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25
ModelTheoretic Methods in Combined Constraint Satisfiability
 Journal of Automated Reasoning
, 2004
"... We extend NelsonOppen combination procedure to the case of theories which are compatible with respect to a common subtheory in the shared signature. The notion of compatibility relies on model completions and related concepts from classical model theory. ..."
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Cited by 40 (10 self)
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We extend NelsonOppen combination procedure to the case of theories which are compatible with respect to a common subtheory in the shared signature. The notion of compatibility relies on model completions and related concepts from classical model theory.
Unions of NonDisjoint Theories and Combinations of Satisfiability Procedures
 THEORETICAL COMPUTER SCIENCE
, 2001
"... In this paper we outline a theoretical framework for the combination of decision procedures for constraint satisfiability. We describe a general combination method which, given a procedure that decides constraint satisfiability with respect to a constraint theory T1 and one that decides constraint s ..."
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Cited by 35 (3 self)
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In this paper we outline a theoretical framework for the combination of decision procedures for constraint satisfiability. We describe a general combination method which, given a procedure that decides constraint satisfiability with respect to a constraint theory T1 and one that decides constraint satisfiability with respect to a constraint theory T2, produces a procedure that (semi)decides constraint satisfiability with respect to the union of T1 and T2. We provide a number of modeltheoretic conditions on the constraint language and the component constraint theories for the method to be sound and complete, with special emphasis on the case in which the signatures of the component theories are nondisjoint. We also describe some general classes of theories to which our combination results apply, and relate our approach to some of the existing combination methods in the field.
Decidability and undecidability results for NelsonOppen and rewritebased decision procedures
 In Proc. IJCAR3, U. Furbach and
, 2006
"... Abstract. In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in arbitrary and in infinite models, respectively. We exhibit a theory T1 such that satisfiability is decidable, but s ..."
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Cited by 22 (15 self)
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Abstract. In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in arbitrary and in infinite models, respectively. We exhibit a theory T1 such that satisfiability is decidable, but satisfiability in infinite models is undecidable. It follows that satisfiability in T1 ∪ T2 is undecidable, whenever T2 has only infinite models, even if signatures are disjoint and satisfiability in T2 is decidable. In the second part of the paper we strengthen the NelsonOppen decidability transfer result, by showing that it applies to theories over disjoint signatures, whose satisfiability problem, in either arbitrary or infinite models, is decidable. We show that this result covers decision procedures based on rewriting, complementing recent work on combination of theories in the rewritebased approach to satisfiability. 1
Connecting manysorted theories
 The Journal of Symbolic Logic
, 2007
"... Abstract. Basically, the connection of two manysorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of ..."
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Cited by 18 (5 self)
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Abstract. Basically, the connection of two manysorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of universal formulae in the component theories transfers to their connection. In addition, we consider variants of the basic connection scheme. 1
Canonization for Disjoint Unions of Theories
, 2003
"... If there exist ecient procedures (canonizers) for reducing terms of two rstorder theories to canonical form, can one use them to construct such a procedure for terms of the disjoint union of the two theories? We prove this is possible whenever the original theories are convex. As an application, w ..."
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Cited by 14 (4 self)
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If there exist ecient procedures (canonizers) for reducing terms of two rstorder theories to canonical form, can one use them to construct such a procedure for terms of the disjoint union of the two theories? We prove this is possible whenever the original theories are convex. As an application, we prove that algorithms for solving equations in the two theories (solvers) cannot be combined in a similar fashion. These results are relevant to the widely used Shostak's method for combining decision procedures for theories. They provide the rst rigorous answers to the questions about the possibility of directly combining canonizers and solvers.
A New Combination Procedure for the Word Problem that Generalizes Fusion Decidability Results in Modal Logics
 In David A. Basin and Michaël Rusinowitch, editors, IJCAR ’04
, 2004
"... Previous results for combining decision procedures for the word problem in the nondisjoint case do not apply to equational theories induced by modal logicswhose combination is not disjoint since they share the theory of Boolean algebras. Conversely, decidability results for the fusion of mod ..."
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Cited by 13 (7 self)
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Previous results for combining decision procedures for the word problem in the nondisjoint case do not apply to equational theories induced by modal logicswhose combination is not disjoint since they share the theory of Boolean algebras. Conversely, decidability results for the fusion of modal logics are strongly tailored towards the special theories at hand, and thus do not generalize to other equational theories.
Hierarchical and modular reasoning in complex theories: The case of local theory extensions
 In Proc. 6th Int. Symp. Frontiers of Combining Systems (FroCos 2007), LNCS 4720
, 2007
"... Abstract. We present an overview of results on hierarchical and modular reasoning in complex theories. We show that for a special type of extensions of a base theory, which we call local, hierarchic reasoning is possible (i.e. proof tasks in the extension can be hierarchically reduced to proof tasks ..."
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Cited by 10 (8 self)
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Abstract. We present an overview of results on hierarchical and modular reasoning in complex theories. We show that for a special type of extensions of a base theory, which we call local, hierarchic reasoning is possible (i.e. proof tasks in the extension can be hierarchically reduced to proof tasks w.r.t. the base theory). Many theories important for computer science or mathematics fall into this class (typical examples are theories of data structures, theories of free or monotone functions, but also functions occurring in mathematical analysis). In fact, it is often necessary to consider complex extensions, in which various types of functions or data structures need to be taken into account at the same time. We show how such local theory extensions can be identified and under which conditions locality is preserved when combining theories, and we investigate possibilities of efficient modular reasoning in such theory combinations. We present several examples of application domains where local theories and local theory extensions occur in a natural way. We show, in particular, that various phenomena analyzed in the verification literature can be explained in a unified way using the notion of locality. 1
Connecting manysorted structures and theories through adjoint functions
 In Proc. 5th FroCoS
, 2005
"... functions ..."
A comprehensive combination framework
, 2006
"... We define a general notion of a fragment within higherorder type theory; a procedure for constraint satisfiability in combined fragments is outlined, following NelsonOppen schema. The procedure is in general only sound, but it becomes terminating and complete when the shared fragment enjoys suitab ..."
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Cited by 5 (2 self)
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We define a general notion of a fragment within higherorder type theory; a procedure for constraint satisfiability in combined fragments is outlined, following NelsonOppen schema. The procedure is in general only sound, but it becomes terminating and complete when the shared fragment enjoys suitable noetherianity conditions and admits an abstract version of a ‘KeislerShelah like’ isomorphism theorem. We show that this general decidability transfer result covers recent work on combination in firstorder theories as well as in various intensional logics such as description, modal, and temporal logics.
Combining Constraint Solving
, 2001
"... this paper. On the one hand, dening a semantics for the combined system may depend on methods and results from formal logic and universal algebra. On the other hand, an ecient combination of the actual constraint solvers often requires the possibility of communication and cooperation between the sol ..."
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Cited by 5 (0 self)
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this paper. On the one hand, dening a semantics for the combined system may depend on methods and results from formal logic and universal algebra. On the other hand, an ecient combination of the actual constraint solvers often requires the possibility of communication and cooperation between the solvers.