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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 70 (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
Definable Relations and FirstOrder Query Languages over Strings
"... We study analogs of classical relational calculus in the context of strings. We start by studying string logics. Taking a classical modeltheoretic approach, we fix a set of string operations and look at the resulting collection of definable relations. These form an algebra  a class of nary relati ..."
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Cited by 23 (8 self)
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We study analogs of classical relational calculus in the context of strings. We start by studying string logics. Taking a classical modeltheoretic approach, we fix a set of string operations and look at the resulting collection of definable relations. These form an algebra  a class of nary relations for every n, closed under projection and Boolean operations. We show that by choosing the string vocabulary carefully, we get string logics that have desirable properties: computable evaluation and normal forms. We identify five distinct models and study the differences in their modeltheory and complexity of evaluation. We identify a subset of these models which have additional attractive properties, such as finite VC dimension and quantifier elimination. Once you have a logic,
A ModelTheoretic Approach to Regular String Relations
, 2001
"... We study algebras of de nable string relations  classes of regular nary relations that arise as the definable sets within a model whose carrier is the set of all strings. We show that the largest such algebra  the collection of regular relations  has some quite undesirable computational and mode ..."
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Cited by 15 (4 self)
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We study algebras of de nable string relations  classes of regular nary relations that arise as the definable sets within a model whose carrier is the set of all strings. We show that the largest such algebra  the collection of regular relations  has some quite undesirable computational and modeltheoretic properties. In contrast, we exhibit several definable relation algebras that have much tamer behavior: for example, they admit quantifier elimination, and have finite VC dimension. We show that the properties of a definable relation algebra are not at all determined by the onedimensional definable sets. We give models whose definable sets are all starfree, but whose binary relations are quite complex, as well as models whose definable sets include all regular sets, but which are much more restricted and tractable than the full algebra of regular relations.
A Survey of Arithmetical Definability
, 2002
"... We survey definability and decidability issues related to firstorder fragments of arithmetic, with a special emphasis on Presburger and Skolem arithmetic and their (un)decidable extensions. ..."
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Cited by 2 (0 self)
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We survey definability and decidability issues related to firstorder fragments of arithmetic, with a special emphasis on Presburger and Skolem arithmetic and their (un)decidable extensions.
Automata and Numeration Systems
"... This article is a short survey on the following problem: given a set X ` ..."
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ON THE EXPANSION (N, +, 2 x) OF PRESBURGER ARITHMETIC FRANÇOISE POINT 1
"... This is based on a preprint ([9]) which appeared in the Proceedings of the fourth ..."
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This is based on a preprint ([9]) which appeared in the Proceedings of the fourth
Analogues of Hilbert’s tenth problem
"... Hilbert’s 10th problem asked: Give a procedure which, in a finite number of steps, can determine whether a polynomial equation (in several variables) with integer coefficients has or does not have integer solutions. The answer by Matiyasevich ([42]), following work of Davis, Putnam and J. Robinson ..."
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Hilbert’s 10th problem asked: Give a procedure which, in a finite number of steps, can determine whether a polynomial equation (in several variables) with integer coefficients has or does not have integer solutions. The answer by Matiyasevich ([42]), following work of Davis, Putnam and J. Robinson, was negative. Analogous questions can be asked for domains other than the ring of integers. Our presentation will consist of four parts. Part A will deal with positive (decidability) results for analogues of Hilbert’s tenth problem for substructures of the integers and for certain local rings. Part B will focus on the ‘parametric problem ’ and the relevance of Hilbert’s tenth problem to conjectures of Lang. Part C will deal with the analogue of Hilbert’s tenth problem for rings of Analytic and Meromorphic functions. Part D will be an informal discussion on the chances of proving a negative (or could it be positive?) answer to the analogue of Hilbert’s tenth problem for the field of rational numbers. Some familiarity with [12], [13], [50], [51] and [58] will be assumed.