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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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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
Constraint Databases: A Survey
 Semantics in Databases, number 1358 in LNCS
, 1998
"... . Constraint databases generalize relational databases by finitely representable infinite relations. This paper surveys the state of the art in constraint databases: known results, remaining open problems and current research directions. The paper also describes a new algebra for databases with inte ..."
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Cited by 25 (3 self)
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. Constraint databases generalize relational databases by finitely representable infinite relations. This paper surveys the state of the art in constraint databases: known results, remaining open problems and current research directions. The paper also describes a new algebra for databases with integer order constraints and a complexity analysis of evaluating queries in this algebra. In memory of Paris C. Kanellakis 1 Introduction There is a growing interest in recent years among database researchers in constraint databases, which are a generalization of relational databases by finitely representable infinite relations. Constraint databases are parametrized by the type of constraint domains and constraint used. The good news is that for many parameters constraint databases leave intact most of the fundamental assumptions of the relational database framework proposed by Codd. In particular, 1. Constraint databases can be queried by constraint query languages that (a) have a semantics ba...
On Presentations of Algebraic Structures
 in Complexity, Logic and Recursion Theory
, 1995
"... This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicat ..."
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Cited by 17 (6 self)
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This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995
Linear quantifier elimination
 In Automated reasoning (IJCAR), volume 5195 of LNCS
, 2008
"... Abstract. This paper presents verified quantifier elimination procedures for dense linear orders (DLO), for real and for integer linear arithmetic. The DLO procedures are new. All procedures are defined and verified in the theorem prover Isabelle/HOL, are executable and can be applied to HOL formula ..."
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Abstract. This paper presents verified quantifier elimination procedures for dense linear orders (DLO), for real and for integer linear arithmetic. The DLO procedures are new. All procedures are defined and verified in the theorem prover Isabelle/HOL, are executable and can be applied to HOL formulae themselves (by reflection). 1
Computing the Wellfounded Semantics for Constraint Extensions of Datalog
 Lecture Notes in Computer Science
, 1997
"... . We present a new technique for computing the wellfounded semantics for constraint extensions of Datalog : . The method is based on tabulated resolution enhanced with a new refinement strategy for deriving negative conclusions. This approach leads to an efficient and terminating query evaluation ..."
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. We present a new technique for computing the wellfounded semantics for constraint extensions of Datalog : . The method is based on tabulated resolution enhanced with a new refinement strategy for deriving negative conclusions. This approach leads to an efficient and terminating query evaluation algorithm that preserves the goaloriented nature of the resolution based methods. 1 Introduction The wellfounded semantics of Datalog : programs [29, 30] provides a robust model for handling negation in deductive databases (and in general logic programming systems). There have been numerous proposals for query evaluation procedures under the wellfounded semantics. The two main approaches to computing the wellfounded semantics of Datalog : programs are the bottomup methods, usually based on the Alternating fixpoint [10, 13, 20, 31], or the topdown methods, based on a variation of SLDresolution [3, 4, 5, 19]. However, in all the cases, the query evaluation algorithms assume that th...
Reflecting Quantifier Elimination for Linear Arithmetic
"... Abstract. This paper formalizes and verifies quantifier elimination procedures for dense linear orders and for real and integer linear arithmetic in the theorem prover ..."
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Abstract. This paper formalizes and verifies quantifier elimination procedures for dense linear orders and for real and integer linear arithmetic in the theorem prover
Combining Theories Sharing Dense Orders
, 2003
"... The NelsonOppen combination method combines decision procedures for firstorder theories satisfying certain conditions into a single decision procedure for the union theory. The NelsonOppen combination method can be applied only if the signatures of the combined theories are disjoint. Combination ..."
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The NelsonOppen combination method combines decision procedures for firstorder theories satisfying certain conditions into a single decision procedure for the union theory. The NelsonOppen combination method can be applied only if the signatures of the combined theories are disjoint. Combination tableaux (Ctableaux) are an extension of Smullyan tableaux for combining firstorder theories whose signatures may not be disjoint. Ctableaux are sound and complete, but not terminating in general. In this paper we show that, when we combine firstorder theories that share the theory of dense order, Ctableaux can be made terminating without sacrificing completeness. Thus, Ctableaux provide a decision procedure for the combination of firstorder theories sharing the theory of dense order.
Quantifier Elimination and Real Closed Ordered Fields with a Predicate for the Powers of Two
, 2005
"... In this thesis we first review the model theory of quantifier elimination and investigate the logical relations among various quantifier elimination tests. In particular we prove the equivalence of two quantifier elimination tests for countable theories. Next we give a procedure for eliminating qu ..."
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In this thesis we first review the model theory of quantifier elimination and investigate the logical relations among various quantifier elimination tests. In particular we prove the equivalence of two quantifier elimination tests for countable theories. Next we give a procedure for eliminating quantifiers for the theory of real closed ordered fields with a predicate for the powers of two. This result was first obtained by van den Dries [20]. His method is modeltheoretic, which provides no apparent bounds on the complexity of a decision procedure. In the last section we give a complete axiomatization of the theory of real closed ordered fields with a predicate for the Fibonacci numbers. 1 Acknowledgements I thank my advisor Jeremy Avigad for the guidance he has provided me. I am grateful to James Cummings and Rami Grossberg for many helpful suggestions. I also thank Chris Miller for suggesting the problem considered in the last section of this thesis. 2 1