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A Minimality Study for Set Unification
, 1997
"... A unification algorithm is said to be minimal for a unification problem if it generates exactly a (minimal) complete set of mostgeneral unifiers, without instances, and without repetitions. The aim of this paper is to present a combinatorial minimality study for a significant collection of sample p ..."
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Cited by 10 (7 self)
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A unification algorithm is said to be minimal for a unification problem if it generates exactly a (minimal) complete set of mostgeneral unifiers, without instances, and without repetitions. The aim of this paper is to present a combinatorial minimality study for a significant collection of sample problems that can be used as benchmarks for testing any setunification algorithm. Based on this combinatorial study, a new SetUnification Algorithm (named SUA) is also described and proved to be minimal for all the analyzed problems. Furthermore, an existing nave setunification algorithm has also been tested to show its bad behavior for most of the sample problems.
Computational Complexity of Simultaneous Elementary Matching Problems (Extended Abstract)
 EDS), PROCEEDINGS 20TH INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE, PRAGUE (CZECH REPUBLIC). LECTURE NOTES IN COMPUTER SCIENCE
, 1995
"... The simultaneous elementary Ematching problem for an equational theory E is to decide whether there is an Ematcher for a given system of equations in which the only function symbols occurring in the terms to be matched are the ones constrained by the equational axioms of E. We study the computat ..."
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Cited by 6 (4 self)
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The simultaneous elementary Ematching problem for an equational theory E is to decide whether there is an Ematcher for a given system of equations in which the only function symbols occurring in the terms to be matched are the ones constrained by the equational axioms of E. We study the computational complexity of simultaneous elementary matching problems for the equational theories A of semigroups, AC of commutative semigroups, and ACU of commutative monoids. In each case, we delineate the boundary between NPcompleteness and solvability in polynomial time by considering two parameters, the number of equations in the systems and the number of constant symbols in the signature. Moreover, we analyze further the intract...
Unification Algorithms Cannot be Combined in Polynomial Time
 in Proceedings of the 13th International Conference on Automated Deduction, M.A. McRobbie and J.K. Slaney (Eds.), Springer LNAI 1104
, 1996
"... . We establish that there is no polynomialtime general combination algorithm for unification in finitary equational theories, unless the complexity class #P of counting problems is contained in the class FP of function problems solvable in polynomialtime. The prevalent view in complexity theory is ..."
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Cited by 4 (0 self)
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. We establish that there is no polynomialtime general combination algorithm for unification in finitary equational theories, unless the complexity class #P of counting problems is contained in the class FP of function problems solvable in polynomialtime. The prevalent view in complexity theory is that such a collapse is extremely unlikely for a number of reasons, including the fact that the containment of #P in FP implies that P = NP. Our main result is obtained by establishing the intractrability of the counting problem for general AGunification, where AG is the equational theory of Abelian groups. Specifically, we show that computing the cardinality of a minimal complete set of unifiers for general AGunification is a #Phard problem. In contrast, AGunification with constants is solvable in polynomial time. Since an algorithm for general AGunification can be obtained as a combination of a polynomialtime algorithm for AGunification with constants and a polynomialtime algorithm...
AntiPattern Matching Modulo
"... Negation is intrinsic to human thinking and most of the time when searching for something, we base our patterns on both positive and negative conditions. In a previous work, we have extended the notion of term to the one of antiterm that may contain complement symbols. Matching such antiterms aga ..."
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Cited by 3 (1 self)
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Negation is intrinsic to human thinking and most of the time when searching for something, we base our patterns on both positive and negative conditions. In a previous work, we have extended the notion of term to the one of antiterm that may contain complement symbols. Matching such antiterms against terms has the nice property of being unitary. Here we generalize the syntactic antipattern matching to antipattern matching modulo an arbitrary equational theory E, and we study the specific and practically very useful case of associativity, possibly with a unity (AU). To this end, based on the syntacticness of associativity, we present a rulebased associative matching algorithm, and we extend it to AU. This algorithm is then used to solve AU antipattern matching problems. This allows us to be generic enough so that for instance, the AllDiff standard predicate of constraint programming becomes simply expressible in this framework. AU antipatterns are implemented in the Tom language and we show some examples of their usage.
On the Complexity of Counting the Hilbert Basis of a Linear Diophantine System
, 1999
"... We investigate the computational complexity of counting the Hilbert basis of a homogeneous system of linear Diophantine equations. We establish lower and upper bounds on the complexity of this problem by showing that counting the Hilbert basis is #Phard and belongs to the class #NP. Moreover, we in ..."
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Cited by 2 (2 self)
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We investigate the computational complexity of counting the Hilbert basis of a homogeneous system of linear Diophantine equations. We establish lower and upper bounds on the complexity of this problem by showing that counting the Hilbert basis is #Phard and belongs to the class #NP. Moreover, we investigate the complexity of variants obtained by restricting the number of occurrences of the variables in the system.
Antipatterns for Rulebased Languages
, 2010
"... Negation is intrinsic to human thinking and most of the time when searching for something, we base our patterns on both positive and negative conditions. This should be naturally reflected in software that provide patternbased searches. We would like for example to specify that we search for white ..."
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Cited by 2 (0 self)
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Negation is intrinsic to human thinking and most of the time when searching for something, we base our patterns on both positive and negative conditions. This should be naturally reflected in software that provide patternbased searches. We would like for example to specify that we search for white cars that are not station wagons, or that we search for a list of objects that does not contain two identical elements. In this paper we extend the notion of pattern to the one of antipattern, i.e. patterns that may contain complement symbols. This concept is appropriate to design powerful extensions to patternbased programming languages like Ml, Asf+Sdf, Stratego, Maude, Elan or Tom and we show how this is used to extend the expressiveness and usability of the Tom language. We further define formally the semantics of antipatterns both in the syntactic case, i.e. when the symbols have no specific theory associated, and modulo an arbitrary equational theory E. We then extend the classical notion of matching between patterns and ground terms to matching between antipatterns and ground terms. Solving such problems can be performed either using general techniques as disunification, which we exemplify in the syntactical case, or more tailored and efficient approaches, which we chose to illustrate on the specific and very useful case of associativity, possibly with a unity. This allows us to be generic enough to give in this framework a very simple and natural expression of, for instance, the AllDiff standard predicate of constraint programming.
Solving, Reasoning, and Programming in Common Logic
"... Abstract. Common Logic (CL) is a recent ISO standard for exchanging logicbased information between disparate computer systems. Sharing and reasoning upon knowledge represented in CL require equation solving over terms of this language. We study computationally wellbehaved fragments of such solving ..."
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Abstract. Common Logic (CL) is a recent ISO standard for exchanging logicbased information between disparate computer systems. Sharing and reasoning upon knowledge represented in CL require equation solving over terms of this language. We study computationally wellbehaved fragments of such solving problems and show how they can influence reasoning in CL and transformations of CL expressions. 1
On the Complexity of Unification and Disunification in Commutative Idempotent Semigroups
 In Principles and Practice of Constraint Programming  CP97, Third International Conference
, 1997
"... . We analyze the computational complexity of elementary unification and disunification problems for the equational theory ACI of commutative idempotent semigroups. From earlier work, it was known that the decision problem for elementary ACIunification is solvable in polynomial time. We show that th ..."
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. We analyze the computational complexity of elementary unification and disunification problems for the equational theory ACI of commutative idempotent semigroups. From earlier work, it was known that the decision problem for elementary ACIunification is solvable in polynomial time. We show that this problem is inherently sequential by establishing that it is complete for polynomial time (Pcomplete) via logarithmicspace reductions. We also investigate the decision problem and the counting problem for elementary ACImatching and observe that the former is solvable in logarithmic space, but the latter is #Pcomplete. After this, we analyze the computational complexity of the decision problem for elementary ground ACIdisunification. Finally, we study the computational complexity of a restricted version of elementary ACImatching, which arises naturally as a setterm matching problem in the context of the logic data language LDL. In both cases, we delineate the boundary between polynomi...
Recueil d’articles
"... Annexes du manuscrit d’HDR — Le calcul de réécritureiiSommaire Présentations du calcul de réécriture 1 Propriétés des calculs à motifs 57 Extensions du calcul de réécriture 75 Expressivité du calcul de réécriture 147 Systèmes de types pour le calcul de réécriture 195 Applications 225Présentations du ..."
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Annexes du manuscrit d’HDR — Le calcul de réécritureiiSommaire Présentations du calcul de réécriture 1 Propriétés des calculs à motifs 57 Extensions du calcul de réécriture 75 Expressivité du calcul de réécriture 147 Systèmes de types pour le calcul de réécriture 195 Applications 225Présentations du calcul de réécriture [CK01] [CKL01a]
Regular Expression OrderSorted Unification and Matching
"... We extend ordersorted unification by permitting regular expression sorts for variables and in the domains of function symbols. The obtained signature corresponds to a finite bottomup unranked tree automaton. We prove that regular expression ordersorted (REOS) unification is of type infinitary and ..."
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We extend ordersorted unification by permitting regular expression sorts for variables and in the domains of function symbols. The obtained signature corresponds to a finite bottomup unranked tree automaton. We prove that regular expression ordersorted (REOS) unification is of type infinitary and decidable. The unification problem generalizes some known problems, such as, e.g., ordersorted unification for ranked terms, sequence unification, and word unification with regular constraints. Decidability of REOS unification implies that sequence unification with regular hedge language constraints is decidable, generalizing the decidability result of word unification with regular constraints to terms. A sort weakening algorithm helps to construct a minimal complete set of REOS unifiers from the solutions of sequence unification problems. We also give a direct procedure to compute the minimal complete set of REOS unifiers. Moreover, we