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HigherOrder rewriting: Framework, Confluence and termination
 Processes, Terms and Cycles: Steps on the road to infinity. Essays Dedicated to Jan Willem Klop on the occasion of his 60th Birthday. LNCS 3838
, 2005
"... Equations are ubiquitous in mathematics and in computer science as well. This first sentence of a survey on firstorder rewriting borrowed again and again characterizes best the fundamental reason why rewriting, as a technology for processing equations, is so important in our discipline [10]. ..."
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Equations are ubiquitous in mathematics and in computer science as well. This first sentence of a survey on firstorder rewriting borrowed again and again characterizes best the fundamental reason why rewriting, as a technology for processing equations, is so important in our discipline [10].
Residuals in higherorder rewriting
 Proceedings of Rewriting Techniques and Applications (RTA’03
, 2003
"... Abstract. Residuals have been studied for various forms of rewriting setting. In this article we study residuals in orthogonal Pattern Rewriting Systems (PRSs). First, the rewrite relation is defined by means of a higherorder rewriting logic, and proof terms are defined that witness reductions. The ..."
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Abstract. Residuals have been studied for various forms of rewriting setting. In this article we study residuals in orthogonal Pattern Rewriting Systems (PRSs). First, the rewrite relation is defined by means of a higherorder rewriting logic, and proof terms are defined that witness reductions. Then, we have the formal machinery to define a residual operator for PRSs, and we will prove that an orthogonal PRS together with the residual operator mentioned above, is a residual system. As a sideeffect, all results of (abstract) residual theory are inherited by orthogonal PRSs, such as confluence, and the notion of permutation equivalence of reductions. 1
Levels of Undecidability in Rewriting
, 2011
"... Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarc ..."
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Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas, and continuing into the analytic hierarchy, where quantification over function variables is allowed. In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies. We consider properties related to normalization (strong normalization, weak normalization and dependency problems) and properties related to confluence (confluence, local confluence and the unique normal form property). For all of these we distinguish between the single term version and the uniform version. Where appropriate, we also distinguish between ground and open terms. Most uniform properties are Π 0 2complete. The particular problem of local confluence turns out to be Π 0 2complete for ground terms, but only Σ 0 1complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be Π 1 1complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications [35, 7]. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.
A higherorder iterative path ordering
 Proceedings of LPAR 2008, volume 5330 of LNAI
, 2008
"... Abstract. The higherorder recursive path ordering (HORPO) defined by Jouannaud and Rubio provides a method to prove termination of higherorder rewriting. We present an iterative version of HORPO by means of an auxiliary term rewriting system, following an approach originally due to Bergstra and K ..."
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Abstract. The higherorder recursive path ordering (HORPO) defined by Jouannaud and Rubio provides a method to prove termination of higherorder rewriting. We present an iterative version of HORPO by means of an auxiliary term rewriting system, following an approach originally due to Bergstra and Klop. We study wellfoundedness of the iterative definition, discuss its relationship with the original HORPO, and point out possible ways to strengthen the ordering. 1
2.2 Residual Theory......................... 4
, 2003
"... Abstract. Residuals have been studied for various forms of rewriting and residual systems have been defined to capture residuals in an abstract setting. In this article we study residuals in orthogonal Pattern Rewriting Systems (PRSs). First, the rewrite relation is defined by means of a higherorde ..."
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Abstract. Residuals have been studied for various forms of rewriting and residual systems have been defined to capture residuals in an abstract setting. In this article we study residuals in orthogonal Pattern Rewriting Systems (PRSs). First, the rewrite relation is defined by means of a higherorder rewriting logic, and proof terms are defined that witness reductions. Then, we have the formal machinery to define a residual operator for PRSs, and we will prove that an orthogonal PRS together with the residual operator mentioned above, is a residual system. As a sideeffect, all results of (abstract) residual theory are inherited by orthogonal PRSs, such as confluence, and the notion of
SecondOrder Equational Logic (Extended Abstract)
"... We provide an extension of universal algebra and its equational logic from first to second order. Conservative extension, soundness, and completeness results are established. ..."
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We provide an extension of universal algebra and its equational logic from first to second order. Conservative extension, soundness, and completeness results are established.
Semantic Labelling for Proving Termination of Combinatory Reduction Systems
"... Abstract. We give a novel transformation method for proving termination of higherorder rewrite rules in Klop’s format called Combinatory Reduction System (CRS). The format CRS essentially covers the usual pure higherorder functional programs such as Haskell. Our method called higherorder semant ..."
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Abstract. We give a novel transformation method for proving termination of higherorder rewrite rules in Klop’s format called Combinatory Reduction System (CRS). The format CRS essentially covers the usual pure higherorder functional programs such as Haskell. Our method called higherorder semantic labelling is an extension of a method known in the theory of term rewriting. This attaches semantics of the arguments to each function symbol. We systematically define the labelling by using the complete algebraic semantics of CRS, Σmonoids. We also examine the power of higherorder semantic labelling by several examples. This includes an interesting example from the viewpoint of functional programming. 1
Contents
, 2003
"... 1.1 Background........................................ 3 1.1.1 Term rewriting systems............................. 3 ..."
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1.1 Background........................................ 3 1.1.1 Term rewriting systems............................. 3
An Eective Proof of the WellFoundedness of the Multiset Path Ordering
, 2005
"... The contribution of this paper is an eective proof of the wellfoundedness of MPO, as a term of the Calculus of Inductive Constructions. This proof is direct, short and simple. It is a sequence of nested inductions and it only requires as preliminary results the transitivity of MPO and the fact tha ..."
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The contribution of this paper is an eective proof of the wellfoundedness of MPO, as a term of the Calculus of Inductive Constructions. This proof is direct, short and simple. It is a sequence of nested inductions and it only requires as preliminary results the transitivity of MPO and the fact that nite multisets whose elements are accessible for the basic relation are themselves accessible for the multiset order. The terms we consider are not supposed to be ground nor the signature to be nite. All the proofs have been carried out in the Coq proofassistant 1