Static Dependency Pair Method for Simply-Typed Term Rewriting and Related Techniques

Kusakari, Keiichirou; Sakai, Masahiko

doi:10.1587/transinf.e92.d.235

Cited by 9 publications

(21 citation statements)

References 24 publications

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“…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%

“…To formulate the notion of safe function-passing in a simple type setting, we introduced notions of peeling types and peeling orders [17]. We extend these notions to RFPs, and construct a strong computability predicate and a safety function.…”

Section: Constructing a Strong Computability Predicate And A Safety Fmentioning

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].…”

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%

“…Definition 4.3). Then we can prove the soundness by a similar story line in [17], although minor adjustments are needed in almost all parts.…”

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: Strong Computability Safety Function and Safe Function-passingmentioning

confidence: 99%

Section: Constructing a Strong Computability Predicate And A Safety Fmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…Definition 4.3). Then we can prove the soundness by a similar story line in [17], although minor adjustments are needed in almost all parts.…”

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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Argument Filterings and Usable Rules for Simply Typed Dependency Pairs

Aoto

Yamada

2009

Frontiers of Combining Systems

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Superposition for Higher-Order Logic

et al. 2023

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Static Dependency Pair Method for Simply-Typed Term Rewriting and Related Techniques

Cited by 9 publications

References 24 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

Argument Filterings and Usable Rules for Simply Typed Dependency Pairs

Superposition for Higher-Order Logic

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