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Combination of Constraint Solving Techniques: An Algebraic Point of View
 In Proceedings of the 6th International Conference on Rewriting Techniques and Applications, volume 914 of Lecture Notes in Computer Science
"... . In a previous paper we have introduced a method that allows one to combine decision procedures for unifiability in disjoint equational theories. Lately, it has turned out that the prerequisite for this method to applynamely that unification with socalled linear constant restrictions is dec ..."
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. In a previous paper we have introduced a method that allows one to combine decision procedures for unifiability in disjoint equational theories. Lately, it has turned out that the prerequisite for this method to applynamely that unification with socalled linear constant restrictions is decidable in the single theoriesis equivalent to requiring decidability of the positive fragment of the first order theory of the equational theories. Thus, the combination method can also be seen as a tool for combining decision procedures for positive theories of free algebras defined by equational theories. Complementing this logical point of view, the present paper isolates an abstract algebraic property of free algebras called combinabilitythat clarifies why our combination method applies to such algebras. We use this algebraic point of view to introduce a new proof method that depends on abstract notions and results from universal algebra, as opposed to technical manipul...
A decision procedure for the existential theory of term algebras with the KnuthBendix ordering
, 2000
"... We show the decidability of the existential theory of term algebras with any KnuthBendix ordering by giving a procedure for solving KnuthBendix ordering constraints. ..."
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Cited by 12 (4 self)
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We show the decidability of the existential theory of term algebras with any KnuthBendix ordering by giving a procedure for solving KnuthBendix ordering constraints.
Orienting rewrite rules with the KnuthBendix order
 Information and Computation
"... 2). We show that both the orientability problem is Pcomplete. Also we show that if a system is orientable using a realvalued instance of KBO, then it is also orientable using an integervalued instance of KBO. Therefore, all our results hold both for the integervalued and the realvalued KBO. 1 I ..."
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Cited by 12 (1 self)
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2). We show that both the orientability problem is Pcomplete. Also we show that if a system is orientable using a realvalued instance of KBO, then it is also orientable using an integervalued instance of KBO. Therefore, all our results hold both for the integervalued and the realvalued KBO. 1 Introduction In this section we give an informal overview of the results proved in this paper. The formal definitions will be given in the next section.
Solved Forms for Path Ordering Constraints
 in `In Proc. 10th International Conference on Rewriting Techniques and Applications (RTA
, 1999
"... . A usual technique in symbolic constraint solving is to apply transformation rules until a solved form is reached for which the problem becomes simple. Ordering constraints are wellknown to be reducible to (a disjunction of) solved forms, but unfortunately no polynomial algorithm deciding the ..."
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. A usual technique in symbolic constraint solving is to apply transformation rules until a solved form is reached for which the problem becomes simple. Ordering constraints are wellknown to be reducible to (a disjunction of) solved forms, but unfortunately no polynomial algorithm deciding the satisfiability of these solved forms is known. Here we deal with a different notion of solved form, where fundamental properties of orderings like transitivity and monotonicity are taken into account. This leads to a new family of constraint solving algorithms for the full recursive path ordering with status (RPOS), and hence as well for other path orderings like LPO, MPO, KNS and RDO, and for all possible total precedences and signatures. Apart from simplicity and elegance from the theoretical point of view, the main contribution of these algorithms is on efficiency in practice. Since guessing is minimized, and, in particular, no linear orderings between the subterms are guessed, ...
KnuthBendix constraint solving is NPcomplete
 IN PROCEEDINGS OF 28TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP), VOLUME 2076 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2000
"... We show that the problem of solving KnuthBendix ordering constraints is NPcomplete, as a corollary we show that the existential firstorder theory of any term algebra with the KnuthBendix ordering is NPcomplete. ..."
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Cited by 11 (3 self)
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We show that the problem of solving KnuthBendix ordering constraints is NPcomplete, as a corollary we show that the existential firstorder theory of any term algebra with the KnuthBendix ordering is NPcomplete.
Algorithms, datastructures, and other issues in efficient automated deduction
 Automated Reasoning. 1st. International Joint Conference, IJCAR 2001, number 2083 in LNAI
, 2001
"... Abstract. Algorithms and datastructures form the kernel of any efficient theorem prover. In this abstract we discuss research on algorithms and datastructures for efficient theorem proving based on our experience with the theorem prover Vampire. We also briefly overview other works related to algori ..."
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Abstract. Algorithms and datastructures form the kernel of any efficient theorem prover. In this abstract we discuss research on algorithms and datastructures for efficient theorem proving based on our experience with the theorem prover Vampire. We also briefly overview other works related to algorithms and datastructures, and to efficient theorem proving in general. 1
Syntactic Unification Problems under Constrained Substitutions
, 1996
"... ... This paper is a collection of results on the decidability and the computational complexity of a syntactic unification problem under constrained substitutions. A number of decidable, undecidable, tractable and intractable results of the problem are presented. Since a unification problem under con ..."
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... This paper is a collection of results on the decidability and the computational complexity of a syntactic unification problem under constrained substitutions. A number of decidable, undecidable, tractable and intractable results of the problem are presented. Since a unification problem under constrained substitutions can be regarded as an ordersorted unification problem with term declarations such that the number of sorts is only one, the results presented in this paper also indicate how the intractability of ordersorted unification problems is reduced by restricting the number of sorts to one.
A Methodological View of Constraint Solving
, 1996
"... Constraints have become very popular during the last decade. Constraints allow to define sets of data by means of logical formulae. Our goal here is to survey the notion of constraint system and to give examples of constraint systems operating on various domains, such as natural, rational or real nu ..."
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Cited by 6 (2 self)
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Constraints have become very popular during the last decade. Constraints allow to define sets of data by means of logical formulae. Our goal here is to survey the notion of constraint system and to give examples of constraint systems operating on various domains, such as natural, rational or real numbers, finite domains, and term domains. We classify the different methods used for solving constraints, syntactic methods based on transformations, semantic methods based on adequate representations of constraints, hybrid methods combining transformations and enumerations. Examples are used throughout the paper to illustrate the concepts and methods. We also discuss applications of constraints to various fields, such as programming, operations research, and theorem proving. y CNRS and LRI, Bat. 490, Universit'e de Paris Sud, 91405 ORSAY Cedex, France fcomon, jouannaudg@lri.lri.fr z COSYTEC, Parc Club Orsay Universit'e, 4 Rue Jean Rostand, 91893 Orsay Cedex, France dincbas@cosytec.fr x ...