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Determinization of Transducers Over Finite and Infinite Words
, 2001
"... We study the determinization of transducers over finite and infinite words. The first part of the paper is devoted to finite words. We recall the characterization of subsequential functions due to Choffrut. We describe here a known algorithm to determinize a transducer. In the case of infinite words ..."
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We study the determinization of transducers over finite and infinite words. The first part of the paper is devoted to finite words. We recall the characterization of subsequential functions due to Choffrut. We describe here a known algorithm to determinize a transducer. In the case of infinite
Recognizable Sets with Multiplicities in the Tropical Semiring
, 1988
"... The last ten years saw the emergence of some results about recognizable subsets of a free monoid with multiplicities in the MinPlus semiring. An interesting aspect of this theoretical body is that its discovery was motivated throughout by applications such as the finite power property, Eggan's ..."
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Cited by 44 (1 self)
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The last ten years saw the emergence of some results about recognizable subsets of a free monoid with multiplicities in the MinPlus semiring. An interesting aspect of this theoretical body is that its discovery was motivated throughout by applications such as the finite power property, Eggan's classical star height problem and the measure of the nondeterministic complexity of finite automata. We review here these results, their applications and point out some open problems. 1 Introduction One of the richest extensions of finite automaton theory is obtained by associating multiplicities to words, edges and states. Perhaps the most intuitive appearence of this concept is obtained by counting for every word the number of successful paths spelling it in a (nondeterministic) finite automaton. This is motivated by the formalization of ambiguity in a finite automaton and leads to the theory of recognizable subsets of a free monoid with multiplicities in the semiring of natural numbers. This...
1.1 Definition of Concatenation Hierarchies.................................. 15 1.2 Alternative Definitions andNormalForms................................ 17
"... Thomas Wilke. Mein Dank gilt ebenso Sven Kosub, Heribert Vollmer, Christiane ..."
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Thomas Wilke. Mein Dank gilt ebenso Sven Kosub, Heribert Vollmer, Christiane
On Semigroups of Matrices over the Tropical Semiring
, 1994
"... The tropical semiring M consists of the set of natural numbers extended with infinity, equipped with the operations of taking minimums (as semiring addition) and addition (as semiring multiplication). We use factorization forests to prove finiteness results related to semigroups of matrices over M. ..."
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The tropical semiring M consists of the set of natural numbers extended with infinity, equipped with the operations of taking minimums (as semiring addition) and addition (as semiring multiplication). We use factorization forests to prove finiteness results related to semigroups of matrices over M. Our method is used to recover results of Hashiguchi, Leung and the author in a unified combinatorial framework.
The Nondeterministic Complexity of a Finite Automaton
, 1990
"... We define the nondeterministic complexity of a finite automaton and show that there exist, for any integer p>=1, automata which need \Theta(k^{1/p}) nondeterministic transitions to spell words of length k. This leads to a subdivision of the family of recognizable Msubsets of a free monoid into a ..."
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Cited by 28 (2 self)
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We define the nondeterministic complexity of a finite automaton and show that there exist, for any integer p>=1, automata which need \Theta(k^{1/p}) nondeterministic transitions to spell words of length k. This leads to a subdivision of the family of recognizable Msubsets of a free monoid into a hierarchy whose members are indexed by polynomials, where M denotes the MinPlus semiring.
Relations over words and logic: a chronology
"... The purpose of this short note is to give credit to the right people who produced original work on the connection between rational relations and logic. Indeed, my experience is that some authors seem to partially ignore the literature or at least neglect to cite it correctly. It is probably due to t ..."
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The purpose of this short note is to give credit to the right people who produced original work on the connection between rational relations and logic. Indeed, my experience is that some authors seem to partially ignore the literature or at least neglect to cite it correctly. It is probably due to the fact that language theory and logic, though having largely filled the original gap which separated them, still have different backgrounds. I hope that recalling the chronology might be of some help. If I had some doubt about the necessity of this reminder, a recent experience proved it was justified. Indeed, I posted an early version of the present work on my web page. Kamal Lodaya from the University of Chennai happened to come across it, got interested and posed a few questions. Doing some bibliographical search he found that the relations which after Läuchli and Savioz I had called “special”, had in fact been introduced three years earlier by D. Angluin and D. N. Hoover as “regular prefix relations”. Now we come to the point. Given n finite, nonempty alphabets Σi, i = 1,..., n, I’m interested in the class of subsets, also called relations, of the direct product Σ ∗ 1 × · · · × Σ∗n which are rational (known as regular in the anglosaxon literature). A simple example: the relation which is the graph of the operation of concatenation of two words and which consists of all triples of the form (u, v, uv) where u, v ∈ Σ ∗ , is defined by the rational expression ∆ ∗ 1 ∆∗ 2 where ∆1 = � a∈Σ(a, 1, a) and ∆2 = � a∈Σ(1, a, a). These relations are also defined via an extension of the finite automata operating on tuples of words rather than on words, introduced by Rabin and Scott in the late fifties, [8]. They were studied by Elgot and Mezei who proved most of their general properties, [6]. The main decision issues were settled by Fischer and Rosenberg, [7]. It just happens that this class does not form a Boolean algebra unless n = 1 or all alphabets Σi’s are reduced to a single symbol. Until the mid eighties, only two subclasses of the rational relations were known to be closed under the Boolean operations, to wit the recognizable and the synchronous relations which are therefore natural candidates for logical definability. A new
Minimizing Subsequential Transducers: A Survey
 Comput. Sci
, 2001
"... This paper deals with the notion of subsequential transducer, i. e., of finite deterministic automaton whose transitions are provided with an "output". Its purpose is twofold. First it is meant to give better access to the result saying, in loose terms, that it is possible to define a noti ..."
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This paper deals with the notion of subsequential transducer, i. e., of finite deterministic automaton whose transitions are provided with an "output". Its purpose is twofold. First it is meant to give better access to the result saying, in loose terms, that it is possible to define a notion of morphism on subsequential transducers in such a way that the ordinary theory of deterministic automata carries over to subsequential transducers. In particular, given a function realized by some subsequential transducer, there exists a "minimal" subsequential transducer realizing it such that all trimmed subsequential transducers map onto it. This is a critical departure from the more general case of rational functions where the existence of such a minimal object is still unsettled, cf. [13]. Thus, though more complex than finite deterministic automata, subsequential transducers enjoy the same "syntactical" property and the techniques for proving this fact are not substantially different. This appeared in my doctorial thesis and in the proceedings of the ICALP'79 conference. To be more precise, this was actually stated in a slightly larger context, since the function was supposed to map the free monoid into the free group. In spite of being more general and easily "downgradable" to free monoids, this formalism has probably confused some readers and obscured the fact that it was dealing with free monoids also.
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