Polymorphic higher-order recursive path orderings

Jouannaud, Jean-Pierre; Rubio, Albert

doi:10.1145/1206035.1206037

Cited by 37 publications

(61 citation statements)

References 28 publications

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“…Recently, an effective and practicable reduction order, namely higher-order recursive path orderings, was introduced [3], [4], [11]. We will import the orderings to RFPs in the future.…”

Section: Discussionmentioning

confidence: 99%

Static Dependency Pair Method in Rewriting Systems for Functional Programs with Product, Algebraic Data, and ML-Polymorphic Types

Kusakari

2013

IEICE Trans. Inf. & Syst.

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SUMMARYFor simply-typed term rewriting systems (STRSs) and higher-order rewrite systems (HRSs)à la Nipkow, we proposed a method for proving termination, namely the static dependency pair method. The method combines the dependency pair method introduced for first-order rewrite systems with the notion of strong computability introduced for typed λ-calculi. This method analyzes a static recursive structure based on definition dependency. By solving suitable constraints generated by the analysis, we can prove termination. In this paper, we extend the method to rewriting systems for functional programs (RFPs) with product, algebraic data, and ML-polymorphic types. Although the type system in STRSs contains only product and simple types and the type system in HRSs contains only simple types, our RFPs allow product types, type constructors (algebraic data types), and type variables (ML-polymorphic types). Hence, our RFPs are more representative of existing functional programs than STRSs and HRSs. Therefore, our result makes a large contribution to applying theoretical rewriting techniques to actual problems, that is, to proving the termination of existing functional programs. key words: rewriting systems for functional programs, termination, static dependency pair method IntroductionVarious extensions of term rewriting systems (TRSs) [24] for handling higher-order functions have been proposed [10], [12], [14], [19], [20]. Simply-typed term rewriting systems (STRSs) introduced by Kusakari [14], and higher-order rewrite systems (HRSs) introduced by Nipkow [19] are two such extensions. In this paper, we introduce rewriting systems for functional programs (RFPs), which is an extension of TRSs with product, algebraic data, and ML-polymorphic types. For example, the typical higher-order function foldl can be represented by the following RFP R foldl :Here we suppose that the function foldl has the type:in which α and β are type variables, and list is a type constructor.The static dependency pair method is a powerful )) xsTo prove the non-loopingness of static recursion components, the notions of subterm criterion and reduction pair have been proposed. The subterm criterion was introduced on TRSs [9], and slightly improved by extending the subterms permitted by the criterion on STRSs [16], and extended on HRSs [18]. Reduction pairs [15] are an abstraction of weak-reduction order [1]. By using the subterm criterion, we can prove the non-loopingness of the above static recursion component from the following fact:cons(x, xs) sub xs (xs is a subterm of cons (x, xs))By recapitulating such a termination proof by the static dependency pair method, we obtain the following claim:The function foldl is explicitly recursively defined on the third argument. Hence, the function foldl is well-defined (terminating).This claim is an assertion of the static dependency pair method, and it may be very natural reasoning. However, it is quite difficult to verify the claim because its reduction may be affected by unanticipated behaviors of functio...

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“…Recently, an effective and practicable reduction order, namely higher-order recursive path orderings, was introduced [3], [4], [11]. We will import the orderings to RFPs in the future.…”

Section: Discussionmentioning

confidence: 99%

Static Dependency Pair Method in Rewriting Systems for Functional Programs with Product, Algebraic Data, and ML-Polymorphic Types

Kusakari

2013

IEICE Trans. Inf. & Syst.

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“…We denote by [[σ]] the computability predicate for candidate terms of type σ. Our definition of computability for candidate terms is standard and it is like the one in [18], but without considering polymorphism.…”

Section: Candidate Interpretationsmentioning

confidence: 99%

“…Due to this, we show its practical usefulness by extending it, using the same techniques as for the higher-order recursive path ordering [18] (HORPO), to rewriting on simply typed higher-order terms union beta-reduction. The resulting ordering, called HORPOLO, can hence be used to prove termination of the so called Algebraic Functional Systems [17] (AFS), and provides a new automatable termination method that allows the user to have polynomial interpretations on some symbols in a higher-order setting.…”

Section: Introductionmentioning

confidence: 99%

“…All the novelties of the ordering are already present in the first-order version, so that the method can be fully understood within this part. Then the ordering is extended to the higher-order case by adapting all proofs for HORPO in [18] to HORPOLO. The paper is structured as follows. In Section 2 some preliminaries on firstorder terms and orderings are given.…”

Section: Introductionmentioning

confidence: 99%

“…However, the relation HORPOLO cannot be proved transitive and moreover HORPOLO The strong normalization proof proceeds like in [18] but uses ideas already introduced for the first-order case when handling polynomial interpretations.…”

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confidence: 99%

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The recursive path and polynomial ordering for first-order and higher-order terms

Bofill¹,

Borralleras²,

Rodríguez-Carbonell

et al. 2012

Journal of Logic and Computation

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Abstract. In most termination tools two ingredients, namely recursive path orderings (RPO) and polynomial interpretation orderings (POLO), are used in a consecutive disjoint way to solve the final constraints generated from the termination problem. In this paper we present a simple ordering that combines both RPO and POLO and defines a family of orderings that includes both, and extend them with the possibility of having, at the same time, an RPO-like treatment for some symbols and a POLO-like treatment for the others. The ordering is extended to higher-order terms, providing a new fully automatable use of polynomial interpretations in combination with betareduction.

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