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A Proof Theoretic View of Constraint Programming
, 1998
"... We provide here a proof theoretic account of constraint programming that attempts to capture the essential ingredients of this programming style. We exemplify it by presenting proof rules for linear constraints over interval domains, and illustrate their use by analyzing the constraint propagation p ..."
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Cited by 26 (8 self)
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We provide here a proof theoretic account of constraint programming that attempts to capture the essential ingredients of this programming style. We exemplify it by presenting proof rules for linear constraints over interval domains, and illustrate their use by analyzing the constraint propagation process for the SEND + MORE = MONEY puzzle. We also show how this approach allows one to build new constraint solvers. 1 Introduction 1.1 Motivation One of the most interesting recent developments in the area of programming has been constraint programming. A prominent instance of it is constraint logic programming exemplified by such programming languages as CLP(R), Prolog III or ECL i PS e . But recently also imperative constraint programming languages emerged, such as 2LP of McAloon & Tretkoff (1995) or CLAIRE of Caseau & Laburthe (1996). (For an overview of this area and related references see Hentenryck, Saraswat & et al. (1996)). The aim of this paper is to explain the essence of t...
AutomataDriven Automated Induction
 Information and Computation
, 1996
"... . This work investigates inductive theorem proving techniques for firstorder functions whose meaning and domains can be specified by Horn Clauses built up from the equality and finitely many unary membership predicates. In contrast with other works in the area, constructors are not assumed to be fr ..."
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Cited by 19 (9 self)
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. This work investigates inductive theorem proving techniques for firstorder functions whose meaning and domains can be specified by Horn Clauses built up from the equality and finitely many unary membership predicates. In contrast with other works in the area, constructors are not assumed to be free. Techniques originating from tree automata are used to describe ground constructor terms in normal form, on which the induction proofs are built up. Validity of (free) constructor clauses is checked by an original technique relying on the recent discovery of a complete axiomatisation of finite trees and their rational subsets. Validity of clauses with defined symbols or nonfree constructor terms is reduced to the latter case by appropriate inference rules using a notion of ground reducibility for these symbols. We show how to check this property by generating proof obligations which can be passed over to the inductive prover. 1 Introduction The need for large formal proofs has lead to t...
RuleBased Constraint Programming
 Fundamenta Informaticae
, 1998
"... In this paper we present a view of constraint programming based on the notion of rewriting controlled by strategies. We argue that this concept allows us to describe in a unified way the constraint solving mechanism as well as the metalanguage needed to manipulate the constraints. This has the a ..."
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Cited by 9 (2 self)
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In this paper we present a view of constraint programming based on the notion of rewriting controlled by strategies. We argue that this concept allows us to describe in a unified way the constraint solving mechanism as well as the metalanguage needed to manipulate the constraints. This has the advantage to provide descriptions that are very close to the proof theoretical setting used now to describe constraint manipulations like unification or numerical constraint solving. We examplify the approach by presenting examples of constraint solvers descriptions and combinations written in the ELAN language. 1
Equational constraint solving using quasisolved forms
 In Proceedings of the 18th International Workshop on Unification (UNIF’04
, 2004
"... Abstract. In this paper, we present a deterministic syntactic method for solving general equational constraints. Our method is based on a class of equational formulas, called answers, which allows restricted universal quantification and for which the satisfiability test is not hard but not trivial. ..."
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Abstract. In this paper, we present a deterministic syntactic method for solving general equational constraints. Our method is based on a class of equational formulas, called answers, which allows restricted universal quantification and for which the satisfiability test is not hard but not trivial. Thus, we consider it as a quasisolved form. The procedure keeps, at each step of quantifier elimination, the inner formula being a disjunction of answers. For that, we only need two basic operations negation and conjunction on answers which can be efficiently implemented. With respect to purely existential solved forms, answers provide two main advantages for efficiency purposes: (1) they are weaker restricted, thus each quantifier elimination step requires less transformational work and (2) they are more compact or represent greater sets of solutions so, each step deals with a smaller quantity of disjuncts. 1
Workshop Programme International Workshop on Unification (UNIF’04)
"... UNIF is the main international meeting on unification. Unification is concerned with the problem of identifying given terms, either syntactically or modulo a given logical theory. Syntactic unification is the basic operation of most automated reasoning systems, and unification modulo theories can be ..."
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UNIF is the main international meeting on unification. Unification is concerned with the problem of identifying given terms, either syntactically or modulo a given logical theory. Syntactic unification is the basic operation of most automated reasoning systems, and unification modulo theories can be used, for instance, to build in special equational theories into theorem provers. The aim of UNIF 2004, as that of the previous meetings, is to to bring together people interested in unification, present recent (even unfinished) work, and discuss new ideas and trends in unification and related fields. In particular, it is intended to offer a good opportunity for young researches and researchers working in related areas to get an overview of the current state of the art in unification theory and get in contact with the experts in the field.
Automated Equational Reasoning in Nondeterministic λCalculi Modulo Theories H*
, 2009
"... In this thesis I study four extensions of untyped λcalculi all under the maximally coarse semantics of the theory H ∗ (observable equality), and implement a system for reasoning about and storing abstract knowledge expressible in languages with these extensions. The extensions are: (1) a semilattic ..."
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In this thesis I study four extensions of untyped λcalculi all under the maximally coarse semantics of the theory H ∗ (observable equality), and implement a system for reasoning about and storing abstract knowledge expressible in languages with these extensions. The extensions are: (1) a semilattice operation J, the join w.r.t the Scott ordering; (2) a random mixture R for stochastic λcalculus; (3) a computational comonad 〈code,apply,eval,quote, {−}〉 for Gödel codes modulo provable equality; and (4) a Π 1 1complete oracle O. I develop three languages from combinations of these extensions. The syntax of these languages is always simple: each is a finitely generated combinatory algebra. The semantics of these languages are various fragments of Dana Scott’s D ∞ models. Although the languages use ideas
Proving Termination of Constraint Solver Programs
"... www.informatik.unimuenchen.de/~fruehwir/ Abstract. We adapt and extend existing approaches to termination in rulebased languages (logic programming and rewriting systems) to prove termination of actually implemented CHR constraint solvers. CHR (Constraint Handling Rules) are a declarative language ..."
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www.informatik.unimuenchen.de/~fruehwir/ Abstract. We adapt and extend existing approaches to termination in rulebased languages (logic programming and rewriting systems) to prove termination of actually implemented CHR constraint solvers. CHR (Constraint Handling Rules) are a declarative language especially designed for writing constraint solvers. CHR are a concurrent constraint logic programming language consisting of multiheaded guarded rules that rewrite constraints into simpler ones until they are solved. The approach allows to prove termination of many constraint solvers, from Boolean and arithmetic to terminological and pathconsistent constraints. Because of multiheads, our termination orders must consider conjunctions, while atomic formulas suÆce in usual approaches. Our results indicate that in practice, proving termination for concurrent constraint logic programs may not be harder than for other classes of logic programming languages, contrary to what has been feared in the literature. 1