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18
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 68 (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
Expansions Of Dense Linear Orders With The Intermediate Value Property
 Consequences of U.S. Roadway Crashes, Accident Analysis and Prevention, Volume 25, Number 5
, 1993
"... this paper. First, we deal with some basic topological results. We then assume that R expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an ..."
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Cited by 14 (3 self)
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this paper. First, we deal with some basic topological results. We then assume that R expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure (Q, <, +, 0, 1, Z). Conventions. From now on, we assume that R has the intermediate value property (IVP for short). Formally adjoin endpoints # and +# to R in the usual fashion and put R# = R # {#}. A set A # R is convex if for all a, b # A with a # b, the set { x # R : a < x < b } is contained in A. An interval (in R) is a convex set I # R such that both inf I and sup I exist in R# (that is, I has endpoints in R# ). The usual notation is employed for the various kinds of intervals. All cartesian powers R n are equipped with the product topology induced by the interval topology on R. (R 0 denotes the onepoint space {#}.) "Definable" means "parametrically definable (in the structure under consideration)". Date: September 12, 2001. To appear in J. Symbolic Logic. 1 1. Basic results Our first result is easy, but important; it will be invoked frequently, usually without mention. Proposition 1.1. If A # R is definable then inf A and sup A exist in R# . Proof. Let # #= A # R be definable and bounded below. We claim that inf A # R. Replacing A by { y # R : #x # A, x # y } we may assume that A is convex and unbounded above. If min A exists, then we're done, so suppose otherwise; then A is open. It su#ces to show that max(R \ A) exists. Suppose not; then R \ A is open. Choose any a, b # R with a < b. Then the definable ...
Expansions Of The Real Line By Open Sets: OMinimality And Open Cores
 Department of Mathematics, The Ohio State University
, 1999
"... . The open core of a structure R := (R; !; : : : ) is defined to be the reduct (in the sense of definability) of R generated by all of its definable open sets. If the open core of R is ominimal, then the topological closure of any definable set has finitely many connected components. We show tha ..."
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Cited by 13 (10 self)
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. The open core of a structure R := (R; !; : : : ) is defined to be the reduct (in the sense of definability) of R generated by all of its definable open sets. If the open core of R is ominimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of R is finite or uncountable, or if R defines addition and multiplication and every definable open subset of R has finitely many connected components, then the open core of R is ominimal. An expansion R of the real line (R; !) is ominimal if every definable subset of R is a finite union of points and open intervals (that is, has finitely many connected components). Such structuresparticularly, ominimal expansions of the field of real numbershave many nice properties, and are of interest not only to model theorists, but to analysts and geometers as well. (See e.g. [D2], [DM] for expositions of the subject.) Conventions. Throughout, given A ` R, "Adef...
STRUCTURES HAVING OMINIMAL OPEN CORE
, 2008
"... The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have ominimal open core are investigated, with emphasis on expansions of densely o ..."
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Cited by 12 (3 self)
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The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have ominimal open core are investigated, with emphasis on expansions of densely ordered groups. The first main result establishes conditions under which an expansion of a densely ordered group has an ominimal open core. Specifically, the following is proved: Let R be an expansion of a densely ordered group (R, <, ∗) that is definably complete and satisfies the uniform finiteness property. Then the open core of R is ominimal. Two examples of classes of structures that are not ominimal yet have ominimal open core are discussed: dense pairs of ominimal expansions of ordered groups, and expansions of ominimal structures by generic predicates. In particular, such structures have open core interdefinable with the original ominimal structure. These examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane, a phenomenon that can occur in dense pairs but not in expansions by generic predicates. The property of having no dense graphs is examined and related to uniform finiteness, definable completeness, and having ominimal open core.
Tameness in Expansions of the Real Field
, 2001
"... What might it mean for a firstorder expansion of the field of real numbers to be tame or well behaved? In recent years, much attention has been paid by model theorists and realanalytic geometers to the ominimal setting: expansions of the real field in which every definable set has finitely many c ..."
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Cited by 9 (1 self)
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What might it mean for a firstorder expansion of the field of real numbers to be tame or well behaved? In recent years, much attention has been paid by model theorists and realanalytic geometers to the ominimal setting: expansions of the real field in which every definable set has finitely many connected components. But there are expansions of the real field that define sets with infinitely many connected components, yet are tame in some welldefined sense (e.g., the topological closure of every definable set has finitely many connected components, or every definable set has countably many connected components). The analysis of such structures often requires a mixture of modeltheoretic, analyticgeometric and descriptive settheoretic techniques. An underlying idea is that firstorder definability, in combination with the field structure, can be used as a tool for determining how complicated are given sets of real numbers. This paper is based on a lecture that I delivered at Logic Colloquium '01 (Vienna). I thank the organizers for inviting me to address the Colloquium. Global conventions. Throughout, m, n and p denote arbitrary elements of N (the nonnegative integers). Given a firstorder structure M, with underlying set M , "definable" (in M) means "definable in M with parameters from M " unless otherwise noted. If no ambient space M n is specified, then "definable set" means "definable subset of some M n ". I use "reduct" and "expansion" in the sense of definability, that is, given structures M 1 and M 2 with underlying set M , I say that M 1 is a reduct of M 2 equivalently, M 2 is an expansion of M 1 , or M 2 expands M 1 if every set definable in M 1 is definable in M 2 . For the most part, we shall be concerned with the definable sets of a struc...
Quantifier elimination for the reals with a predicate for the powers of two
 Theoretical Computer Science
"... Abstract. In [5], van den Dries showed that the theory of the reals with a predicate for the integer powers of two admits quantifier elimination in an expanded language, and is hence decidable. He gave a modeltheoretic argument, which provides no apparent bounds on the complexity of a decision proc ..."
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Abstract. In [5], van den Dries showed that the theory of the reals with a predicate for the integer powers of two admits quantifier elimination in an expanded language, and is hence decidable. He gave a modeltheoretic argument, which provides no apparent bounds on the complexity of a decision procedure. We provide a syntactic argument that yields a procedure that is primitive recursive, although not elementary. In particular, we show that it is possible to eliminate a single block of existential quantifiers in time 20, where n is the O(n) length of the input formula and 2x k denotes kfold iterated exponentiation. 1.
Dependent pairs
 Preprint 146 on MODNET Preprint server
, 2008
"... Abstract. We prove that certain pairs of ordered structures are dependent. Among these structures are dense and tame pairs of ominimal structures and further the real field with a multiplicative subgroup with the Mann property. 1. ..."
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Abstract. We prove that certain pairs of ordered structures are dependent. Among these structures are dense and tame pairs of ominimal structures and further the real field with a multiplicative subgroup with the Mann property. 1.
TAME GEOMETRIES IN MODEL THEORY by
, 2006
"... Abstract. — In this Proceedings paper we describe a new notion of tame geometry by the author and F. Loeser, named bminimality, put it in context, and compare it with other notions like ominimality, Cminimality, pminimality, and so on. Proofs are given elsewhere. 1. ..."
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Cited by 1 (1 self)
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Abstract. — In this Proceedings paper we describe a new notion of tame geometry by the author and F. Loeser, named bminimality, put it in context, and compare it with other notions like ominimality, Cminimality, pminimality, and so on. Proofs are given elsewhere. 1.
EXPANSIONS WHICH INTRODUCE NO NEW OPEN SETS
"... Abstract. We consider the question of when an expansion of a topological structure has the property that every open set definable in the expansion is definable in the original structure. This question is related to and inspired by recent work of Dolich, Miller and Steinhorn on the property of having ..."
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Abstract. We consider the question of when an expansion of a topological structure has the property that every open set definable in the expansion is definable in the original structure. This question is related to and inspired by recent work of Dolich, Miller and Steinhorn on the property of having ominimal open core. We answer the question in a fairly general setting and provide conditions which in practice are often easy to check. We give a further characterisation in the special case of an expansion by a generic predicate. 1.