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Logic and precognizable sets of integers
 Bull. Belg. Math. Soc
, 1994
"... We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given ..."
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Cited by 66 (4 self)
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We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of CobhamSemenov, the original proof being published in Russian. 1
Logical definability and query languages over ranked and unranked trees
 ACM TOCL
"... We study relations on trees defined by firstorder constraints over a vocabulary that includes the tree extension relation T ≺ T ′ , holding if and only if every branch of T extends to a branch of T ′, unary nodetests, and a binary relation checking if the domains of two trees are equal. We conside ..."
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Cited by 7 (4 self)
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We study relations on trees defined by firstorder constraints over a vocabulary that includes the tree extension relation T ≺ T ′ , holding if and only if every branch of T extends to a branch of T ′, unary nodetests, and a binary relation checking if the domains of two trees are equal. We consider both ranked and unranked trees. These are trees with and without a restriction on the number of children of nodes. We adopt the modeltheoretic approach to tree relations and study relations definable over the structure consisting of the set of all trees and the above predicates. We relate definability of sets and relations of trees to computability by tree automata. We show that some natural restrictions correspond to familiar logics in the more classical setting, where every tree is a structure over a fixed vocabulary, and to logics studied in the context of XML pattern languages. We then look at relational calculi over collections of trees, and obtain quantifierrestriction results that give us bounds on the expressive power and complexity. As unrestricted relational calculi can express problems complete for each level of the polynomial hierarchy, we look at their restrictions, corresponding to the restricted logics over the family of all unranked trees, and find several calculi with low (NC 1) data complexity, while still expressing properties important for database and
TRANSFORMING STRUCTURES BY SET INTERPRETATIONS
"... We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic secondorder (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of eleme ..."
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Cited by 6 (1 self)
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We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic secondorder (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of elements of the orignal structure. The definition of these interpretations directly implies that they send structures with a decidable WMSO theory to structures with a decidable firstorder theory. In this paper, we investigate the expressive power of such interpretations applied to infinite deterministic trees. The results can be used in the study of automatic and treeautomatic structures.
Logical Definability and Query Languages over Unranked Trees
 Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science (LICS
, 2003
"... Unranked trees, that is, trees with no restriction on the number of children of nodes, have recently attracted much attention, primarily as an abstraction of XML documents. In this paper, we study logical definability over unranked trees, as well as collections of unranked trees, that can be viewed ..."
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Cited by 4 (1 self)
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Unranked trees, that is, trees with no restriction on the number of children of nodes, have recently attracted much attention, primarily as an abstraction of XML documents. In this paper, we study logical definability over unranked trees, as well as collections of unranked trees, that can be viewed as databases of XML documents. The traditional approach to definability is to view each tree as a structure of a fixed vocabulary, and study the expressive power of various logics on trees. A different approach, based on model theory, considers a structure whose universe is the set of all trees, and studies definable sets and relations; this approach extends smoothly to the setting of definability over collections of trees. We study the latter, modeltheoretic approach. We find sets of operations on unranked trees that define regular tree languages, and show that some natural restrictions correspond to logics studied in the context of XML pattern languages. We then look at relational calculi over collections of unranked trees, and obtain quantifierrestriction results that give us bounds on the expressive power and complexity. As unrestricted relational calculi can express problems complete for each level of the polynomial hierarchy, we look at their restrictions, corresponding to the restricted logics over the family of all unranked trees, and find several calculi with low (NC ) data complexity, that can express important XML properties like DTD validation and XPath evaluation.
On rational trees
 20TH INTERNATIONAL WORKSHOP ON COMPUTER SCIENCE LOGIC (CSL'06), SZEGED: HUNGARY
, 2006
"... Rational graphs are a family of graphs defined using labelled rational transducers. Unlike automatic graphs (defined using synchronized transducers) the first order theory of these graphs is undecidable, there is even a rational graph with an undecidable first order theory. In this paper we conside ..."
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Cited by 3 (2 self)
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Rational graphs are a family of graphs defined using labelled rational transducers. Unlike automatic graphs (defined using synchronized transducers) the first order theory of these graphs is undecidable, there is even a rational graph with an undecidable first order theory. In this paper we consider the family of rational trees, that is rational graphs which are trees. We prove that first order theory is decidable for this family. We also present counter examples showing that this result cannot be significantly extended both in terms of logic and of structure.
Bertrand Numeration Systems and Recognizability
, 1995
"... . There exist various wellknown characterizations of sets of numbers recognizable by a finite automaton, when they are represented in some integer base p 2. We show how to modify these characterizations, when integer bases p are replaced by linear numeration systems whose characteristic polynomial ..."
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. There exist various wellknown characterizations of sets of numbers recognizable by a finite automaton, when they are represented in some integer base p 2. We show how to modify these characterizations, when integer bases p are replaced by linear numeration systems whose characteristic polynomial is the minimal polynomial of a Pisot number. We also prove some related interesting properties. 1 Introduction Since the work of [9], sets of integers recognizable by finite automata have been studied in numerous papers. One of the jewels in this topic is the famous Cobham's theorem [11]: the only sets of numbers recognizable by finite automata, independently of the base of representation, are those which are ultimately periodic. Other studies are concerned with computation models equivalent to finite automata in the recognition of sets of integers. The proposed models use firstorder logical formulae [9], uniform substitutions [12], algebraic formal series [10]. We refer the reader to the...
BOUNDEDNESS IN LANGUAGES OF INFINITE WORDS
"... Abstract. We define a new class of languages of ωwords, strictly extending ωregular languages. One way to present this new class is by a type of regular expressions. The new expressions are an extension of ωregular expressions where two new variants of the Kleene star L ∗ are added: L B and L S. ..."
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Abstract. We define a new class of languages of ωwords, strictly extending ωregular languages. One way to present this new class is by a type of regular expressions. The new expressions are an extension of ωregular expressions where two new variants of the Kleene star L ∗ are added: L B and L S. These new exponents are used to say that parts of the input word have bounded size, and that parts of the input can have arbitrarily large sizes, respectively. For instance, the expression (a B b) ω represents the language of infinite words over the letters a, b where there is a common bound on the number of consecutive letters a. The expression (a S b) ω represents a similar language, but this time the distance between consecutive b’s is required to tend toward the infinite. We develop a theory for these languages, with a focus on decidability and closure. We define an equivalent automaton model, extending Büchi automata. The main technical result is a complementation lemma that works for languages where only one type of exponent—either L B or L S —is used. We use the closure and decidability results to obtain partial decidability results for the logic MSOLB, a logic obtained by extending monadic secondorder logic with new quantifiers that speak about the size of sets. 1.
TOWARDS A CHARACTERIZATION OF THE STARFREE SETS OF INTEGERS
, 2001
"... Abstract. Let U be a numeration system, a set X ⊆ N is Ustarfree if the set made up of the Urepresentations of the elements in X is a starfree regular language. Answering a question of A. de Luca and A. Restivo [10], we obtain a complete logical characterization of the Ustarfree sets of intege ..."
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Abstract. Let U be a numeration system, a set X ⊆ N is Ustarfree if the set made up of the Urepresentations of the elements in X is a starfree regular language. Answering a question of A. de Luca and A. Restivo [10], we obtain a complete logical characterization of the Ustarfree sets of integers for suitable numeration systems related to a Pisot number and in particular for integer base systems. For these latter systems, we study as well the problem of the base dependence. Finally, the case of kadic systems is also investigated. 1.