Tyrolean termination tool: Techniques and features

Hirokawa, Nao; Middeldorp, Aart

doi:10.1016/j.ic.2006.08.010

Cited by 87 publications

(157 citation statements)

References 35 publications

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“…The notion of usable rules optimize a constraint generated by the dependency pair method. This analysis was first conducted for TRSs [6], [9], and has been extended to STRSs [17], [21], and to HRSs [22]. In the future we will extend the notion on RFPs.…”

Section: Discussionmentioning

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: Discussionmentioning

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%

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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“…However, cost function often include polynomials of higher degrees since they describe the complexity of the overall program. Unlike our approach, these approaches are restricted to polynomials, however, there are some techniques [19] that allow using expressions of the form max{0, P } where P is a polynomial, they basically reduce to the previous case using approximations. We believe that the advantage of our approach, over these approaches, is in its modular nature.…”

Section: Other Approaches and Related Workmentioning

confidence: 99%

A practical comparator of cost functions and its applications

Albert

Arenas

Genaim

et al. 2015

Science of Computer Programming

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Automatic cost analysis has significantly advanced in the last few years. Nowadays, a number of cost analyzers exist which automatically produce upperand/or lower-bounds on the amount of resources required to execute a program. Cost analysis has a number of important applications such as resource-usage verification and program synthesis and optimization. For such applications to be successful, it is not sufficient to have automatic cost analysis. It is also required to have automated means for handling the analysis results, which are in the form of Cost Functions (CFs for short) i.e., non-recursive expressions composed of a relatively small number of types of basic expressions. In particular, we need automated means for comparing CFs in order to prove that a CF is smaller than or equal to another one for all input values of interest. General function comparison is a hard mathematical problem. Rather than attacking the general problem, in this work we focus on comparing CFs by exploiting their syntactic properties and we present, to the best of our knowledge, the first practical CF comparator which opens the door to fully automated applications of cost analysis. We have implemented the comparator and made its source code available online, so that any cost analyzer can use it.

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“…6, +, −, and * may also occur on non-Pol -increasing positions. But as shown in [17,20], this is possible since the rules for these symbols are equivalent w.r.t. ≈ Pol .…”

Section: Now [N/m]mentioning

confidence: 99%

“…This conditional constraint is a conjunction of formulas like ϕ s t , ϕ s≻t , and ϕ s c for DPs s → t as well as ℓ r for usable rules ℓ → r. Moreover, one has to ensure that the I-interpretation is chosen in such a way that Pol is proper for the right-hand sides of the DPs and the usable rules. 20 Compared to our earlier work in [11], the only additional difficulty is that now we really consider arbitrary max-polynomial interpretations over the integers where [t] Pol can also be negative for any ground term t.…”

Section: Generating I-interpretationsmentioning

confidence: 99%