Narrowing-based simulation of term rewriting systems with extra variables and its Termination Proof

Nishida, Naoki; Sakai, Masahiko; Sakabe, Toshiki

doi:10.1016/s1571-0661(04)80693-5

Cited by 11 publications

(28 citation statements)

References 9 publications

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“…Nishida, Sakai and Sakabe [57] adapted the dependency pair method for proving the termination of narrowing for TRSs with extra variables. Nishida and Miura [56] then extended the method with the use of a dependency graph.…”

Section: Related Workmentioning

confidence: 99%

“…These methods constitute a direct adaptation of the dependency pair approach to the case of narrowing. The main results in [57,56] consider TRSs with the TRAT property. Although the class of TRSs with the TRAT property is slightly more general than the class of left-linear constructor systems, our notion is simpler, easy to check, and still widely applicable.…”

Section: Related Workmentioning

confidence: 99%

“…The main differences w.r.t. [57,56] are the following: we have considered the termination of narrowing via the termination of rewriting rather than defining a direct termination analysis for narrowing (which greatly simplifies the formal proofs), we have presented also a transformational approach based on the argument filtering transformation, we have presented several refinements of the basic technique (cf. Sec-tion 6), and we have implemented a tool for the termination analysis of narrowing.…”

Section: Related Workmentioning

confidence: 99%

“…Sec-tion 6), and we have implemented a tool for the termination analysis of narrowing. Furthermore, regarding the dependency pair approach, our technique and that of [57,56] are in principle not comparable since one can find examples where our technique is not applicable (e.g., when the TRS is not a left-linear constructor TRS) and the technique of [57,56] is, and examples where our technique is able to prove termination (e.g., thanks to the removal of some extra variables which are replaced by ⊥) and the technique of [57,56] is not.…”

Section: Related Workmentioning

confidence: 99%

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Termination of narrowing via termination of rewriting

Nishida

Vidal

2010

AAECC

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Narrowing extends rewriting with logic capabilities by allowing logic variables in terms and by replacing matching with unification. Narrowing has been widely used in different contexts, ranging from theorem proving (e.g., protocol verification) to language design (e.g., it forms the basis of functional logic languages). Surprisingly, the termination of narrowing has been mostly overlooked. In this work, we present a novel approach for analyzing the termination of narrowing in left-linear constructor systems-a widely accepted class of systems-that allows us to reuse existing methods in the literature on termination of rewriting.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

See 2 more Smart Citations

Termination of narrowing via termination of rewriting

Nishida

Vidal

2010

AAECC

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“…In [Nishida and Miura, 2006;Nishida et al, 2003], the approach of [Arts and Giesl , 2000] is adapted to the termination of narrowing, whereas [Vidal , 2008;Nishida and Vidal , 2010] proves termination of narrowing w.r.t. a given set of initial queries, relying on a restricted form of termination of rewriting based on relative termination.…”

Section: Automatic Terminationmentioning

confidence: 99%

Termination of Narrowing: Automated Proofs and Modularity Properties

López¹

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In 1936, Alan Turing proved that the halting problem, that is, deciding whether a program terminates, is an undecidable problem for most practical programming languages. Even so, termination is so relevant that a vast number of techniques for proving the termination of programs have been researched in the recent decades. Term rewriting systems provide an abstract theoretical framework ideally suited for the study of termination. In term rewriting systems, evaluation of a term corresponds to the left-to-right, non-deterministic application of rewriting rules.Narrowing is a generalization of term rewriting that provides a mechanism for automated reasoning. For instance, given a set of rules defining addition and multiplication with natural numbers, rewriting allows to perform evaluation of arithmetic expressions, whereas narrowing allows to solve equations with variables over arithmetic expressions. This thesis constitutes an in-depth study of the termination properties of narrowing. The contributions are as follows.First, we identify classes of systems where narrowing behaves well, in the sense that it always terminates for any system belonging to these classes. Many methods of analysis, such as the semantics-based analysis of program properties by means of narrowing, benefit from this characterization.Second, we develop an automatic method to prove termination of narrowing for a given system, based on the abstract framework of dependency pairs. For the first time, such an automatic method is not restricted to specific classes of TRSs and hence is universally applicable.Third, we propose another automatic method to prove termination of narrowing from a given term, also on top of the framework of dependency pairs. Termination from a given term is relevant for many applications, including the termination analysis of programming languages. Our method generalizes the current state-of-the-art, enabling the study of termination of logic programs in terms of the termination of narrowing, something which was not possible previously.Fourth and last, the modularity properties of the termination of narrowing are considered in detail. That is, given two systems A and B where narrowing is terminating, determine whether it is still terminating in the union system. This notion of modularity has implications in many of the applications of narrowing. We develop the case of combining systems for equational unification.Furthermore, the automated approaches of the second and third contributions above have been implemented in Narradar, our tool for automatic proofs of termination of narrowing. ResumenEn 1936 Alan Turing demostró que el halting problem, esto es, el problema de decidir si un programa termina o no, es un problema indecidible para la inmensa mayoría de los lenguajes de programación. A pesar de ello, la terminación es un problema tan relevante que en lasúltimas décadas un gran número de técnicas han sido desarrolladas para demostrar la terminación de forma automática de la máxima cantidad posible de programas. Los...

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Reducing Relative Termination to Dependency Pair Problems

Iborra

Nishida

Vidal

et al. 2015

Automated Deduction - CADE-25

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Relative termination, a generalized notion of termination, has been used in a number of different contexts like proving the confluence of rewrite systems or analyzing the termination of narrowing. In this paper, we introduce a new technique to prove relative termination by reducing it to dependency pair problems. To the best of our knowledge, this is the first significant contribution to Problem #106 of the RTA List of Open Problems. The practical significance of our method is illustrated by means of an experimental evaluation.

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Narrowing-based simulation of term rewriting systems with extra variables and its Termination Proof

Cited by 11 publications

References 9 publications

Termination of narrowing via termination of rewriting

Termination of narrowing via termination of rewriting

Termination of Narrowing: Automated Proofs and Modularity Properties

Reducing Relative Termination to Dependency Pair Problems

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