17,000 research outputs found

    On probabilistic analog automata

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    We consider probabilistic automata on a general state space and study their computational power. The model is based on the concept of language recognition by probabilistic automata due to Rabin and models of analog computation in a noisy environment suggested by Maass and Orponen, and Maass and Sontag. Our main result is a generalization of Rabin's reduction theorem that implies that under very mild conditions, the computational power of the automaton is limited to regular languages

    Limited automata and unary languages

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    Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. When d = 1 these models characterize regular languages. An exponential gap between the size of limited automata accepting unary languages and the size of equivalent finite automata is proved. Since a similar gap was already known from unary contextfree grammars to finite automata, also the conversion of such grammars into limited automata is investigated. It is proved that from each unary context-free grammar it is possible to obtain an equivalent 1-limited automaton whose description has a size which is polynomial in the size of the grammar. Furthermore, despite the exponential gap between the sizes of limited automata and of equivalent unary finite automata, there are unary regular languages for which d-limited automata cannot be significantly smaller than equivalent finite automata, for any arbitrarily large d

    Context-dependent nondeterminism for pushdown automata

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    AbstractPushdown automata using a limited and unlimited amount of nondeterminism are investigated. Moreover, nondeterministic steps are allowed only within certain contexts, i.e., in configurations that meet particular conditions. The relationships of the accepted language families with closures of the deterministic context-free languages (DCFL) under regular operations are studied. For example, automata with unbounded nondeterminism that have to empty their pushdown store up to the initial symbol in order to make a guess are characterized by the regular closure of DCFL. Automata that additionally have to reenter the initial state are (almost) characterized by the Kleene star closure of the union closure of the prefix-free deterministic context-free languages. Pushdown automata with bounded nondeterminism are characterized by the union closure of DCFL in any of the considered contexts. Proper inclusions between all language classes discussed are shown. Finally, closure properties of these families under AFL operations are investigated

    Forgetting 1-Limited Automata

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    We introduce and investigate forgetting 1-limited automata, which are single-tape Turing machines that, when visiting a cell for the first time, replace the input symbol in it by a fixed symbol, so forgetting the original contents. These devices have the same computational power as finite automata, namely they characterize the class of regular languages. We study the cost in size of the conversions of forgetting 1-limited automata, in both nondeterministic and deterministic cases, into equivalent one-way nondeterministic and deterministic automata, providing optimal bounds in terms of exponential or superpolynomial functions. We also discuss the size relationships with two-way finite automata. In this respect, we prove the existence of a language for which forgetting 1-limited automata are exponentially larger than equivalent minimal deterministic two-way automata.Comment: In Proceedings NCMA 2023, arXiv:2309.0733

    Once-Marking and Always-Marking 1-Limited Automata

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    Single-tape nondeterministic Turing machines that are allowed to replace the symbol in each tape cell only when it is scanned for the first time are also known as 1-limited automata. These devices characterize, exactly as finite automata, the class of regular languages. However, they can be extremely more succinct. Indeed, in the worst case the size gap from 1-limited automata to one-way deterministic finite automata is double exponential. Here we introduce two restricted versions of 1-limited automata, once-marking 1-limited automata and always-marking 1-limited automata, and study their descriptional complexity. We prove that once-marking 1-limited automata still exhibit a double exponential size gap to one-way deterministic finite automata. However, their deterministic restriction is polynomially related in size to two-way deterministic finite automata, in contrast to deterministic 1-limited automata, whose equivalent two-way deterministic finite automata in the worst case are exponentially larger. For always-marking 1-limited automata, we prove that the size gap to one-way deterministic finite automata is only a single exponential. The gap remains exponential even in the case the given machine is deterministic. We obtain other size relationships between different variants of these machines and finite automata and we present some problems that deserve investigation.Comment: In Proceedings AFL 2023, arXiv:2309.0112

    One-Tape Turing Machine Variants and Language Recognition

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    We present two restricted versions of one-tape Turing machines. Both characterize the class of context-free languages. In the first version, proposed by Hibbard in 1967 and called limited automata, each tape cell can be rewritten only in the first dd visits, for a fixed constant d≥2d\geq 2. Furthermore, for d=2d=2 deterministic limited automata are equivalent to deterministic pushdown automata, namely they characterize deterministic context-free languages. Further restricting the possible operations, we consider strongly limited automata. These models still characterize context-free languages. However, the deterministic version is less powerful than the deterministic version of limited automata. In fact, there exist deterministic context-free languages that are not accepted by any deterministic strongly limited automaton.Comment: 20 pages. This article will appear in the Complexity Theory Column of the September 2015 issue of SIGACT New

    From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity

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    The equivalence of finite automata and regular expressions dates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the present paper we tour a fragment of the literature and summarize results on upper and lower bounds on the conversion of finite automata to regular expressions and vice versa. We also briefly recall the known bounds for the removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free nondeterministic devices. Moreover, we report on recent results on the average case descriptional complexity bounds for the conversion of regular expressions to finite automata and brand new developments on the state elimination algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527

    Formal Languages in Dynamical Systems

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    We treat here the interrelation between formal languages and those dynamical systems that can be described by cellular automata (CA). There is a well-known injective map which identifies any CA-invariant subshift with a central formal language. However, in the special case of a symbolic dynamics, i.e. where the CA is just the shift map, one gets a stronger result: the identification map can be extended to a functor between the categories of symbolic dynamics and formal languages. This functor additionally maps topological conjugacies between subshifts to empty-string-limited generalized sequential machines between languages. If the periodic points form a dense set, a case which arises in a commonly used notion of chaotic dynamics, then an even more natural map to assign a formal language to a subshift is offered. This map extends to a functor, too. The Chomsky hierarchy measuring the complexity of formal languages can be transferred via either of these functors from formal languages to symbolic dynamics and proves to be a conjugacy invariant there. In this way it acquires a dynamical meaning. After reviewing some results of the complexity of CA-invariant subshifts, special attention is given to a new kind of invariant subshift: the trapped set, which originates from the theory of chaotic scattering and for which one can study complexity transitions.Comment: 23 pages, LaTe
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