Jacques Sakarovitch - Academia.edu (original) (raw)

Papers by Jacques Sakarovitch

In this paper we investigate how it is possible to recover an automaton from a rational expressio... more In this paper we investigate how it is possible to recover an automaton from a rational expression that has been computed from that automaton. The notion of derived term of an expression, introduced by Antimirov, appears to be instrumental in this problem. The second important ingredient is the co-minimization of an automaton, a dual and generalized Moore algorithm on non-deterministic automata. We show here that if an automaton is then sufficiently “decorated”, the combination of these two algorithms gives the desired result. Reducing the amount of “decoration” is still the object of ongoing investigation. 1991 Mathematics Subject Classification. 68Q45, 68Q70. dedicated to Imre Simon on the occasion of his 60th birtday 1. A natural question Kleene’s theorem states the equality of two families of languages: the family of langagues described by rational (i.e. regular) expressions coincides with the family of languages accepted (or recognized) by finite automata — equality which is of...

J. Autom. Lang. Comb., 2010

Bounds are given on the number of broken derived terms (a variant of Antimirov’s ‘partial derivat... more Bounds are given on the number of broken derived terms (a variant of Antimirov’s ‘partial derivatives’) of a rational expression E. It is shown that this number is less than or equal to 2`(E) + 1 in the general case, where `(E) is the literal length of the expression E, and that the classical bound `(E) + 1 which holds for partial derivatives also holds for broken derived terms if E is in star normal form. In a second part of the paper, the influence of the bracketing of an expression on the number of its derived terms is also discussed.

Theoretical Computer Science, 2017

We present here the notion of signature of trees and of languages, and its relationships with the... more We present here the notion of signature of trees and of languages, and its relationships with the theory of numeration systems. The signature of an ordered infinite tree (of bounded degree) is an infinite (bounded) sequence of integers, the sequence of the degrees of the nodes taken in the visit order of the canonical breadth-first traversal of the tree. A prefix-closed language defines such a tree augmented with labels on arcs, hence is associated with a signature. This way of 'traversing' a language is related to the notion of abstract numeration system, due to Lecomte and Rigo. After having set in detail the framework of signature, we study and characterise the signatures of rational languages. Using a known construction from numeration system theory, we show that these signatures form a special subclass of morphic words. We then use this framework to give an alternative definition to morphic numeration systems (also called Dumont-Thomas numeration systems). We finally highlight that the classes of morphic numeration systems and of (prefix-closed) rational abstract numeration systems are essentially the same.

Lecture Notes in Computer Science, 2006

This survey paper reviews the means that allow to go from one representation of the languages to ... more This survey paper reviews the means that allow to go from one representation of the languages to the other and how, and to what extend, one can keep them small. Some emphasis is put on the comparison between the expressions that can be computed from a given automaton and on the construction of the derived term automaton of an expression.

Lecture Notes in Computer Science

LATIN 2010: Theoretical Informatics, 2010

We prove that the radix cross-section of a rational set for a length morphism, and more generally... more We prove that the radix cross-section of a rational set for a length morphism, and more generally for a rational function from a free monoid into N, is rational, a property that does not hold any more if the image of the function is a subset of a free monoid with two or more generators. proceedings short version 1 The purpose of this paper is to give a positive answer to a problem left open in an old paper by the second author ([11]) and to prove the following property, a refinement of the Cross-Section Theorem ([3]): Proposition 1. The radix cross-section of a rational set for a length morphism is rational.

Lecture Notes in Computer Science

It is established here that it is decidable whether a rational set of a free partially commutativ... more It is established here that it is decidable whether a rational set of a free partially commutative monoid (i.e. trace monoid) is recognizable or not if and only if the commutation relation is transitive (i.e. if the trace monoid is isomorphic to a free product of free commutative monoids). The bulk of the paper consists in a characterization of recognizable sets of free products via generalized finite automata.

The Mathematical Foundation of Informatics - Proceedings of the Conference, 2005

We define the notion of K-covering of automata with multiplicity (in a semiring K) that extend th... more We define the notion of K-covering of automata with multiplicity (in a semiring K) that extend the one of covering of automata. We make use of this notion, together with the Schützenberger construct that we have explained in a previous work and that we briefly recall here, in order to give a direct and constructive proof of a fundamental theorem on N-rational power series.

Computer Science – Theory and Applications, 2006

We show that two equivalent K-automata are conjugate to a third one, when K is equal to B, N, Z, ... more We show that two equivalent K-automata are conjugate to a third one, when K is equal to B, N, Z, or any (skew) field and that the same holds true for functional tranducers as well.

Automata, Languages and Programming, 2005

We prove that two automata with multiplicity in Z are equivalent, i.e. define the same rational s... more We prove that two automata with multiplicity in Z are equivalent, i.e. define the same rational series, if and only if there is a sequence of Z-coverings, co-Z-coverings, and circulations of −1, which transforms one automaton into the other. Moreover, the construction of these transformations is effective. This is obtained by combining two results: the first one relates coverings to conjugacy of automata, and is modeled after a theorem from symbolic dynamics; the second one is an adaptation of Schützenberger's reduction algorithm of representations in a field to representations in an Euclidean domain (and thus in Z).

Finite-State Methods and Natural Language Processing, 2008

We present an XML format that allows to describe a large class of finite weighted automata and tr... more We present an XML format that allows to describe a large class of finite weighted automata and transducers. Our design choices stem from our policy of making the implementation as simple as possible. This format has been tested for the communication between the modules of our automata manipulation platform Vaucanson, but this document is less an experiment report than a

Lecture Notes in Computer Science, 2013

It is decidable if a set of numbers, whose representation in a base b is a regular language, is u... more It is decidable if a set of numbers, whose representation in a base b is a regular language, is ultimately periodic. This was established by Honkala in 1986. We give here a structural description of minimal automata that accept an ultimately periodic set of numbers. We then show that it can be verified in linear time if a given minimal automaton meets this description. This yields a O(n log(n)) procedure for deciding whether a general deterministic automaton accepts an ultimately periodic set of numbers.

Lecture Notes in Computer Science, 2013

Vaucanson is an open source C ++ platform dedicated to the computation with finite weighted autom... more Vaucanson is an open source C ++ platform dedicated to the computation with finite weighted automata. It is generic: it allows to write algorithms that apply on a wide set of mathematical objects. Initiated ten years ago, several shortcomings were discovered along the years, especially problems related to code complexity and obfuscation as well as performance issues. This paper presents the concepts underlying Vaucanson 2, a complete rewrite of the platform that addresses these issues.

Lecture Notes in Computer Science, 2008

We give a new and conceptually different proof for the decidability of k-valuedness of transducer... more We give a new and conceptually different proof for the decidability of k-valuedness of transducers (a result due to Gurari and Ibarra), without resorting to any other kind of machines than transducers. In contrast with the previous proof, our algorithm takes into account the structure of the analysed transducers and yields better complexity bounds. With the same techniques, we also present a new proof, hopefully more easily understandable, for the decidability of bounded valuedness (a result due to Weber).

Automata theory lies at the foundation of computer science, and is vital to a theoretical underst... more Automata theory lies at the foundation of computer science, and is vital to a theoretical understanding of how computers work and what constitutes formal methods. This treatise gives a rigorous account of the topic and illuminates its real meaning by looking at the subject in a variety of ways. The first part of the book is organised around notions of rationality and recognisability. The second part deals with relations between words realised by finite automata, which not only exemplifies the automata theory but also illustrates the variety of its methods and its fields of application. Many exercises are included, ranging from those that test the reader, to those that are technical results, to those that extend ideas presented in the text. Solutions or answers to many of these are included in the book.

Theoretical Computer Science, 1985

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Theoretical Computer Science, 2005

This paper addresses the problem of turning a rational (i.e. regular) expression into a finite au... more This paper addresses the problem of turning a rational (i.e. regular) expression into a finite automaton. We formalize and generalize the idea of "partial derivatives" introduced in 1995 by Antimirov, in order to obtain a construction of an automaton with multiplicity from a rational expression describing a formal power series with coefficients in a semiring. We first define precisely what is such a rational expression with multiplicity and which hypothesis should be put on the semiring of coefficients in order to keep the usual identities. We then define the derivative of such a rational expression as a linear combination of expressions called derived terms and we show that all derivatives of a given expression are generated by a finite set of derived terms, that yields a finite automaton with multiplicity whose behaviour is the series denoted by the expression. We also prove that this automaton is a quotient of the standard (or Glushkov) automaton of the expression. Finally, we propose and discuss some possible modifications to our definition of derivation.

Theoretical Computer Science, 2006

Theoretical Computer Science, 1999

This paper presents properties of relations between words that are realized by defermini.svtic fi... more This paper presents properties of relations between words that are realized by defermini.svtic finite 2-tape automata. It has been made as complete as possible, and is structured by the systematic use of the matrix representation of automata. It is first shown that deterministic 2-tape automata are characterized as those which can be given a prefix matrix representation. Sc~~tzenberger construct on representations, the one that gives semi-monomial represen~tions for rational functions of words, is then applied to this prefix representation in order to obtain a new proof of the fact that the lexicographic selection of a deterministic rational relation on words is a rational function.

Theoretical Computer Science, 1998

We show how a construction on matrix representations of two tape automata proposed by Schiitzenbe... more We show how a construction on matrix representations of two tape automata proposed by Schiitzenberger to prove that rational functions are unambiguous can be given a central rble in the theory of relations and functions realized by finite automata, in such a way that the other basic results such as the "Cross-Section Theorem", its dual the theorem of rational uniformisation, or the decomposition theorem of rational functions into sequential functions, appear as direct and formal consequences of it.

In this paper we investigate how it is possible to recover an automaton from a rational expressio... more In this paper we investigate how it is possible to recover an automaton from a rational expression that has been computed from that automaton. The notion of derived term of an expression, introduced by Antimirov, appears to be instrumental in this problem. The second important ingredient is the co-minimization of an automaton, a dual and generalized Moore algorithm on non-deterministic automata. We show here that if an automaton is then sufficiently “decorated”, the combination of these two algorithms gives the desired result. Reducing the amount of “decoration” is still the object of ongoing investigation. 1991 Mathematics Subject Classification. 68Q45, 68Q70. dedicated to Imre Simon on the occasion of his 60th birtday 1. A natural question Kleene’s theorem states the equality of two families of languages: the family of langagues described by rational (i.e. regular) expressions coincides with the family of languages accepted (or recognized) by finite automata — equality which is of...

J. Autom. Lang. Comb., 2010

Bounds are given on the number of broken derived terms (a variant of Antimirov’s ‘partial derivat... more Bounds are given on the number of broken derived terms (a variant of Antimirov’s ‘partial derivatives’) of a rational expression E. It is shown that this number is less than or equal to 2`(E) + 1 in the general case, where `(E) is the literal length of the expression E, and that the classical bound `(E) + 1 which holds for partial derivatives also holds for broken derived terms if E is in star normal form. In a second part of the paper, the influence of the bracketing of an expression on the number of its derived terms is also discussed.

Theoretical Computer Science, 2017

We present here the notion of signature of trees and of languages, and its relationships with the... more We present here the notion of signature of trees and of languages, and its relationships with the theory of numeration systems. The signature of an ordered infinite tree (of bounded degree) is an infinite (bounded) sequence of integers, the sequence of the degrees of the nodes taken in the visit order of the canonical breadth-first traversal of the tree. A prefix-closed language defines such a tree augmented with labels on arcs, hence is associated with a signature. This way of 'traversing' a language is related to the notion of abstract numeration system, due to Lecomte and Rigo. After having set in detail the framework of signature, we study and characterise the signatures of rational languages. Using a known construction from numeration system theory, we show that these signatures form a special subclass of morphic words. We then use this framework to give an alternative definition to morphic numeration systems (also called Dumont-Thomas numeration systems). We finally highlight that the classes of morphic numeration systems and of (prefix-closed) rational abstract numeration systems are essentially the same.

Lecture Notes in Computer Science, 2006

This survey paper reviews the means that allow to go from one representation of the languages to ... more This survey paper reviews the means that allow to go from one representation of the languages to the other and how, and to what extend, one can keep them small. Some emphasis is put on the comparison between the expressions that can be computed from a given automaton and on the construction of the derived term automaton of an expression.

Lecture Notes in Computer Science

LATIN 2010: Theoretical Informatics, 2010

We prove that the radix cross-section of a rational set for a length morphism, and more generally... more We prove that the radix cross-section of a rational set for a length morphism, and more generally for a rational function from a free monoid into N, is rational, a property that does not hold any more if the image of the function is a subset of a free monoid with two or more generators. proceedings short version 1 The purpose of this paper is to give a positive answer to a problem left open in an old paper by the second author ([11]) and to prove the following property, a refinement of the Cross-Section Theorem ([3]): Proposition 1. The radix cross-section of a rational set for a length morphism is rational.

Lecture Notes in Computer Science

It is established here that it is decidable whether a rational set of a free partially commutativ... more It is established here that it is decidable whether a rational set of a free partially commutative monoid (i.e. trace monoid) is recognizable or not if and only if the commutation relation is transitive (i.e. if the trace monoid is isomorphic to a free product of free commutative monoids). The bulk of the paper consists in a characterization of recognizable sets of free products via generalized finite automata.

The Mathematical Foundation of Informatics - Proceedings of the Conference, 2005

We define the notion of K-covering of automata with multiplicity (in a semiring K) that extend th... more We define the notion of K-covering of automata with multiplicity (in a semiring K) that extend the one of covering of automata. We make use of this notion, together with the Schützenberger construct that we have explained in a previous work and that we briefly recall here, in order to give a direct and constructive proof of a fundamental theorem on N-rational power series.

Computer Science – Theory and Applications, 2006

We show that two equivalent K-automata are conjugate to a third one, when K is equal to B, N, Z, ... more We show that two equivalent K-automata are conjugate to a third one, when K is equal to B, N, Z, or any (skew) field and that the same holds true for functional tranducers as well.

Automata, Languages and Programming, 2005

We prove that two automata with multiplicity in Z are equivalent, i.e. define the same rational s... more We prove that two automata with multiplicity in Z are equivalent, i.e. define the same rational series, if and only if there is a sequence of Z-coverings, co-Z-coverings, and circulations of −1, which transforms one automaton into the other. Moreover, the construction of these transformations is effective. This is obtained by combining two results: the first one relates coverings to conjugacy of automata, and is modeled after a theorem from symbolic dynamics; the second one is an adaptation of Schützenberger's reduction algorithm of representations in a field to representations in an Euclidean domain (and thus in Z).

Finite-State Methods and Natural Language Processing, 2008

We present an XML format that allows to describe a large class of finite weighted automata and tr... more We present an XML format that allows to describe a large class of finite weighted automata and transducers. Our design choices stem from our policy of making the implementation as simple as possible. This format has been tested for the communication between the modules of our automata manipulation platform Vaucanson, but this document is less an experiment report than a

Lecture Notes in Computer Science, 2013

It is decidable if a set of numbers, whose representation in a base b is a regular language, is u... more It is decidable if a set of numbers, whose representation in a base b is a regular language, is ultimately periodic. This was established by Honkala in 1986. We give here a structural description of minimal automata that accept an ultimately periodic set of numbers. We then show that it can be verified in linear time if a given minimal automaton meets this description. This yields a O(n log(n)) procedure for deciding whether a general deterministic automaton accepts an ultimately periodic set of numbers.

Lecture Notes in Computer Science, 2013

Vaucanson is an open source C ++ platform dedicated to the computation with finite weighted autom... more Vaucanson is an open source C ++ platform dedicated to the computation with finite weighted automata. It is generic: it allows to write algorithms that apply on a wide set of mathematical objects. Initiated ten years ago, several shortcomings were discovered along the years, especially problems related to code complexity and obfuscation as well as performance issues. This paper presents the concepts underlying Vaucanson 2, a complete rewrite of the platform that addresses these issues.

Lecture Notes in Computer Science, 2008

We give a new and conceptually different proof for the decidability of k-valuedness of transducer... more We give a new and conceptually different proof for the decidability of k-valuedness of transducers (a result due to Gurari and Ibarra), without resorting to any other kind of machines than transducers. In contrast with the previous proof, our algorithm takes into account the structure of the analysed transducers and yields better complexity bounds. With the same techniques, we also present a new proof, hopefully more easily understandable, for the decidability of bounded valuedness (a result due to Weber).

Automata theory lies at the foundation of computer science, and is vital to a theoretical underst... more Automata theory lies at the foundation of computer science, and is vital to a theoretical understanding of how computers work and what constitutes formal methods. This treatise gives a rigorous account of the topic and illuminates its real meaning by looking at the subject in a variety of ways. The first part of the book is organised around notions of rationality and recognisability. The second part deals with relations between words realised by finite automata, which not only exemplifies the automata theory but also illustrates the variety of its methods and its fields of application. Many exercises are included, ranging from those that test the reader, to those that are technical results, to those that extend ideas presented in the text. Solutions or answers to many of these are included in the book.

Theoretical Computer Science, 1985

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Theoretical Computer Science, 2005

This paper addresses the problem of turning a rational (i.e. regular) expression into a finite au... more This paper addresses the problem of turning a rational (i.e. regular) expression into a finite automaton. We formalize and generalize the idea of "partial derivatives" introduced in 1995 by Antimirov, in order to obtain a construction of an automaton with multiplicity from a rational expression describing a formal power series with coefficients in a semiring. We first define precisely what is such a rational expression with multiplicity and which hypothesis should be put on the semiring of coefficients in order to keep the usual identities. We then define the derivative of such a rational expression as a linear combination of expressions called derived terms and we show that all derivatives of a given expression are generated by a finite set of derived terms, that yields a finite automaton with multiplicity whose behaviour is the series denoted by the expression. We also prove that this automaton is a quotient of the standard (or Glushkov) automaton of the expression. Finally, we propose and discuss some possible modifications to our definition of derivation.

Theoretical Computer Science, 2006

Theoretical Computer Science, 1999

This paper presents properties of relations between words that are realized by defermini.svtic fi... more This paper presents properties of relations between words that are realized by defermini.svtic finite 2-tape automata. It has been made as complete as possible, and is structured by the systematic use of the matrix representation of automata. It is first shown that deterministic 2-tape automata are characterized as those which can be given a prefix matrix representation. Sc~~tzenberger construct on representations, the one that gives semi-monomial represen~tions for rational functions of words, is then applied to this prefix representation in order to obtain a new proof of the fact that the lexicographic selection of a deterministic rational relation on words is a rational function.

Theoretical Computer Science, 1998

We show how a construction on matrix representations of two tape automata proposed by Schiitzenbe... more We show how a construction on matrix representations of two tape automata proposed by Schiitzenberger to prove that rational functions are unambiguous can be given a central rble in the theory of relations and functions realized by finite automata, in such a way that the other basic results such as the "Cross-Section Theorem", its dual the theorem of rational uniformisation, or the decomposition theorem of rational functions into sequential functions, appear as direct and formal consequences of it.