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Decidable Approximations of Sets of Descendants and Sets of Normal Forms
, 1997
"... We present here decidable approximations of sets of descendants and sets of normal forms of Term Rewriting Systems, based on specific tree automata techniques. In the context of rewriting logic, a Term Rewriting System is a program, and a normal form is a result of the program. Thus, approximations ..."
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Cited by 54 (18 self)
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We present here decidable approximations of sets of descendants and sets of normal forms of Term Rewriting Systems, based on specific tree automata techniques. In the context of rewriting logic, a Term Rewriting System is a program, and a normal form is a result of the program. Thus, approximations of sets of descendants and sets of normal forms provide tools for analysing a few properties of programs: we show how to compute a superset of results, to prove the sufficient completeness property, or to find a criterion for proving termination under a specific strategy, the sequential reduction strategy.
Unification in extensions of shallow equational theories
 REWRITING TECHNIQUES AND APPLICATIONS, 9TH INTERNATIONAL CONFERENCE, RTA98', VOL. 1379 OF LNCS
, 1998
"... We show that unification in certain extensions of shallow equational theories is decidable. Our extensions generalize the known classes of shallow or standard equational theories. In order to prove decidability of unification in the extensions, a class of Horn clause sets called sorted shallow equa ..."
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Cited by 10 (2 self)
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We show that unification in certain extensions of shallow equational theories is decidable. Our extensions generalize the known classes of shallow or standard equational theories. In order to prove decidability of unification in the extensions, a class of Horn clause sets called sorted shallow equational theories is introduced. This class is a natural extension of tree automata with equality constraints between brother subterms as well as shallow sort theories. We show that saturation under sorted superposition is effective on sorted shallow equational theories. So called semilinear equational theories can be e ectively transformed into equivalent sorted shallow equational theories and generalize the classes of shallow and standard equational theories.
A New Result about the Decidability of the Existential OneStep Rewriting Theory
, 1998
"... . We give a decision procedure for the whole existential fragment of onestep rewriting firstorder theory, in the case where rewrite systems are linear, non leftleftoverlapping (i.e. without critical pairs), and non fflleftrightoverlapping (i.e. no lefthandside overlaps on top with the right ..."
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Cited by 5 (2 self)
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. We give a decision procedure for the whole existential fragment of onestep rewriting firstorder theory, in the case where rewrite systems are linear, non leftleftoverlapping (i.e. without critical pairs), and non fflleftrightoverlapping (i.e. no lefthandside overlaps on top with the righthandside of the same rewrite rule 2 ). The procedure is defined by means of treetuple synchronized grammars. 1 Introduction Given a signature \Sigma , the theory of onestep rewriting for a finite rewrite system is the first order theory over the universe of ground \Sigma terms that uses the only predicate symbol !, where x ! y means x rewrites into y by one step. It has been shown undecidable in [11]. Sharper undecidability results have been obtained for some subclasses of rewrite systems, about the 9 8 fragment [10, 8] and the 9 8 9 fragment [12]. It has been shown decidable for the positive existential fragment [9], in the case of unary signatures [3], in the case ...
On Partial Validation of Logic Programs
 proc of the 6th Conf. on Algebraic Methodology and Software Technology, Sydney (Australia), volume 1349 of LNCS
, 1997
"... . In this paper, we propose a method allowing us to compare the result of an execution of a logic program and a specification of the intended semantics. This approach is particularly interesting when the set of answers cannot be computed in finite time with usual prolog interpreters. We compute, usi ..."
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Cited by 5 (1 self)
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. In this paper, we propose a method allowing us to compare the result of an execution of a logic program and a specification of the intended semantics. This approach is particularly interesting when the set of answers cannot be computed in finite time with usual prolog interpreters. We compute, using a special operational mechanism, a finite set of rewrite rules synthesizing the whole set of answers w.r.t. a goal. Then, we use some tree tuple grammar based techniques to express the languages of the computed answers. An algorithm allows us to compare this language with the intended semantics language which is extracted from a user's specification. This method can be considered as a partial validation mechanism for logic programs. 1 Introduction Validation, partial validation and verification constitute central issues for full logic programming environment building [5]. By partial validation we mean here the comparison between the operational semantics of the program for a given goal, ...
Basic Syntactic Mutation
"... We give a set of inference rules for Eunification, similar to the inference rules for Syntactic Mutation. If the E is finitely saturated by paramodulation, then we can block certain terms from further inferences. Therefore, ..."
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Cited by 5 (1 self)
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We give a set of inference rules for Eunification, similar to the inference rules for Syntactic Mutation. If the E is finitely saturated by paramodulation, then we can block certain terms from further inferences. Therefore,
Solving Disequations modulo some Class of Rewrite Systems
, 1998
"... . This paper gives a procedure for solving disequations modulo equational theories, and to decide existence of solutions. For this, we assume that the equational theory is specified by a confluent and constructorbased rewrite system, and that four additional restrictions are satisfied. The procedur ..."
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Cited by 4 (2 self)
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. This paper gives a procedure for solving disequations modulo equational theories, and to decide existence of solutions. For this, we assume that the equational theory is specified by a confluent and constructorbased rewrite system, and that four additional restrictions are satisfied. The procedure represents the possibly infinite set of solutions thanks to a grammar, and decides existence of solutions thanks to an emptiness test. As a consequence, checking whether a linear equality is an inductive theorem is decidable, if assuming moreover sufficient completeness. 1 Introduction The problem that consists in solving symbolic equations modulo a theory is called equational unification. A lot of work has already studied this subject in a theoretical way to know when the problem can be decided, as well as in a practical way to find efficient algorithms that solve the problem. Another interesting problem consists in solving the negation of equations, called disequations, i.e. in finding ...
Synchronized Tree Languages for Reachability in Nonrightlinear Term Rewrite Systems
"... Abstract. Overapproximating the descendants (successors) of an initial set of terms under a rewrite system is used in reachability analysis. The success of such methods depends on the quality of the approximation. Regular approximations (i.e. those using finite tree automata) have been successful ..."
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Abstract. Overapproximating the descendants (successors) of an initial set of terms under a rewrite system is used in reachability analysis. The success of such methods depends on the quality of the approximation. Regular approximations (i.e. those using finite tree automata) have been successfully applied to protocol verification and Java program analysis. In [9, 2], nonregular approximations have been shown more precise than regular ones. In [3] (fixed version of [2]), we have shown that sound overapproximations using synchronized tree languages can be computed for leftandrightlinear term rewriting systems (TRS). In this paper, we present two new contributions extending [3]. Firstly, we show how to compute at least all innermost descendants for any leftlinear TRS. Secondly, a procedure is introduced for computing overapproximations independently of the applied rewrite strategy for any leftlinear TRS.
Rapport No 9812 A New Result about the Decidability of the Existential Onestep Rewriting Theory
, 1999
"... Universit'e d'Orl'eans, LIFO ..."
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Synchronized Grammars and Primal Grammars
, 2001
"... Tree languages are powerful tools for the representation and schematization of infinite sets of terms for various purposes (unification theory, verification and specification...). In order to extend the regular tree language framework, more complex formalisms have been developed. In this paper, we f ..."
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Tree languages are powerful tools for the representation and schematization of infinite sets of terms for various purposes (unification theory, verification and specification...). In order to extend the regular tree language framework, more complex formalisms have been developed. In this paper, we focus on Tree Synchronized Grammars and Primal Grammars which introduce specific control structures to represent non regular sets of terms. We propose a common unified framework in order to achieve the membership test for these particular languages. Thanks to a proof system, we provide a full operational framework, that allows us to transform tree grammars into Prolog programs (as it already exists for word grammars with DCG) whose goal is to recognize terms of the corresponding language.
Weakly Regular Relations and Applications
"... Abstract A new class of treetuple languages is introduced: the weakly regular relations. It is an extension of the regular case (regular relations) and a restriction of treetuple synchronized languages, that has all usual nice properties, except closure by complement. Two applications are presented ..."
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Abstract A new class of treetuple languages is introduced: the weakly regular relations. It is an extension of the regular case (regular relations) and a restriction of treetuple synchronized languages, that has all usual nice properties, except closure by complement. Two applications are presented: to unification modulo a rewrite system, and to onestep rewriting. 1 Introduction Several classes of treetuple languages (also viewed as tree relations), have been defined by means of automata or grammars. In particular, a simple one, the Regular Relations (RR), consists in defining regularity as being the one of the tree language (over the productalphabet) obtained by overlapping the tuple components.