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Argument Filtering Transformation
 In Proc. 1st PPDP, LNCS 1702
, 1999
"... To simplify the task of proving termination of term rewriting systems, several elimination methods, such as the dummy elimination, the distribution elimination, the general dummy elimination and the improved general dummy elimination, have been proposed. In this paper, we show that the argument lter ..."
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To simplify the task of proving termination of term rewriting systems, several elimination methods, such as the dummy elimination, the distribution elimination, the general dummy elimination and the improved general dummy elimination, have been proposed. In this paper, we show that the argument ltering method combining with the dependency pair technique is essential in all the above elimination methods. We present remarkable simple proofs for the soundness of these elimination methods based on this observation. Moreover, we propose a new elimination method, called the argument ltering transformation, which is not only more powerful than all the other elimination methods but also especially useful to make clear the essential relation hidden behind these methods.
Termination, ACTermination and Dependency Pairs of Term Rewriting Systems
 Ph.D. thesis, JAIST
, 2000
"... Copyright c ○ 2000 by Keiichirou KUSAKARI Recently, Arts and Giesl introduced the notion of dependency pairs, which gives effective methods for proving termination of term rewriting systems (TRSs). In this thesis, we extend the notion of dependency pairs to ACTRSs, and introduce new methods for eff ..."
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Copyright c ○ 2000 by Keiichirou KUSAKARI Recently, Arts and Giesl introduced the notion of dependency pairs, which gives effective methods for proving termination of term rewriting systems (TRSs). In this thesis, we extend the notion of dependency pairs to ACTRSs, and introduce new methods for effectively proving ACtermination. Since it is impossible to directly apply the notion of dependency pairs to ACTRSs, we introduce the head parts in terms and show an analogy between the root positions in infinite reduction sequences by TRSs and the head positions in those by ACTRSs. Indeed, this analogy is essential for the extension of dependency pairs to ACTRSs. Based on this analogy, we define ACdependency pairs. To simplify the task of proving termination and ACtermination, several elimination transformations such as the dummy elimination, the distribution elimination, the general dummy elimination and the improved general dummy elimination, have been proposed. In this thesis, we show that the argument filtering method combined with the ACdependency pair technique is essential in all the elimination transformations above. We present remarkable simple proofs for the soundness of these elimination transformations based on this observation. Moreover, we propose a new elimination transformation, called the argument filtering transformation, which is not only more powerful than all the other elimination transformations but also especially useful to make clear an essential relationship among them.
Termination of rewriting strategies: a generic approach
"... Abstract. We propose a generic termination proof method for rewriting under strategies, based on an explicit induction on the termination property. Rewriting trees on ground terms are modelized by proof trees, generated by alternatively applying narrowing and abstracting steps. The induction princip ..."
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Abstract. We propose a generic termination proof method for rewriting under strategies, based on an explicit induction on the termination property. Rewriting trees on ground terms are modelized by proof trees, generated by alternatively applying narrowing and abstracting steps. The induction principle is applied through the abstraction mechanism, where terms are replaced by variables representing any of their normal forms. The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof. The generic method is then instantiated for the innermost strategy, as an example. 1 Introducing the problem Rewriting techniques are now widely used in automated deduction, especially to handle equality, as well as in programming, in functional, logical or rulebased languages. Termination of rewriting is a crucial property, important in itself to guarantee a result in a finite number of steps, but it is also required to decide properties like confluence and sufficient completeness, or to allow proofs by consistency. Existing methods for proving termination of term rewriting systems (TRS in short) essentially tackle the termination problem on the free term algebra, for rewriting without strategies. Most are based on syntactic or semantic noetherian orderings containing the rewriting relation induced by the TRS; other ones consist in transforming the termination problem of a TRS R into the decreasingness problem of another TRS or of pairs of terms, then handled with
The Hierarchy of Dependency Pairs
 RESEARCH REPORT ISRR990007F, SCHOOL OF INFORMATION SCIENCE, JAIST
, 1999
"... ... this paper, we study the relation for the termination property between a term rewriting system, the set of its dependency pairs and the union of them. ..."
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... this paper, we study the relation for the termination property between a term rewriting system, the set of its dependency pairs and the union of them.
BOLIC AND ALGEBRAIC MANIPULATION]: Languages and Systems—Evaluation strategies,
"... We propose a synthesis of three induction based algorithms, we already have given to prove termination of rewrite rule based programs, respectively for the innermost, the outermost and the local strategies. A generic inference principle is presented, based on an explicit induction on the termination ..."
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We propose a synthesis of three induction based algorithms, we already have given to prove termination of rewrite rule based programs, respectively for the innermost, the outermost and the local strategies. A generic inference principle is presented, based on an explicit induction on the termination property, which genetates ordering constraints, defining the induction relation. The generic inference principle is then instantiated to provide proof procedures for the three specific considered strategies.