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Tractable Disjunctions of Linear Constraints: Basic Results and Applications to Temporal Reasoning
 Theoretical Computer Science
, 1996
"... We study the problems of deciding consistency and performing variable elimination for disjunctions of linear inequalities and disequations with at most one inequality per disjunction. This new class of constraints extends the class of generalized linear constraints originally studied by Lassez an ..."
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Cited by 49 (2 self)
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We study the problems of deciding consistency and performing variable elimination for disjunctions of linear inequalities and disequations with at most one inequality per disjunction. This new class of constraints extends the class of generalized linear constraints originally studied by Lassez and McAloon. We show that deciding consistency of a set of constraints in this class can be done in polynomial time. We also present a variable elimination algorithm which is similar to Fourier's algorithm for linear inequalities. Finally, we use these results to provide new temporal reasoning algorithms for the OrdHorn subclass of Allen's interval formalism. We also show that there is no low level of local consistency that can guarantee global consistency for the OrdHorn subclass. This property distinguishes the OrdHorn subclass from the pointizable subclass (for which strong 5consistency is sufficient to guarantee global consistency), and the continuous endpoint subclass (for whi...
From Local to Global Consistency in Temporal Constraint Networks
 In Proceedings of the 1st International Conference on Principles and Practice of Constraint Programming (CP'95), volume 976 of LNCS
, 1995
"... We study the problem of global consistency for several classes of quantitative temporal constraints which include inequalities, inequations and disjunctions of inequations. In all cases that we consider we identify the level of local consistency that is necessary and sufficient for achieving global ..."
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Cited by 16 (5 self)
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We study the problem of global consistency for several classes of quantitative temporal constraints which include inequalities, inequations and disjunctions of inequations. In all cases that we consider we identify the level of local consistency that is necessary and sufficient for achieving global consistency and present an algorithm which achieves this level. As a byproduct of our analysis, we also develop an interesting minimal network algorithm. 1 Introduction One of the most important notions found in the constraint satisfaction literature is global consistency [Fre78]. In a globally consistent constraint set all interesting constraints are explicitly represented and the projection of the solution set on any subset of the variables can be computed by simply collecting the constraints involving these variables. An important consequence of this property is that a solution can be found by backtrackfree search [Fre82]. Enforcing global consistency can take an exponential amount of ti...
Abstracting Numerical Values in CLP(H,N)
, 1994
"... ing Numerical Values in CLP(H,N) Gerda Janssens 1 , Maurice Bruynooghe 1 , Vincent Englebert 2 1 Department of Computer Science, K.U. Leuven Celestijnenlaan 200A, B3001 Heverlee, Belgium 2 Institut d'Informatique, Facult'es Universitaires Notre Dame de la Paix rue GrandGagnage 21, B5000 Na ..."
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Cited by 6 (2 self)
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ing Numerical Values in CLP(H,N) Gerda Janssens 1 , Maurice Bruynooghe 1 , Vincent Englebert 2 1 Department of Computer Science, K.U. Leuven Celestijnenlaan 200A, B3001 Heverlee, Belgium 2 Institut d'Informatique, Facult'es Universitaires Notre Dame de la Paix rue GrandGagnage 21, B5000 Namur, Belgium Abstract. The paper defines approximations for the numerical leaves of variables in CLP(H,N) constraint systems. The abstractions are based on intervals which are computed by narrowing rules. The novelty of this approach lays in the fact that intervals are used as abstraction and that narrowing rules do not only correspond to numerical constraints but also to unification constraints. In the first abstraction the impact of the narrowing rules is limited. A prototype implementation has been developed and the obtained results are sufficiently precise to recognise (future) redundant constraints. The abstraction can be extended (1) by incorporating the narrowing rules more globally (...
Tractable Disjunctions of Linear Constraints
 Theoretical Computer Science
, 1996
"... We study the problems of deciding consistency and performing variable elimination for disjunctions of linear inequalities and inequations with at most one inequality per disjunction. This new class of constraints extends the class of generalized linear constraints originally studied by Lassez and ..."
Abstract
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We study the problems of deciding consistency and performing variable elimination for disjunctions of linear inequalities and inequations with at most one inequality per disjunction. This new class of constraints extends the class of generalized linear constraints originally studied by Lassez and McAloon. We show that deciding consistency of a set of constraints in this class can be done in polynomial time. We also present a variable elimination algorithm which is similar to Fourier's algorithm for linear inequalities. Finally, we use these results to improve the best known temporal reasoning algorithms for the OrdHorn subclass of Allen's interval formalism. This answers an open question posed by Nebel and Burckert. 1 Introduction Linear constraints over the reals have recently been studied in depth by researchers in constraint logic programming (CLP) and constraint databases (CDB) [JM94, KKR95, Kou94c]. Two very important operations in CLP and CDB systems are deciding cons...
Eliminating Variables in General Constraint Logic ∗
"... Drawing inferences from a set of general constraint clauses is known as a difficult problem. A general approach is based on the idea of eliminating some or all variables involved. In the particular case of propositional logic, this approach leads to a simple procedure that incorporates the wellknow ..."
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Drawing inferences from a set of general constraint clauses is known as a difficult problem. A general approach is based on the idea of eliminating some or all variables involved. In the particular case of propositional logic, this approach leads to a simple procedure that incorporates the wellknown resolution principle. The purpose of this paper is to show how the resolution principle can be extended to constraint logic where the knowledge is given as a set of constraint clauses. The result is a general variable elimination method. The paper shows that the elimination problem can always be reduced to the problem of eliminating the variable from a (conjunctive) set of atomic constraints. Variabele elimination has a number of possible applications such as satisfiability testing, hypotheses testing, constraint solving, argumentative reasoning, and many others. 1