Enhancing dependency pair method using strong computability in simply-typed term rewriting

Kusakari, Keiichirou; Sakai, Masahiko

doi:10.1007/s00200-007-0046-9

Cited by 15 publications

(37 citation statements)

References 26 publications

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“…The static dependency pair method is a powerful method to prove termination, which was introduced on STRSs [16], [17], and extended to HRSs [18], [22]. In this section, we extend the method to RFPs.…”

Section: Static Dependency Pair Methodsmentioning

confidence: 99%

“…From this observation, we proposed notions of plain function-passing [16] and of safe function-passing [17], under which the static dependency pair method works well. In this section, we extend the notion of safe function-passing to RFPs, with the notions of a strong computability predicate and a safety function.…”

Section: Strong Computability Safety Function and Safe Function-passingmentioning

confidence: 99%

“…method to prove termination, which was introduced on STRSs [16], [17], and extended to HRSs [18], [22]. The method combines the dependency pair method introduced for first-order rewrite systems [1] with the notion of strong computability introduced for typed λ-calculi [7], [23].…”

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confidence: 99%

“…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].…”

mentioning

confidence: 99%

“…This example indicates that such a claim does not hold in general. As a class in which the static dependency pair method is sound, we founded the class of plain function-passing [16], and extended this class to the class of safe function-passing [17].…”

mentioning

confidence: 99%

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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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Section: Static Dependency Pair Methodsmentioning

confidence: 99%

Section: Strong Computability Safety Function and Safe Function-passingmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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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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Uncurrying for Termination

Hirokawa

Middeldorp

Zankl

2008

Logic for Programming, Artificial Intelligence, and Reasoning

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Abstract. First-order applicative term rewrite systems provide a natural framework for modeling higher-order aspects. In this paper we present a transformation from untyped applicative term rewrite systems to functional term rewrite systems that preserves and reflects termination. Our transformation is less restrictive than other approaches. In particular, head variables in right-hand sides of rewrite rules can be handled. To further increase the applicability of our transformation, we present a version for dependency pairs.

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Argument Filterings and Usable Rules for Simply Typed Dependency Pairs

Aoto

Yamada

2009

Frontiers of Combining Systems

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Enhancing dependency pair method using strong computability in simply-typed term rewriting

Cited by 15 publications

References 26 publications

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

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

Uncurrying for Termination

Argument Filterings and Usable Rules for Simply Typed Dependency Pairs

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