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Evaluation Strategies for Functional Logic Programming
 Journal of Symbolic Computation
, 2001
"... . Recent advances in the foundations and the development of functional logic programming languages originate from farreaching results on narrowing evaluation strategies. Narrowing is a computation similar to rewriting which yields substitutions in addition to normal forms. In functional logic pr ..."
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. Recent advances in the foundations and the development of functional logic programming languages originate from farreaching results on narrowing evaluation strategies. Narrowing is a computation similar to rewriting which yields substitutions in addition to normal forms. In functional logic programming, the classes of rewrite systems to which narrowing is applied are, for the most part, subclasses of the constructorbased, possibly conditional, rewrite systems. Many interesting narrowing strategies, particularly for the smallest subclasses of the constructorbased rewrite systems, are generalizations of wellknown rewrite strategies. However, some strategies for larger nonconfluents subclasses have been developed just for functional logic computations. In this paper, I will discuss the elements that play a relevant role in evaluation strategies for functional logic programming, describe some important classes of rewrite systems that model functional logic programs, show examples of the differences in expressiveness provided by these classes, and review the characteristics of narrowing strategies proposed for each class of rewrite systems. 1
Termination and Reduction Checking for HigherOrder Logic Programs
 In First International Joint Conference on Automated Reasoning (IJCAR
"... . In this paper, we present a syntaxdirected termination and reduction checker for higherorder logic programs. The reduction checker verifies parametric higherorder subterm orderings describing input and output relations. These reduction orderings are exploited during termination checking to ..."
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. In this paper, we present a syntaxdirected termination and reduction checker for higherorder logic programs. The reduction checker verifies parametric higherorder subterm orderings describing input and output relations. These reduction orderings are exploited during termination checking to infer that a specified termination order holds. To reason about parametric higherorder subterm orderings, we introduce a deductive system as a logical foundation for proving termination. This allows the study of prooftheoretical properties, such as consistency, local soundness and completeness and decidability. We concentrate here on proving consistency of the presented inference system. The termination and reduction checker are implemented as part of the Twelf system and enables us to verify proofs by complete induction. 1 Introduction One of the central problems in verifying specifications and checking proofs about them is the need to prove termination. Several automated methods...
A Monotonic HigherOrder Semantic Path Ordering
 Proc. of 8th Int. Conf. on Logic for Programming, Artificial Intelligence and Reasoning (LPAR'01) LNAI 2250:531547
, 2000
"... . There is an increasing use of (first and higherorder) rewrite rules in many programming languages and logical systems. The recursive path ordering (RPO) is a wellknown tool for proving termination of such rewrite rules in the firstorder case [Der82]. However, RPO has some weaknesses. For i ..."
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Cited by 5 (3 self)
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. There is an increasing use of (first and higherorder) rewrite rules in many programming languages and logical systems. The recursive path ordering (RPO) is a wellknown tool for proving termination of such rewrite rules in the firstorder case [Der82]. However, RPO has some weaknesses. For instance, since it is a simplification ordering, it can only handle simply terminating systems. Several techniques have been developed for overcoming these weaknesses of RPO. A very recent such technique is the monotonic semantic path ordering (MSPO) [BFR00], a simple and easily automatizable ordering which generalizes other more adhoc methods. Another recent extension of RPO is its higherorder version HORPO [JR99]. HORPO is an ordering on terms of a typed lambdacalculus generated by a signature of higherorder function symbols. Althgough many interesting examples can be proved terminating using HORPO, it inherits the weaknesses of the firstorder RPO. Therefore, there is an obvious need for higherorder termination orderings without these weaknesses. Here we define the first such ordering, the monotonic higherorder semantic path ordering (MHOSPO), which is still automatizable like MSPO. We give evidence of its power by means of several natural and nontrivial examples which cannot be handled by HORPO. 1
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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Cited by 2 (1 self)
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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.
ChurchRosser Properties of Normal Rewriting, in "Computer Science Logic
 LIPIcs, Dagstuhl Publishing, September 2012
"... We prove a general purpose abstract ChurchRosser result that captures most existing such results that rely on termination of computations. This is achieved by studying abstract normal rewriting in a way that allows to incorporate positions at the abstract level. New concrete ChurchRosser results a ..."
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We prove a general purpose abstract ChurchRosser result that captures most existing such results that rely on termination of computations. This is achieved by studying abstract normal rewriting in a way that allows to incorporate positions at the abstract level. New concrete ChurchRosser results are obtained, in particular for higherorder rewriting at higher types.