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Thom Frühwirth - One of the best experts on this subject based on the ideXlab platform.
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Complexity of the CHR Rational Tree Equation Solver
2006Co-Authors: Marc Meister, Thom FrühwirthAbstract:Constraint Handling Rules (CHR) is a concurrent, committed-choice, rule-based language. One of the first CHR programs is the classic constraint solver for Syntactic Equality of rational trees that performs unification [7, 4, 14]. The worst-case time (and space) complexity of this short and elegant solver so far was an open problem [8] and assumed to be polynomial. In this paper we show that under the standard operational semantics of CHR there exist particular computations with n occurrences of variables and function symbols that produce O(2) constraints, thus leading to exponential time and space complexity. We also show that the standard implementation of the solver in CHR libraries for Prolog may not terminate due to the Prolog built-in order used in comparing terms. Complexity can be improved to be quadratic for any term order under both standard and refined CHR semantics without changing the equation solver, when equations are transformed into flat normal form.
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Rational Trees RT
Cognitive Technologies, 2003Co-Authors: Thom Frühwirth, Slim AbdennadherAbstract:We have already introduced the constraint system E dealing with Herbrand terms (or: first-order terms, finite trees) and the Syntactic Equality constraint. Here, we consider an important variation of E.
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CSCLP - Complexity of a CHR solver for existentially quantified conjunctions of equations over trees
Lecture Notes in Computer Science, 1Co-Authors: Marc Meister, Khalil Djelloul, Thom FrühwirthAbstract:Constraint Handling Rules (CHR) is a concurrent, committed-choice, rule-based language. One of the first CHR programs is the classic constraint solver for Syntactic Equality of rational trees that performs unification. We first prove its exponential complexity in time and space for non-flat equations and deduce from this proof a quadratic complexity for flat equations. We then present an extended CHR solver for solving existentially quantified conjunctions of non-flat equations in the theory of finite or infinite trees. We reach a quadratic complexity by first flattening the equations and introducing new existentially quantified variables, then using the classic solver, and finally eliminating particular equations and quantified variables.
Marc Meister - One of the best experts on this subject based on the ideXlab platform.
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Complexity of a CHR solver for existentially quantified conjunctions of equations over trees.
2007Co-Authors: Marc Meister, Khalil Djelloul, Thom FruehwirthAbstract:Constraint Handling Rules (CHR) is a concurrent, committed- choice, rule-based language. One of the first CHR programs is the classic constraint solver for Syntactic Equality of rational trees that performs unification. We first prove its exponential complexity in time and space for non-flat equations and deduce from this proof a quadratic complexity for flat equations. We then present an extended CHR solver for solving existentially quantified conjunctions of non-flat equations in the theory of finite or infinite trees. We reach a quadratic complexity by first flattening the equations and introducing new existentially quantified variables, then using the classic solver, and finally eliminating particular equations and quantified variables.
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Complexity of the CHR Rational Tree Equation Solver
2006Co-Authors: Marc Meister, Thom FrühwirthAbstract:Constraint Handling Rules (CHR) is a concurrent, committed-choice, rule-based language. One of the first CHR programs is the classic constraint solver for Syntactic Equality of rational trees that performs unification [7, 4, 14]. The worst-case time (and space) complexity of this short and elegant solver so far was an open problem [8] and assumed to be polynomial. In this paper we show that under the standard operational semantics of CHR there exist particular computations with n occurrences of variables and function symbols that produce O(2) constraints, thus leading to exponential time and space complexity. We also show that the standard implementation of the solver in CHR libraries for Prolog may not terminate due to the Prolog built-in order used in comparing terms. Complexity can be improved to be quadratic for any term order under both standard and refined CHR semantics without changing the equation solver, when equations are transformed into flat normal form.
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CSCLP - Complexity of a CHR solver for existentially quantified conjunctions of equations over trees
Lecture Notes in Computer Science, 1Co-Authors: Marc Meister, Khalil Djelloul, Thom FrühwirthAbstract:Constraint Handling Rules (CHR) is a concurrent, committed-choice, rule-based language. One of the first CHR programs is the classic constraint solver for Syntactic Equality of rational trees that performs unification. We first prove its exponential complexity in time and space for non-flat equations and deduce from this proof a quadratic complexity for flat equations. We then present an extended CHR solver for solving existentially quantified conjunctions of non-flat equations in the theory of finite or infinite trees. We reach a quadratic complexity by first flattening the equations and introducing new existentially quantified variables, then using the classic solver, and finally eliminating particular equations and quantified variables.
Florent Jacquemard - One of the best experts on this subject based on the ideXlab platform.
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Automated Induction with Constrained Tree Automata
2017Co-Authors: Adel Bouhoula, Florent JacquemardAbstract:We propose a procedure for automated implicit inductive theorem proving for equational specifications made of rewrite rules with conditions and constraints. The constraints are interpreted over constructor terms (representing data values), and may express Syntactic Equality, disEquality, ordering and also membership in a fixed tree language. Constrained equational axioms between constructor terms are supported and can be used in order to specify complex data structures like sets, sorted lists, trees, powerlists... Our procedure is based on tree grammars with constraints, a formalism which can describe exactly the initial model of the given specification (when it is sufficiently complete and terminating). They are used in the inductive proofs first as an induction scheme for the generation of subgoals at induction steps, second for checking validity and redundancy criteria by reduction to an emptiness problem, and third for defining and solving membership constraints. We show that the procedure is sound and refutationally complete. It generalizes former test set induction techniques and yields natural proofs for several non-trivial examples presented in the paper, these examples are difficult (if not impossible) to specify and carry on automatically with other induction procedures.
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Automated Induction for Complex Data Structures
arXiv: Logic in Computer Science, 2008Co-Authors: Adel Bouhoula, Florent JacquemardAbstract:We propose a procedure for automated implicit inductive theorem proving for equational specifications made of rewrite rules with conditions and constraints. The constraints are interpreted over constructor terms (representing data values), and may express Syntactic Equality, disEquality, ordering and also membership in a fixed tree language. Constrained equational axioms between constructor terms are supported and can be used in order to specify complex data structures like sets, sorted lists, trees, powerlists... Our procedure is based on tree grammars with constraints, a formalism which can describe exactly the initial model of the given specification (when it is sufficiently complete and terminating). They are used in the inductive proofs first as an induction scheme for the generation of subgoals at induction steps, second for checking validity and redundancy criteria by reduction to an emptiness problem, and third for defining and solving membership constraints. We show that the procedure is sound and refutationally complete. It generalizes former test set induction techniques and yields natural proofs for several non-trivial examples presented in the paper, these examples are difficult to specify and carry on automatically with related induction procedures.
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IJCAR - Automated Induction with Constrained Tree Automata
Automated Reasoning, 1Co-Authors: Adel Bouhoula, Florent JacquemardAbstract:We propose a procedure for automated implicit inductive theorem proving for equational specifications made of rewrite rules with conditions and constraints. The constraints are interpreted over constructor terms (representing data values), and may express Syntactic Equality, disEquality, ordering and also membership in a fixed tree language. Constrained equational axioms between constructor terms are supported and can be used in order to specify complex data structures like sets, sorted lists, trees, powerlists... Our procedure is based on tree grammars with constraints, a formalism which can describe exactly the initial model of the given specification (when it is sufficiently complete and terminating). They are used in the inductive proofs first as an induction scheme for the generation of subgoals at induction steps, second for checking validity and redundancy criteria by reduction to an emptiness problem, and third for defining and solving membership constraints. We show that the procedure is sound and refutationally complete. It generalizes former test set induction techniques and yields natural proofs for several non-trivial examples presented in the paper, these examples are difficult (if not impossible) to specify and carry on automatically with other induction procedures.
Reinhard Pichler - One of the best experts on this subject based on the ideXlab platform.
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On the complexity of equational problems in CNF
Journal of Symbolic Computation, 2003Co-Authors: Reinhard PichlerAbstract:AbstractEquational problems (i.e. first-order formulae with quantifier prefix ∃∗∀∗, whose only predicate symbol is Syntactic Equality) are an important tool in many areas of automated deduction, e.g. restricting the set of ground instances of a clause via equational constraints allows the definition of stronger redundancy criteria and hence, in general, of more efficient theorem provers. Moreover, the inference rules themselves can be restricted via constraints. In automated model building, equational problems play an important role both in the definition of an appropriate model representation and in the evaluation of clauses in such models. Also, many problems in the area of logic programming can be reduced to equational problem solving. Finally, equational problems over a finite domain correspond to the evaluation of certain queries over relational databases.The goal of this work is a complexity analysis of the satisfiability problem of equational problems. The main results will be a proof of the NP-completeness (and, in particular, the NP-membership) of equational problems in CNF over an infinite domain and of the Σ2p-completeness in the case of CNF over a finite domain
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Explicit versus implicit representations of subsets of the Herbrand universe
Theoretical Computer Science, 2003Co-Authors: Reinhard PichlerAbstract:In Lassez and Marriott (J. Automat. Reson. 3 (3) (1987) 301-317), explicit and implicit generalizations were studied as representations of subsets of some fixed Herbrand universe H. An explicit generalization E = r1V ... V rl represents all ground terms that are instances of at least one of the terms ti, whereas an implicit generalization I = t/t1 V ... V tm represents all H-ground instances of t that are not instances of any term ti. More generally, a disjunction I = I1 V ... V In of implicit generalizations contains all ground terms that are contained in at least one of the implicit generalizations Ij.Implicit generalizations have applications to many areas of Computer Science like machine learning, unification, specification of abstract data types, logic programming, functional programming, etc. In these areas, the so-called finite explicit representability problem plays an important role, i.e. given a disjunction of implicit generalizations I =I1 V ... V In, does there exist an explicit generalization E, s.t. I and E are equivalent? We shall prove the coNP-completeness of this decision problem.Implicit generalizations can be represented as equational formulae, i.e., first-order formulae whose only predicate symbol is Syntactic Equality. Closely related to the finite explicit representability problem is the so-called negation elimination problem of equational formulae, i.e. given an arbitrary equational formula p is p semantically equivalent to an equational formula without universal quantifiers and negation. In this work we study the negation elimination problem of equational formulae with purely existential quantifier prefix. We prove the coNP-completeness for such formulae in DNF and the Π2p -hardness in case of CNF.
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ICALP - Negation Elimination from Simple Equational Formulae
Automata Languages and Programming, 2000Co-Authors: Reinhard PichlerAbstract:Equational formulae are first-order formulae over an alphabet F of function symbols whose only predicate symbol is Syntactic Equality. Unification problems are an important special case of equational formulae, where no universal quantifiers and no negation occur. By the negation elimination problem we mean the problem of deciding whether a given equational formula is semantically equivalent to a unification problem. This decision problem has many interesting applications in machine learning, logic programming, functional programming, constrained rewriting, etc. In this work we present a new algorithm for the negation elimination problem of equational formulae with purely existential quantifier prefix. Moreover, we prove the coNP-completeness for equational formulae in DNF and the Π2p-hardness in case of CNF.
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CADE - Solving Equational Problems Efficiently
Automated Deduction — CADE-16, 1999Co-Authors: Reinhard PichlerAbstract:Equational problems (i.e.: first-order formulae with quantifier prefix ∃* ∀*, whose only predicate symbol is Syntactic Equality) are an important tool in many areas of automated deduction, e.g.: restricting the set of ground instances of a clause via equational constraints allows the definition of stronger redundancy criteria and hence, in general, of more efficient theorem provers. Moreover, also the inference rules themselves can be restricted via constraints. In automated model building, equational problems play an important role both in the definition of an appropriate model representation and in the evaluation of clauses in such models. Also, many problems in the area of logic programming can be reduced to equational problem solving. The goal of this work is a complexity analysis of the satisfiability problem of equational problems in CNF over an infinite Herbrand universe. The main result will be a proof of the NP-completeness (and, in particular, of the NP-membership) of this problem.
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Solving equational problems efficiently
Lecture Notes in Computer Science, 1999Co-Authors: Reinhard PichlerAbstract:Equational problems (i.e.: first-order formulae with quantifier prefix ∃ * ∀ * , whose only predicate symbol is Syntactic Equality) are an important tool in many areas of automated deduction, e.g.: restricting the set of ground instances of a clause via equational constraints allows the definition of stronger redundancy criteria and hence, in general, of more efficient theorem provers. Moreover, also the inference rules themselves can be restricted via constraints. In automated model building, equational problems play an important role both in the definition of an appropriate model representation and in the evaluation of clauses in such models. Also, many problems in the area of logic programming can be reduced to equational problem solving. The goal of this work is a complexity analysis of the satisfiability problem of equational problems in CNF over an infinite Herbrand universe. The main result will be a proof of the NP-completeness (and, in particular, of the NP-membership) of this problem.
Uwe Nestmann - One of the best experts on this subject based on the ideXlab platform.
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This work is licensed under the Creative Commons Attribution License. Matching in the Pi-Calculus
2016Co-Authors: C K. Peters, Kirstin Peters, Tsvetelina Yonova-karbe, Uwe NestmannAbstract:We study whether, in the pi-calculus, the match prefix—a conditional operator testing two names for (Syntactic) Equality—is expressible via the other operators. Previously, Carbone and Maffeis proved that matching is not expressible this way under rather strong requirements (preservation and reflection of observables). Later on, Gorla developed a by now widely-tested set of criteria for en-codings that allows much more freedom (e.g. instead of direct translations of observables it allows comparison of calculi with respect to reachability of successful states). In this paper, we offer a con-siderably stronger separation result on the non-expressibility of matching using only Gorla’s relaxed requirements.
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EXPRESS/SOS - Matching in the Pi-Calculus.
Electronic Proceedings in Theoretical Computer Science, 2014Co-Authors: Kirstin Peters, Tsvetelina Yonova-karbe, Uwe NestmannAbstract:We study whether, in the pi-calculus, the match prefix-a conditional operator testing two names for (Syntactic) Equality-is expressible via the other operators. Previously, Carbone and Maffeis proved that matching is not expressible this way under rather strong requirements (preservation and reflection of observables). Later on, Gorla developed a by now widely-tested set of criteria for encodings that allows much more freedom (e.g. instead of direct translations of observables it allows comparison of calculi with respect to reachability of successful states). In this paper, we offer a considerably stronger separation result on the non-expressibility of matching using only Gorla's relaxed requirements.
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Matching in the Pi-Calculus (Technical Report)
arXiv: Logic in Computer Science, 2014Co-Authors: Kirstin Peters, Tsvetelina Yonova-karbe, Uwe NestmannAbstract:We study whether, in the pi-calculus, the match prefix---a conditional operator testing two names for (Syntactic) Equality---is expressible via the other operators. Previously, Carbone and Maffeis proved that matching is not expressible this way under rather strong requirements (preservation and reflection of observables). Later on, Gorla developed a by now widely-tested set of criteria for encodings that allows much more freedom (e.g. instead of direct translations of observables it allows comparison of calculi with respect to reachability of successful states). In this paper, we offer a considerably stronger separation result on the non-expressibility of matching using only Gorla's relaxed requirements.
