The automaton accepts the input string if this probability exceeds some rational threshold (usually equal to 1 2) In the literature, the transition probabilities are often restricted to f0,1 2,1g [Fre81]. This restriction does not change the complexity classes we deal with; but for expositional reasons we prefer to work in the apparently more general setting of probabilistic automata having more general configuration transition probabilities (for example the set of rationals in [0,1] Moreover, ....

Freivalds, R. Probabilistic two-way machines. Proceedings, Int. Sympos. Math. Found. of Comput. Sci., LNCS 118, 1981, pp. 33-45.

.... (2qfa s) 2qfa s are the quantum analogue of deterministic, nondeterministic and probabilistic 2 way finite state automata (2dfa s, 2nfa s and 2pfa s) While 2dfa s and 2nfa s are known to be equivalent in power to ordinary (1 way) deterministic automata [20, 21, 14] it was shown by Freivalds in [11] that the non regular language fa m b m jm 1g could be recognized by a 2pfa with arbitrarily small error. However, the 2pfa s for fa m b m g defined by Freivalds require exponential expected time, and it was subsequently shown by Dwork and Stockmeyer [9, 10] and independently by Kaneps ....

R. Freivalds. Probabilistic two-way machines. In Proceedings of the International Symposium on Mathematical Foundations of Computer Science, volume 188 of Lecture Notes in Computer Science, pages 33--45. Springer-Verlag, 1981.

....2 way nite automaton with n states are proved to be strongly recognized by a Monte Carlo 2 way nite automaton with n O(n) states. This improves dramatically over the previously known result by M.Karpinski and R. Verbeek [10] which is also nontrivial since these languages can be nonregular [5]. For tally languages the increase in the number of states is proved to be only polynomial, and these languages are regular. 1 Introduction In most of papers on Computational Complexity the authors do not distinguish between the complexity of the recognition of a language and its complement. ....

....Council of Science y Research partially supported by the International Computer Science Institute, Berkeley, California, by the DFG grant KA 673 4 1, and by the ESPRIT BR Grants 7079 and ECUS030 their arguments are valid for the weak recognition as well. On the other hand, R. Freivalds proved in [5] that there is a nonregular language strongly recognized by a Monte Carlo 2 way nite automaton. M.Karpinski and R.Verbeek [10] proved that if a language is weakly recognized by a Monte Carlo 2 way nite automaton with n states then the language can be strongly recognized by a Monte Carlo 2 way ....

R.Freivalds, Probabilistic two-way machines. Lecture Notes in Computer Science, 118(1981), 33-45.

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R¯usi¸ns Freivalds. Probabilistic two-way machines. Lecture Notes in Computer Science, 188:33-45, 1981

....For instance, it was proved in [SHL65] that DLOGLOGSPACE contains nonregular languages but every language recognizable in o(log log n) space even by nondeterministic Turing machines is regular. However there are regular languages recognizable by Monte Carlo 2 way Turing machines in constant space[Fr81], and there are nontrivial space classes defined by log log log n, log Delta Delta Delta log n, log n for Monte Carlo 2 way Turing machines[KV87] In this section we prove some results motivating the naturalness of our approach to sublinear time complexity. Theorem 3.1 If n g(n) is the ....

Freivalds, R., Probabilistic two-way machines, LNCS, 118(1981), pp.33-45

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R. Freivalds. Probabilistic two way machines. In Proc. International Symposium on Mathemarical Foundations of Computer Science, volume 118, pages 33--45. SpringerVerlag, 1981.

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Freivalds, R. Probabilistic two-way machines. Proceedings, Int. Sympos. Math. Found. of Comput. Sci. 1981, LNCS 118, 1981, pp. 33-45.

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Rusins Freivalds. Probabilistic two-way machines. In Proceedings, Math. Found. Comput. Sci. '81, pages 33--45, New York/Berlin, 1981. Lecture Notes in Computer Science Vol. 118, Springer-Verlag.

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