Primitive Inductive Theorems Bridge Implicit Induction Methods and Inductive Theorems in Higher-Order Rewriting

Kusakari, Keiichirou; Sakai, Masahiko; Sakabe, Toshiki

doi:10.1093/ietisy/e88-d.12.2715

Cited by 5 publications

(10 citation statements)

References 13 publications

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“…Developing a termination prover for STRSs based on our results will also be future work. We also hope to see the results of this research applied to inductive reasoning [20] and the Knuth-Bendix procedure [21] on STRSs.…”

Section: Discussionmentioning

confidence: 96%

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

Kusakari

Sakai

2009

IEICE Trans. Inf. & Syst.

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Keiichirou KUSAKARI†a) and Masahiko SAKAI †b) , Members SUMMARY A static dependency pair method, proposed by us, can effectively prove termination of simply-typed term rewriting systems (STRSs). The theoretical basis is given by the notion of strong computability. This method analyzes a static recursive structure based on definition dependency. By solving suitable constraints generated by the analysis result, we can prove the termination. Since this method is not applicable to every system, we proposed a class, namely, plain function-passing, as a restriction. In this paper, we first propose the class of safe function-passing, which relaxes the restriction by plain function-passing. To solve constraints, we often use the notion of reduction pairs, which is designed from a reduction order by the argument filtering method. Next, we improve the argument filtering method for STRSs. Our argument filtering method does not destroy type structure unlike the existing method for STRSs. Hence, our method can effectively apply reduction orders which make use of type information. To reduce constraints, the notion of usable rules is proposed. Finally, we enhance the effectiveness of reducing constraints by incorporating argument filtering into usable rules for STRSs.

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Section: Discussionmentioning

confidence: 96%

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

Kusakari

Sakai

2009

IEICE Trans. Inf. & Syst.

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“…We expect that the equivalent transformation technique can be extended to higher-order TRSs without much difficulties. However, an integration of automated verification methods for properties such as sufficient completeness, and an incorporation of higher-order inductive theorem proving [1], [17] † Available from http://www.jaist.ac.jp/ chiba/RAPT/ seem yet not clear. The extension toward this direction remains as a future work.…”

Section: Raptmentioning

confidence: 99%

Program Transformation Templates for Tupling Based on Term Rewriting

Chiba

Aoto

Toyama

2010

IEICE Trans. Inf. & Syst.

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SUMMARYChiba et al. (2006) proposed a framework of program transformation of term rewriting systems by developed templates. Contrast to the previous framework of program transformation by templates based on lambda calculus, this framework provides a method to verify the correctness of transformation automatically. Tupling (Bird, 1980) is a well-known technique to eliminate redundant recursive calls for improving efficiency of programs. In Chiba et al.'s framework, however, one can not use tuple symbols to construct developed templates. Thus their framework is not capable of tupling transformations. In this paper, we propose a more flexible notion of templates so that a wider variety of transformations, including tupling transformations, can be handled.

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“…Furthermore, to enable an equality like Rev = Frev to be handled in a higher-order framework, it is also necessary to incorporate the concept of extensionality in inductive theorems. We here introduce the definition of inductive theorems in STRSs given in [18]. In general, automated reasoning for inductive theorems is not so easy.…”

Section: Inductionless Inductionmentioning

confidence: 99%

“…Toyama made clear an essence of the method [21]. Recently, Kusakari, Sakai and Sakabe extended the method to higher-order settings [18]. Proposition 4.2 [18] Let R and R be STRSs, and E be a set of equations.…”

Section: Inductionless Inductionmentioning

confidence: 99%

“…For this reason, most properties that a program must guarantee are formalized as inductive theorems, and as a result, a method for automatically proving inductive theorems is essential for establishing an automatic program verification method. Various methods for automatically proving inductive theorems have been proposed in the research of TRS as a computational model of functional programming languages [6], [7], [10], [15], [18], [19], [21]. In this regard, we present the application of a higher-order KB procedure to the results in [18], which extends the results of inductionless induction in [15], [21] to an STRS framework.…”

Section: Introductionmentioning

confidence: 99%

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A Higher-Order Knuth-Bendix Procedure and Its Applications

Kusakari¹,

Chiba²

2007

IEICE Transactions on Information and Systems

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The completeness (i.e. confluent and terminating) property is an important concept when using a term rewriting system (TRS) as a computational model of functional programming languages. Knuth and Bendix have proposed a procedure known as the KB procedure for generating a complete TRS. A TRS cannot, however, directly handle higher-order functions that are widely used in functional programming languages. In this paper, we propose a higher-order KB procedure that extends the KB procedure to the framework of a simply-typed term rewriting system (STRS) as an extended TRS that can handle higher-order functions. We discuss the application of this higher-order KB procedure to a certification technique called inductionless induction used in program verification, and its application to fusion transformation, a typical kind of program transformation.

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Primitive Inductive Theorems Bridge Implicit Induction Methods and Inductive Theorems in Higher-Order Rewriting

Cited by 5 publications

References 13 publications

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

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

Program Transformation Templates for Tupling Based on Term Rewriting

A Higher-Order Knuth-Bendix Procedure and Its Applications

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