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37
Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
 J. Amer. Math. Soc
, 1996
"... Recall that a subset of R n is called semialgebraic if it can be represented as a (finite) boolean combination of sets of the form {�α ∈ R n: p(�α) =0}, {�α ∈R n: q(�α)>0}where p(�x), q(�x) arenvariable polynomials with real coefficients. A map from R n to R m is called semialgebraic if its gr ..."
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Cited by 128 (4 self)
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Recall that a subset of R n is called semialgebraic if it can be represented as a (finite) boolean combination of sets of the form {�α ∈ R n: p(�α) =0}, {�α ∈R n: q(�α)>0}where p(�x), q(�x) arenvariable polynomials with real coefficients. A map from R n to R m is called semialgebraic if its graph, considered
Definable sets in ordered structures
 Bull. Amer. Math. Soc. (N.S
, 1984
"... Abstract. This paper introduces and begins the study of a wellbehaved class of linearly ordered structures, the ^minimal structures. The definition of this class and the corresponding class of theories, the strongly ©minimal theories, is made in analogy with the notions from stability theory of m ..."
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Cited by 127 (8 self)
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Abstract. This paper introduces and begins the study of a wellbehaved class of linearly ordered structures, the ^minimal structures. The definition of this class and the corresponding class of theories, the strongly ©minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizations of Cminimal ordered groups and rings. Several other simple results are collected in §3. The primary tool in the analysis of ¿¡minimal structures is a strong analogue of "forking symmetry, " given by Theorem 4.2. This result states that any (parametrically) definable unary function in an (5minimal structure is piecewise either constant or an orderpreserving or reversing bijection of intervals. The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all N0categorical ¿¡¡minimal structures (Theorem 6.1). 1. Introduction. The
Finitely Representable Databases
, 1995
"... : We study classes of infinite but finitely representable databases based on constraints, motivated by new database applications such as geographical databases. We formally define these notions and introduce the concept of query which generalizes queries over classical relational databases. We prove ..."
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Cited by 56 (8 self)
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: We study classes of infinite but finitely representable databases based on constraints, motivated by new database applications such as geographical databases. We formally define these notions and introduce the concept of query which generalizes queries over classical relational databases. We prove that in this context the basic properties of queries (satisfiability, containment, equivalence, etc.) are nonrecursive. We investigate the theory of finitely representable models and prove that it differs strongly from both classical model theory and finite model theory. In particular, we show that most of the well known theorems of either one fail (compactness, completeness, locality, 0/1 laws, etc.). An immediate consequence is the lack of tools to consider the definability of queries in the relational calculus over finitely representable databases. We illustrate this very challenging problem through some classical examples. We then mainly concentrate on dense order databases, and exhibit...
Weakly ominimal structures and real closed fields
 Trans. Amer. Math. Soc
"... Abstract. A linearly ordered structure is weakly ominimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly ominimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results ..."
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Cited by 40 (6 self)
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Abstract. A linearly ordered structure is weakly ominimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly ominimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly ominimal structures. Foremost among these, we show that every weakly ominimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly ominimal structures, patterned, as much as possible, after that for ominimal structures. 1.
Exponentiation is hard to avoid
 Proc. Amer. Math. Soc
, 1994
"... Abstract. Let 3t be an Ominimal expansion of the field of real numbers. If 31 is not polynomially bounded, then the exponential function is definable (without parameters) in 32. If 31 is polynomially bounded, then for every definable function / : R — ► R, / not ultimately identically 0, there exis ..."
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Cited by 37 (3 self)
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Abstract. Let 3t be an Ominimal expansion of the field of real numbers. If 31 is not polynomially bounded, then the exponential function is definable (without parameters) in 32. If 31 is polynomially bounded, then for every definable function / : R — ► R, / not ultimately identically 0, there exist c, r e R, c ^ 0, such that x> » xT: (0, +oo)> R is definable in 31 and limJC_+oc. f(x)/xr = c. In the following, let 31: = (R, <, 0, 1,+,•,...) be an expansion of the ordered field of real numbers. "Definable " means firstorder definable in ¿% with parameters from R. A function f: X — ► R, AT ç R, is said to be definable if its graph is definable. We may, whenever convenient, assume that we deal with totally defined functions by setting partial functions equal to 0 off their domain of definition. For the rest of this note, all functions mentioned are of one variable. We say that ¿ÏÏ is polynomially bounded if, for every definable function /, there exists N £ N such that ultimately /(x)  < xN. ("Ultimately " abbreviates
Querying Aggregate Data
, 1999
"... We introduce a firstorder language with real polynomial arithmetic and aggregation operators (count, iterated sum and multiply), which is well suited for the definition of aggregate queries involving complex statistical functions. It offers a good tradeoff between expressive power and complexity, ..."
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Cited by 32 (2 self)
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We introduce a firstorder language with real polynomial arithmetic and aggregation operators (count, iterated sum and multiply), which is well suited for the definition of aggregate queries involving complex statistical functions. It offers a good tradeoff between expressive power and complexity, with a tractable data complexity. Interestingly, some fundamental properties of firstorder with real arithmetic are preserved in the presence of aggregates. In particular, there is an effective quantifier elimination for formulae with aggregation. We consider the problem of querying data that has already been aggregated in aggregate views, and focus on queries with an aggregation over a conjunctive query. Our main conceptual contribution is the introduction of a new equivalence relation among conjunctive queries, the isomorphism modulo a product. We prove that the equivalence of aggregate queries such as for instance averages reduces to it. Deciding if two queries are isomorphic modulo a p...
Queries with Arithmetical Constraints
 Theoretical Computer Science
, 1997
"... In this paper, we study the expressive power and the complexity of firstorder logic with arithmetic, as a query language over relational and constraint databases. We consider constraints over various domains (N, Z, Q, and R), and with various arithmetical operations (6, +, \Theta, etc.). We first c ..."
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Cited by 29 (3 self)
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In this paper, we study the expressive power and the complexity of firstorder logic with arithmetic, as a query language over relational and constraint databases. We consider constraints over various domains (N, Z, Q, and R), and with various arithmetical operations (6, +, \Theta, etc.). We first consider the data complexity of firstorder queries. We prove in particular that linear queries can be evaluated in AC 0 over finite integer databases, and in NC 1 over linear constraint databases. This improves previously known bounds. We also show that over all domains, enough arithmetic lead to arithmetical queries, therefore, showing the frontiers of constraints for database purposes. We then tackle the problem of the expressive power, with the definability of the parity and the connectivity, which are the most classical examples of queries not expressible in firstorder logic over finite structures. We prove that these two queries are firstorder definable in presence of (enough) ari...
Safe constraint queries
 In PODS'98
"... We extend some of the classical characterization theorems of relational database theory  particularly those related to query safety  to the context where database elements come with xed interpreted structure, and where formulae over elements of that structure can be used in queries. We show that ..."
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Cited by 26 (7 self)
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We extend some of the classical characterization theorems of relational database theory  particularly those related to query safety  to the context where database elements come with xed interpreted structure, and where formulae over elements of that structure can be used in queries. We show that the addition of common interpreted functions such as real addition and multiplication to the relational calculus preserves important characterization theorems of the relational calculus, and also preserves certain combinatorial properties of queries. Our main result of the rst kind is that there is a syntactic characterization of the collection of safe queries over the relational calculus supplemented by a wide class of interpreted functions  a class that includes addition, multiplication, and exponentiation  and that this characterization gives us an interpreted analog of the concept of rangerestricted query from the uninterpreted setting. Furthermore, our rangerestricted queries are particularly intuitive for the relational calculus with real arithmetic, and give a natural syntax for safe queries in the presence of polynomial functions. We use these characterizations to show that safety is decidable for Boolean combinations of conjunctive queries for a large class of interpreted structures. We show a dichotomy theorem that sets a polynomial bound on the growth of the output of a query that might refer to addition, multiplication and exponentiation. We apply the above results for nite databases to get results on constraint databases, representing potentially innite objects. We start by getting syntactic characterizations of the queries on constraint databases that preserve geometric conditions in the constraint data model. We consider classes of convex polytopes, polyhedra, and compact semilinear sets, the latter corresponding to many spatial applications. We show how to give an eective syntax to safe queries, and prove that for conjunctive queries the preservation properties are decidable. 1