Converting two-way nondeterministic unary automata into simpler automata

Geffert, Viliam; Mereghetti, Carlo; Pighizzini, Giovanni

doi:10.1016/s0304-3975(02)00403-6

Cited by 64 publications

(33 citation statements)

References 8 publications

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“…The first study of unary 2DFAs was undertaken by Chrobak [4], who has sketched an argument that an n-state 2DFA over a unary alphabet can be simulated by a Θ(G(n))-state 1DFA, where G(n) = e (1+o(1)) √ n ln n is the maximal order of a permutation on n elements, known as Landau's function [9]. Further work in this direction was done by Mereghetti and Pighizzini [10] and by Geffert, Mereghetti and Pighizzini [5], who similarly estimated the 2NFA-1DFA tradeoff. Their approach lies with considering only the periodic part of the language and using a general upper bound on the starting point of its periodicity, and thus leads only to asymptotic succinctness tradeoffs.…”

Section: Introductionmentioning

confidence: 99%

Describing Periodicity in Two-Way Deterministic Finite Automata Using Transformation Semigroups

Kunc

Okhotin

2011

Developments in Language Theory

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Abstract. A framework for the study of periodic behaviour of twoway deterministic finite automata (2DFA) is developed. Computations of 2DFAs are represented by a two-way analogue of transformation semigroups, every element of which describes the behaviour of a 2DFA on a certain string x. A subsemigroup generated by this element represents the behaviour on strings in x + . The main contribution of this paper is a description of all such monogenic subsemigroups up to isomorphism. This characterization is then used to show that transforming an n-state 2DFA over a one-letter alphabet to an equivalent sweeping 2DFA requires exactly n + 1 states, and transforming it to a one-way automaton requires exactly max 0 n G(n − ) + + 1 states, where G(k) is the maximum order of a permutation of k elements.

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Section: Introductionmentioning

confidence: 99%

Describing Periodicity in Two-Way Deterministic Finite Automata Using Transformation Semigroups

Kunc

Okhotin

2011

Developments in Language Theory

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“…Hence, the conversion does not significantly increase the number of the states. 3 A generalization of the Chrobak normal form to the two-way case has been obtained by Geffert, Mereghetti, and Pighizzini [13], and it is given in the next Theorem 5.1. In order to present it, it is useful to relax the notion of equivalence between automata, by allowing a finite number of "errors".…”

Section: Normal Forms For Unary Nondeterministic Automatamentioning

confidence: 95%

“…An inspection to the proof of Theorem 5.1 [13,15] shows that M and its computations have a very simple structure. In particular, in each sweep M uses a deterministic loop to count the input length modulo one integer.…”

Section: Normal Forms For Unary Nondeterministic Automatamentioning

confidence: 99%

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Two-Way Finite Automata: Old and Recent Results

Pighizzini

2013

Fundamenta Informaticae

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The notion of two-way automata was introduced at the very beginning of automata theory. In 1959, Rabin and Scott and, independently, Shepherdson, proved that these models, both in the deterministic and in the nondeterministic versions, have the same power of one-way automata, namely, they characterize the class of regular languages. In 1978, Sakoda and Sipser posed the question of the costs, in the number of the states, of the simulations of one-way and two-way nondeterministic automata by two-way deterministic automata. They conjectured that these costs are exponential. In spite of all attempts to solve it, this question is still open. In the last ten years the problem of Sakoda and Sipser was widely reconsidered and many new results related to it have been obtained. In this work we discuss some of them. In particular, we focus on the restriction to the unary case, namely the case of automata defined over the one letter input alphabet, and on the connections with open questions in space complexity.

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“…Hence, all (x j ) (k) are negative. 7 Here α and β (without subscripts) denote subsets of [h]. These names preserve notational symmetry, and should cause no confusion with the names (with subscripts) of the two partial functions in the inner behavior of M. If ϑ i x j ϑ i is live, then some path a * b * connects the two outer columns, for a * , b * ∈ [h] (Fig.…”

Section: The Hard Instancesmentioning

confidence: 99%