The Experts below are selected from a list of 246 Experts worldwide ranked by ideXlab platform
Mateu Villaret - One of the best experts on this subject based on the ideXlab platform.
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Simplifying the signature in second-order unification
Applicable Algebra in Engineering Communication and Computing, 2009Co-Authors: Jordi Levy, Mateu VillaretAbstract:Second-Order Unification is undecidable even for very specialized fragments. The signature plays a crucial role in these fragments. If all Symbols are monadic, then the problem is NP-complete, whereas it is enough to have just one binary constant to lose decidability. In this work we reduce Second-Order Unification to Second-Order Unification with a signature that contains just one binary Function Symbol and constants. The reduction is based on partially currying the equations by using the binary Function Symbol for explicit application @. Our work simplifies the study of Second-Order Unification and some of its variants, like Context Unification and Bounded Second-Order Unification.
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RTA - Currying Second-Order Unification Problems
Rewriting Techniques and Applications, 2002Co-Authors: Jordi Levy, Mateu VillaretAbstract:The Curry form of a term, like f(a, b), allows us to write it, using just a single binary Function Symbol, as @(@(f, a), b). Using this technique we prove that the signature is not relevant in second-order unification, and conclude that one binary Symbol is enough. By currying variable applications, like X(a), as @(X, a), we can transform second-order terms into first-order terms, but we have to add beta-reduction as a theory. This is roughly what it is done in explicit unification. We prove that by currying only constant applications we can reduce second-order unification to second-order unification with just one binary Function Symbol. Both problems are already known to be undecidable, but applying the same idea to context unification, for which decidability is still unknown, we reduce the problem to context unification with just one binary Function Symbol.We also discuss about the difficulties ofapplying the same ideas to third or higher order unification.
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Currying second-order unification problems
Lecture Notes in Computer Science, 2002Co-Authors: Jordi Levy, Mateu VillaretAbstract:The Curry form of a term, like f(a,b), allows us to write it, using just a single binary Function Symbol, as @(@(f,a),b). Using this technique we prove that the signature is not relevant in second-order unification, and conclude that one binary Symbol is enough. By currying variable applications, like X(a), as @(X,a), we can transform second-order terms into first-order terms, but we have to add betareduction as a theory. This is roughly what it is done in explicit unification. We prove that by currying only constant applications we can reduce second-order unification to second-order unification with just one binary Function Symbol. Both problems are already known to be undecidable, but applying the same idea to context unification, for which decidability is still unknown, we reduce the problem to context unification with just one binary Function Symbol. We also discuss about the difficulties of applying the same ideas to third or higher order unification.
Ondrej Klíma - One of the best experts on this subject based on the ideXlab platform.
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MFCS - Unification Modulo Associativity and Idempotency Is NP-complete
Lecture Notes in Computer Science, 2002Co-Authors: Ondrej KlímaAbstract:We show that the unification problem for the theory of one associative and idempotent Function Symbol (AI-unification), i.e. solving word equations in free idempotent semigroups, is NP-complete.
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Unification modulo associativity and idempotency is NP-complete
Lecture Notes in Computer Science, 2002Co-Authors: Ondrej KlímaAbstract:We show that the unification problem for the theory of one associative and idempotent Function Symbol (AI-unification), i.e. solving word equations in free idempotent semigroups, is NP-complete.
Nicolas Peltier - One of the best experts on this subject based on the ideXlab platform.
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TABLEAUX - Prenex Separation Logic with One Selector Field.
Lecture Notes in Computer Science, 2019Co-Authors: Mnacho Echenim, Radu Iosif, Nicolas PeltierAbstract:We show that infinite satisfiability can be reduced to finite satisfiability for all prenex formulas of Separation Logic with \(k\ge 1\) selector fields (\(\mathsf {SL}^{\!\scriptstyle {k}}\)). This fact entails the decidability of the finite and infinite satisfiability problems for the class of prenex formulas of \(\mathsf {SL}^{\!\scriptstyle {1}}\), by reduction to the first-order theory of a single unary Function Symbol and an arbitrary number of unary predicate Symbols. We also prove that the complexity of this fragment is not elementary recursive, by reduction from the first-order theory of one unary Function Symbol. Finally, we prove that the Bernays-Schonfinkel-Ramsey fragment of prenex \(\mathsf {SL}^{\!\scriptstyle {1}}\) formulas with quantifier prefix in the language Open image in new window is PSPACE-complete.
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Prenex Separation Logic with One Selector Field
2019Co-Authors: Mnacho Echenim, Radu Iosif, Nicolas PeltierAbstract:We show that infinite satisfiability can be reduced to finite satisfiabil-ity for all prenex formulas of Separation Logic with k ≥ 1 selector fields (SL k). This fact entails the decidability of the finite and infinite satisfiability problems for the class of prenex formulas of SL 1 , by reduction to the first-order theory of a single unary Function Symbol and an arbitrary number of unary predicate Symbols. We also prove that the complexity of this fragment is not elementary recursive, by reduction from the first-order theory of one unary Function Symbol. Finally, we prove that the Bernays-Schönfinkel-Ramsey fragment of prenex SL 1 formulas with quantifier prefix in the language ∃ * ∀ * is PSPACE-complete.
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The Complexity of Prenex Separation Logic with One Selector
2018Co-Authors: Mnacho Echenim, Radu Iosif, Nicolas PeltierAbstract:We first show that infinite satisfiability can be reduced to finite satisfiability for all prenex formulas of Separation Logic with $k\geq1$ selector fields ($\seplogk{k}$). Second, we show that this entails the decidability of the finite and infinite satisfiability problem for the class of prenex formulas of $\seplogk{1}$, by reduction to the first-order theory of one unary Function Symbol and unary predicate Symbols. We also prove that the complexity is not elementary, by reduction from the first-order theory of one unary Function Symbol. Finally, we prove that the Bernays-Sch\"onfinkel-Ramsey fragment of prenex $\seplogk{1}$ formulae with quantifier prefix in the language $\exists^*\forall^*$ is \pspace-complete. The definition of a complete (hierarchical) classification of the complexity of prenex $\seplogk{1}$, according to the quantifier alternation depth is left as an open problem.
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Completeness and Decidability Results for First-order Clauses with Indices
2013Co-Authors: Abdelkader Kersani, Nicolas PeltierAbstract:We define a proof procedure that allows for a limited form of inductive reasoning. The first argument of a Function Symbol is allowed to belong to an inductive type. We will call such an argument an index. We enhance the standard superposition calculus with a loop detection rule, in order to encode a particular form of mathematical induction. The satisfiability problem is not semi-decidable, but some classes of clause sets are identified for which the proposed procedure is complete and/or terminating.
Z. Zhou - One of the best experts on this subject based on the ideXlab platform.
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Multiway Decision Graphs for Automated Hardware Verification
Formal Methods in System Design, 1997Co-Authors: F. Corella, Z. Zhou, X. Song, M. Langevin, E. CernyAbstract:Traditional ROBDD-based methods of automated verification suffer from the drawback that they require a binary representation of the circuit. To overcome this limitation we propose a broader class of decision graphs, called Multiway Decision Graphs (MDGs), of which ROBDDs are a special case. With MDGs, a data value is represented by a single variable of abstract type, rather than by 32 or 64 boolean variables, and a data operation is represented by an uninterpreted Function Symbol. MDGs are thus much more compact than ROBDDs, and this greatly increases the range of circuits that can be verified. We give algorithms for MDG manipulation, and for implicit state enumeration using MDGs. We have implemented an MDG package and provide experimental results.
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multiway decision graphs for automated hardware verification
Formal Methods, 1997Co-Authors: F. Corella, Z. Zhou, X. Song, M. Langevin, E. CernyAbstract:Traditional ROBDD-based methods of automated verification suffer from the drawback that they require a binary representation of the circuit. To overcome this limitation we propose a broader class of decision graphs, called {\em Multiway Decision Graphs} (MDGs), of which ROBDDs are a special case. With MDGs, a data value is represented by a single variable of abstract type, rather than by 32 or 64 boolean variables, and a data operation is represented by an uninterpreted Function Symbol. MDGs are thus much more compact than ROBDDs, and this greatly increases the range of circuits that can be verified. We give algorithms for MDG manipulation, and for implicit state enumeration using MDGs. We have implemented an MDG package and provide experimental results.
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CAV - MDG Tools for the Verification of RTL Designs
Computer Aided Verification, 1996Co-Authors: K. D. Anon, F. Corella, X. Song, M. Langevin, E. Cerny, N. Boulerice, Sofiène Tahar, Ying Xu, Z. ZhouAbstract:Although ROBDDs [1, 2] have proved to be a powerful tool for automated hardware verification, they require a Boolean representation of the circuit. Since the size of an ROBDD grows, sometimes exponentially, with the number of Boolean variables, ROBDD-based verification cannot be directly applied to circuits with complex datapaths. We have recently proposed a new class of decision graphs, called Multiway Decision Graphs (MDGs) [3], that comprises, 1)ut is much broader than, the class of ROBDDs. The underlying logic of MDGs is a subset of many-sorted first-order logic with a distinction between concrete and abstract sorts. A concrete sort has an enumeration while an abstract sort does not. Hence a data value can be represented by a single variable of abstract sort, rather than by a vector of Boolean variables, and a data operation can be viewed as a black box and represented by an uninterpreted Function Symbol. MDGs are thus much more compact than ROBDDs for designs containing a datapath, and this greatly increases the range of applications. We have developed a collection of MDG tools that include implementations of the basic MDG operators and verification procedures for RTL designs.
Jordi Levy - One of the best experts on this subject based on the ideXlab platform.
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Simplifying the signature in second-order unification
Applicable Algebra in Engineering Communication and Computing, 2009Co-Authors: Jordi Levy, Mateu VillaretAbstract:Second-Order Unification is undecidable even for very specialized fragments. The signature plays a crucial role in these fragments. If all Symbols are monadic, then the problem is NP-complete, whereas it is enough to have just one binary constant to lose decidability. In this work we reduce Second-Order Unification to Second-Order Unification with a signature that contains just one binary Function Symbol and constants. The reduction is based on partially currying the equations by using the binary Function Symbol for explicit application @. Our work simplifies the study of Second-Order Unification and some of its variants, like Context Unification and Bounded Second-Order Unification.
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RTA - Currying Second-Order Unification Problems
Rewriting Techniques and Applications, 2002Co-Authors: Jordi Levy, Mateu VillaretAbstract:The Curry form of a term, like f(a, b), allows us to write it, using just a single binary Function Symbol, as @(@(f, a), b). Using this technique we prove that the signature is not relevant in second-order unification, and conclude that one binary Symbol is enough. By currying variable applications, like X(a), as @(X, a), we can transform second-order terms into first-order terms, but we have to add beta-reduction as a theory. This is roughly what it is done in explicit unification. We prove that by currying only constant applications we can reduce second-order unification to second-order unification with just one binary Function Symbol. Both problems are already known to be undecidable, but applying the same idea to context unification, for which decidability is still unknown, we reduce the problem to context unification with just one binary Function Symbol.We also discuss about the difficulties ofapplying the same ideas to third or higher order unification.
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Currying second-order unification problems
Lecture Notes in Computer Science, 2002Co-Authors: Jordi Levy, Mateu VillaretAbstract:The Curry form of a term, like f(a,b), allows us to write it, using just a single binary Function Symbol, as @(@(f,a),b). Using this technique we prove that the signature is not relevant in second-order unification, and conclude that one binary Symbol is enough. By currying variable applications, like X(a), as @(X,a), we can transform second-order terms into first-order terms, but we have to add betareduction as a theory. This is roughly what it is done in explicit unification. We prove that by currying only constant applications we can reduce second-order unification to second-order unification with just one binary Function Symbol. Both problems are already known to be undecidable, but applying the same idea to context unification, for which decidability is still unknown, we reduce the problem to context unification with just one binary Function Symbol. We also discuss about the difficulties of applying the same ideas to third or higher order unification.
