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Yoshihito Toyama - One of the best experts on this subject based on the ideXlab platform.
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correctness of context moving transformations for Term Rewriting Systems
Logic-based Program Synthesis and Transformation, 2015Co-Authors: Koichi Sato, Takahito Aoto, Kentaro Kikuchi, Yoshihito ToyamaAbstract:Proofs by induction are often incompatible with functions in tail-recursive form as the accumulator changes in the course of unfolding the definitions. Context-moving and context-splitting Giesl, 2000 for functional programs transform tail-recursive programs into non tail-recursive ones which are more suitable for proofs by induction and thus for verification. In this paper, we formulate context-moving and context-splitting transformations in the framework of Term Rewriting Systems, and prove their correctness with respect to both eager evaluation semantics and initial algebra semantics under some conditions on the programs to be transformed. The conditions for the correctness with respect to initial algebra semantics can be checked by automated methods for inductive theorem proving developed in the field of Term Rewriting Systems.
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A Reduction-Preserving Completion for Proving Confluence of Non-Terminating Term Rewriting Systems
Logical Methods in Computer Science, 2012Co-Authors: Takahito Aoto, Yoshihito ToyamaAbstract:We give a method to prove confluence of Term Rewriting Systems that contain non-Terminating rewrite rules such as commutativity and associativity. Usually, confluence of Term Rewriting Systems containing such rules is proved by treating them as equational Term Rewriting Systems and considering E-critical pairs and/or Termination modulo E. In contrast, our method is based solely on usual critical pairs and it also (partially) works even if the system is not Terminating modulo E. We first present confluence criteria for Term Rewriting Systems whose rewrite rules can be partitioned into a Terminating part and a possibly non-Terminating part. We then give a reduction-preserving completion procedure so that the applicability of the criteria is enhanced. In contrast to the well-known Knuth-Bendix completion procedure which preserves the equivalence relation of the system, our completion procedure preserves the reduction relation of the system, by which confluence of the original system is inferred from that of the completed system.
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a reduction preserving completion for proving confluence of non Terminating Term Rewriting Systems
Rewriting Techniques and Applications, 2011Co-Authors: Takahito Aoto, Yoshihito ToyamaAbstract:We give a method to prove confluence of Term Rewriting Systems that contain non-Terminating rewrite rules such as commutativity and associativity. Usually, confluence of Term Rewriting Systems containing such rules is proved by treating them as equational Term Rewriting Systems and considering E-critical pairs and/or Termination modulo E. In contrast, our method is based solely on usual critical pairs and usual Termination. We first present confluence criteria for Term Rewriting Systems whose rewrite rules can be partitioned into Terminating part and possibly non-Terminating part. We then give a reduction-preserving completion procedure so that the applicability of the criteria is enhanced. In contrast to the well-known Knuth-Bendix completion procedure which preserves the equivalence relation of the system, our completion procedure preserves the reduction relation of the system, by which confluence of the original system is inferred from that of the completed system.
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proving confluence of Term Rewriting Systems automatically
Rewriting Techniques and Applications, 2009Co-Authors: Takahito Aoto, Junichi Yoshida, Yoshihito ToyamaAbstract:We have developed an automated confluence prover for Term Rewriting Systems (TRSs). This paper presents theoretical and technical ingredients that have been used in our prover. A distinctive feature of our prover is incorporation of several divide---and---conquer criteria such as those for commutative (Toyama, 1988), layer-preserving (Ohlebusch, 1994) and persistent (Aoto & Toyama, 1997) combinations. For a TRS to which direct confluence criteria do not apply, the prover decomposes it into components and tries to apply direct confluence criteria to each component. Then the prover combines these results to infer the (non-)confluence of the whole system. To the best of our knowledge, an automated confluence prover based on such an approach has been unknown.
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RTA - Confluent Term Rewriting Systems
Lecture Notes in Computer Science, 2005Co-Authors: Yoshihito ToyamaAbstract:The confluence property is one of the most important properties of Term Rewriting Systems, and various sufficient criteria for proving this property have been widely investigated. A necessary and sufficient criterion for confluence of Terminating Term Rewriting Systems, in which every reduction must Terminate, was demonstrated by Knuth and Bendix (1970). For non-Terminating Term Rewriting Systems, Rosen (1973) proved that left-linear and non-overlapping Term Rewriting Systems (i.e., no variable occurs twice or more in the left-hand side of a Rewriting rule and two left-hand sides of Rewriting rules must not overlap) are confluent, and the non-overlapping limitation was somewhat relaxed by Huet (1980), Toyama (1988), and van Oostrom (1997). However, few criteria have been proposed for confluence of Term Rewriting Systems that are non-left-linear and non-Terminating. Thus, it is still worth while extending criteria for these Systems. A powerful technique for showing confluence of a non-left-linear non-Terminating Term Rewriting system is a divide-and-conquer method based on modularity by Toyama (1987) or persistency by Zantema (1994), Aoto and Toyama (1997). The method guarantees that if the system is decomposed into small subSystems and each of them is confluent then this system has the confluence property. Another useful technique is a transformational method based on conditional-linearization by Klop and de Vrijer (1991), Toyama and Oyamaguchi (1994), or a labelling technique. In this method we apply a non-confluence preserving transformation on a Term Rewriting system. Then the Term Rewriting system is confluent if the transformed system is confluent, because of non-confluence preserving. In this talk we will illustrate these techniques through various examples and discuss the relation among them.
Bas Luttik - One of the best experts on this subject based on the ideXlab platform.
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Remarks on Thatte's transformation of Term Rewriting Systems
Information & Computation, 2004Co-Authors: Bas Luttik, Piet Rodenburg, Rakesh M VermaAbstract:We carry out a detailed analysis of Thatte's transformation of Term Rewriting Systems. We refute an earlier claim that this transformation preserves confluence for weakly persistent Systems. We prove the preservation of weak normalization, and of confluence in weakly normalizing Systems and in nonoverlapping Systems with linear subtemplates. We conclude by proving that weak persistence is an undecidable property of Term Rewriting Systems.
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Remarks on Thatte’s transformation of Term Rewriting Systems
Information and Computation, 2004Co-Authors: Bas Luttik, Piet Rodenburg, Rakesh VermaAbstract:AbstractWe carry out a detailed analysis of Thatte’s transformation of Term Rewriting Systems. We refute an earlier claim that this transformation preserves confluence for weakly persistent Systems. We prove the preservation of weak normalization, and of confluence in weakly normalizing Systems and in nonoverlapping Systems with linear subtemplates. We conclude by proving that weak persistence is an undecidable property of Term Rewriting Systems
Takahito Aoto - One of the best experts on this subject based on the ideXlab platform.
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correctness of context moving transformations for Term Rewriting Systems
Logic-based Program Synthesis and Transformation, 2015Co-Authors: Koichi Sato, Takahito Aoto, Kentaro Kikuchi, Yoshihito ToyamaAbstract:Proofs by induction are often incompatible with functions in tail-recursive form as the accumulator changes in the course of unfolding the definitions. Context-moving and context-splitting Giesl, 2000 for functional programs transform tail-recursive programs into non tail-recursive ones which are more suitable for proofs by induction and thus for verification. In this paper, we formulate context-moving and context-splitting transformations in the framework of Term Rewriting Systems, and prove their correctness with respect to both eager evaluation semantics and initial algebra semantics under some conditions on the programs to be transformed. The conditions for the correctness with respect to initial algebra semantics can be checked by automated methods for inductive theorem proving developed in the field of Term Rewriting Systems.
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disproving confluence of Term Rewriting Systems by interpretation and ordering
Frontiers of Combining Systems, 2013Co-Authors: Takahito AotoAbstract:In order to disprove confluence of Term Rewriting Systems, we develop new criteria for ensuring non-joinability of Terms based on interpretation and ordering. We present some instances of the criteria which are amenable for automation, and report on an implementation of a confluence disproving procedure based on these instances. The experiments reveal that our method is successfully applied to automatically disprove confluence of some Term Rewriting Systems, on which state-of-the-art automated confluence provers fail. A key idea to make our method effective is the introduction of usable rules—this allows one to decompose the constraint on rewrite rules into smaller components that depend on starting Terms.
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A Reduction-Preserving Completion for Proving Confluence of Non-Terminating Term Rewriting Systems
Logical Methods in Computer Science, 2012Co-Authors: Takahito Aoto, Yoshihito ToyamaAbstract:We give a method to prove confluence of Term Rewriting Systems that contain non-Terminating rewrite rules such as commutativity and associativity. Usually, confluence of Term Rewriting Systems containing such rules is proved by treating them as equational Term Rewriting Systems and considering E-critical pairs and/or Termination modulo E. In contrast, our method is based solely on usual critical pairs and it also (partially) works even if the system is not Terminating modulo E. We first present confluence criteria for Term Rewriting Systems whose rewrite rules can be partitioned into a Terminating part and a possibly non-Terminating part. We then give a reduction-preserving completion procedure so that the applicability of the criteria is enhanced. In contrast to the well-known Knuth-Bendix completion procedure which preserves the equivalence relation of the system, our completion procedure preserves the reduction relation of the system, by which confluence of the original system is inferred from that of the completed system.
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a reduction preserving completion for proving confluence of non Terminating Term Rewriting Systems
Rewriting Techniques and Applications, 2011Co-Authors: Takahito Aoto, Yoshihito ToyamaAbstract:We give a method to prove confluence of Term Rewriting Systems that contain non-Terminating rewrite rules such as commutativity and associativity. Usually, confluence of Term Rewriting Systems containing such rules is proved by treating them as equational Term Rewriting Systems and considering E-critical pairs and/or Termination modulo E. In contrast, our method is based solely on usual critical pairs and usual Termination. We first present confluence criteria for Term Rewriting Systems whose rewrite rules can be partitioned into Terminating part and possibly non-Terminating part. We then give a reduction-preserving completion procedure so that the applicability of the criteria is enhanced. In contrast to the well-known Knuth-Bendix completion procedure which preserves the equivalence relation of the system, our completion procedure preserves the reduction relation of the system, by which confluence of the original system is inferred from that of the completed system.
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proving confluence of Term Rewriting Systems automatically
Rewriting Techniques and Applications, 2009Co-Authors: Takahito Aoto, Junichi Yoshida, Yoshihito ToyamaAbstract:We have developed an automated confluence prover for Term Rewriting Systems (TRSs). This paper presents theoretical and technical ingredients that have been used in our prover. A distinctive feature of our prover is incorporation of several divide---and---conquer criteria such as those for commutative (Toyama, 1988), layer-preserving (Ohlebusch, 1994) and persistent (Aoto & Toyama, 1997) combinations. For a TRS to which direct confluence criteria do not apply, the prover decomposes it into components and tries to apply direct confluence criteria to each component. Then the prover combines these results to infer the (non-)confluence of the whole system. To the best of our knowledge, an automated confluence prover based on such an approach has been unknown.
Munehiro Iwami - One of the best experts on this subject based on the ideXlab platform.
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Persistence of Termination for Term Rewriting Systems with Ordered Sorts
World Academy of Science Engineering and Technology International Journal of Mathematical Computational Physical Electrical and Computer Engineering, 2007Co-Authors: Munehiro IwamiAbstract:A property is persistent if for any many-sorted Term Rewriting system , has the property if and only if Term Rewriting system , which results from by omitting its sort information, has the property . Zantema showed that Termination is persistent for Term Rewriting Systems without collapsing or duplicating rules. In this paper, we show that the Zantema’s result can be extended to Term Rewriting Systems on ordered sorts, i.e., Termination is persistent for Term Rewriting Systems on ordered sorts without collapsing, decreasing or duplicating rules. Furthermore we give the example as application of this result. Also we obtain that completeness is persistent for this class of Term Rewriting Systems.
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LAPTEC - An Improved Recursive Decomposition Ordering for Term Rewriting Systems Revisited
2005Co-Authors: Munehiro IwamiAbstract:Simplification orderings, like the recursive path ordering and the improved recursive decomposition ordering, are widely used for proving the Termination property of Term Rewriting Systems. The improved recursive decomposition ordering is known as the most powerful simplification ordering. In this paper, we investigate the improved recursive decomposition ordering for proving Termination of Term Rewriting Systems. We completely show that the improved recursive decomposition ordering is closed under substitutions.
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An improved recursive decomposition ordering for Term Rewriting Systems revisited
2005Co-Authors: Munehiro IwamiAbstract:Simplification orderings, like the recursive path ordering and the improved recursive decomposition ordering, are widely used for proving the Termination property of Term Rewriting Systems. The improved recursive decomposition ordering is known as the most powerful simplification ordering. In this paper, we investigate the improved recursive decomposition ordering for proving Termination of Term Rewriting Systems. We completely show that the improved recursive decomposition ordering is closed under substitutions.
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Artificial Intelligence and Applications - Persistence of Termination for Term Rewriting Systems with Ordered Sorts.
2005Co-Authors: Munehiro IwamiAbstract:A property is persistent if for any many-sorted Term Rewriting system , has the property if and only if Term Rewriting system , which results from by omitting its sort information, has the property . Zantema showed that Termination is persistent for Term Rewriting Systems without collapsing or duplicating rules. In this paper, we show that the Zantema’s result can be extended to Term Rewriting Systems on ordered sorts, i.e., Termination is persistent for Term Rewriting Systems on ordered sorts without collapsing, decreasing or duplicating rules. Furthermore we give the example as application of this result. Also we obtain that completeness is persistent for this class of Term Rewriting Systems.
Rakesh Verma - One of the best experts on this subject based on the ideXlab platform.
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Remarks on Thatte’s transformation of Term Rewriting Systems
Information and Computation, 2004Co-Authors: Bas Luttik, Piet Rodenburg, Rakesh VermaAbstract:AbstractWe carry out a detailed analysis of Thatte’s transformation of Term Rewriting Systems. We refute an earlier claim that this transformation preserves confluence for weakly persistent Systems. We prove the preservation of weak normalization, and of confluence in weakly normalizing Systems and in nonoverlapping Systems with linear subtemplates. We conclude by proving that weak persistence is an undecidable property of Term Rewriting Systems
