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Mechanizing and Improving Dependency Pairs
 Journal of Automated Reasoning
, 2006
"... Abstract. The dependency pair technique [1, 11, 12] is a powerful method for automated termination and innermost termination proofs of term rewrite systems (TRSs). For any TRS, it generates inequality constraints that have to be satisfied by wellfounded orders. We improve the dependency pair techni ..."
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Cited by 100 (38 self)
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Abstract. The dependency pair technique [1, 11, 12] is a powerful method for automated termination and innermost termination proofs of term rewrite systems (TRSs). For any TRS, it generates inequality constraints that have to be satisfied by wellfounded orders. We improve the dependency pair technique by considerably reducing the number of constraints produced for (innermost) termination proofs. Moreover, we extend transformation techniques to manipulate dependency pairs which simplify (innermost) termination proofs significantly. In order to fully mechanize the approach, we show how transformations and the search for suitable orders can be mechanized efficiently. We implemented our results in the automated termination prover AProVE and evaluated them on large collections of examples.
The dependency pair framework: Combining techniques for automated termination proofs
 In Proc. LPAR ’04, LNAI 3452
, 2005
"... Abstract. The dependency pair approach is one of the most powerful techniques for automated termination proofs of term rewrite systems. Up to now, it was regarded as one of several possible methods to prove termination. In this paper, we show that dependency pairs can instead be used as a general co ..."
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Cited by 84 (30 self)
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Abstract. The dependency pair approach is one of the most powerful techniques for automated termination proofs of term rewrite systems. Up to now, it was regarded as one of several possible methods to prove termination. In this paper, we show that dependency pairs can instead be used as a general concept to integrate arbitrary techniques for termination analysis. In this way, the benefits of different techniques can be combined and their modularity and power are increased significantly. We refer to this new concept as the “dependency pair framework ” to distinguish it from the old “dependency pair approach”. Moreover, this framework facilitates the development of new methods for termination analysis. To demonstrate this, we present several new techniques within the dependency pair framework which simplify termination problems considerably. We implemented the dependency pair framework in our termination prover AProVE and evaluated it on large collections of examples. 1
Tyrolean Termination Tool: Techniques and Features
 Proc. of IJCAR’08, LNCS 5195
"... Abstract The Tyrolean Termination Tool (T T T for short) is a powerful tool for automatically proving termination of rewrite systems. It incorporates several new refinements of the dependency pair method that are easy to implement, increase the power of the method, result in simpler termination pro ..."
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Cited by 59 (12 self)
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Abstract The Tyrolean Termination Tool (T T T for short) is a powerful tool for automatically proving termination of rewrite systems. It incorporates several new refinements of the dependency pair method that are easy to implement, increase the power of the method, result in simpler termination proofs, and make the method more efficient. T T T employs polynomial interpretations with negative coefficients, like x − 1 for a unary function symbol or x − y for a binary function symbol, which are useful for extending the class of rewrite systems that can be proved terminating automatically. Besides a detailed account of these techniques, we describe the convenient web interface of T T T and provide some implementation details.
Certification of automated termination proofs
 In Proc. FroCoS’07
, 2007
"... 2 CÉDRIC – Conservatoire national des arts et métiers Abstract. Nowadays, formal methods rely on tools of different kinds: proof assistants with which the user interacts to discover a proof step by step; and fully automated tools which make use of (intricate) decision procedures. But while some pro ..."
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Cited by 26 (6 self)
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2 CÉDRIC – Conservatoire national des arts et métiers Abstract. Nowadays, formal methods rely on tools of different kinds: proof assistants with which the user interacts to discover a proof step by step; and fully automated tools which make use of (intricate) decision procedures. But while some proof assistants can check the soundness of a proof, they lack automation. Regarding automated tools, one still has to be satisfied with their answers Yes/No/Donotknow, the validity of which can be subject to question, in particular because of the increasing size and complexity of these tools. In the context of rewriting techniques, we aim at bridging the gap between proof assistants that yield formal guarantees of reliability and highly automated tools one has to trust. We present an approach making use of both shallow and deep embeddings. We illustrate this approach with a prototype based on the CiME rewriting toolbox, which can discover involved termination proofs that can be certified by the COQ proof assistant, using the COCCINELLE library for rewriting. 1
Tsukuba termination tool
 In Proc. 14th RTA, LNCS 2706
, 2003
"... We present a tool for automatically proving termination of firstorder rewrite systems. The tool is based on the dependency pair method of Arts and Giesl [1]. It incorporates several new ideas that make the method more efficient. The tool produces highquality output and has a convenient web interfa ..."
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Cited by 20 (2 self)
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We present a tool for automatically proving termination of firstorder rewrite systems. The tool is based on the dependency pair method of Arts and Giesl [1]. It incorporates several new ideas that make the method more efficient. The tool produces highquality output and has a convenient web interface. If T T T succeeds in proving termination, it outputs a proof script which explains in considerable detail how termination was proved. This script is available in both HTML and L ATEX format. In the latter, the approximated dependency graph is visualized using the dot tool of the Graphviz toolkit. T T T is written in Objective Caml. We tested the various options of T T T on numerous examples. The results, as well as a comparison with other tools that implement the dependency pair method and some implementation details, can be found in [2, 3]. We describe some of the features of the tool (T T T in the sequel) by means of its web interface, displayed in Fig. 1. TRS The user inputs a TRS by typing the rules into the upper right text area or by uploading a file via the browse button. The exact input format is obtained by clicking the TRS link. Comment and Bibtex Anything typed into the upper right text area will appear as a footnote in the generated L ATEX code. This is useful to identify TRSs. L ATEX \cite commands may be included. In order for this to work correctly, a corresponding bibtex entry should be supplied. This can be done by typing the entry into the appropriate text area or by uploading an appropriate bibtex file via the browse button. Base Order The current version of T T T supports the following three base orders: LPO with strict precedence, LPO with quasiprecedence, and KBO with strict precedence. Dependency Pairs T T T supports the basic features of the dependency pair technique (argument filtering, dependency graph, cycle analysis) described below. Advanced features like narrowing, rewriting, and instantiation are not yet available. Also innermost termination analysis is not yet implemented.
TALP: A Tool for the Termination Analysis of Logic Programs
 In Proc. 11th RTA, LNCS 1833
, 2000
"... Introduction In the last decade, the automatic termination analysis of logic programs has been receiving increasing attention. Among other methods, techniques have been proposed that transform a wellmoded logic program into a term rewriting system (TRS) so that termination of the TRS implies termi ..."
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Cited by 17 (0 self)
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Introduction In the last decade, the automatic termination analysis of logic programs has been receiving increasing attention. Among other methods, techniques have been proposed that transform a wellmoded logic program into a term rewriting system (TRS) so that termination of the TRS implies termination of the logic program under Prolog's selection rule. In [Ohl99] it has been shown that the twostage transformation obtained by combining the transformations of [GW93] into deterministic conditional TRSs (CTRSs) with a further transformation into TRSs [CR93] yields the transformation proposed in [AZ96], and that these three transformations are equally powerful. In most cases simplification orderings are not sufficient to prove termination of the TRSs obtained by the twostage transformation. However, if one uses the dependency pair method [AG00] in combination with polynomial interpretations instead, then most of the examples described in the literature can automatically be pro
A3PAT, an Approach for Certified Automated Termination Proofs
 In ACM SIGPLAN Workshop on Partial Evaluation and Program Manipulation (PEPM 10
, 2010
"... Software engineering, automated reasoning, rulebased programming or specifications often use rewriting systems for which termination, among other properties, may have to be ensured. This paper presents the approach developed in Project A3PAT to discover and moreover certify, with full automation, t ..."
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Software engineering, automated reasoning, rulebased programming or specifications often use rewriting systems for which termination, among other properties, may have to be ensured. This paper presents the approach developed in Project A3PAT to discover and moreover certify, with full automation, termination proofs for term rewriting systems. It consists of two developments: the COCCINELLE library formalises numerous rewriting techniques and termination criteria for the COQ proof assistant; the CiME3 rewriting tool translates termination proofs (discovered by itself or other tools) into traces that are certified by COQ assisted by COCCINELLE. The abstraction level of our formalisation allowed us to weaken premises of some theorems known in the literature, thus yielding new termination criteria, such as an extension of the powerful subterm criterion (for which we propose the first full COQ formalisation). Techniques employed in CiME3 also improve on previous works on formalisation and analysis of dependency graphs. Categories and Subject Descriptors F.3.1 [Logics and Meaning
Improving ContextSensitive Dependency Pairs
, 2008
"... Contextsensitive dependency pairs (CSDPs) are currently the most powerful method for automated termination analysis of contextsensitive rewriting. However, compared to DPs for ordinary rewriting, CSDPs suffer from two main drawbacks: (a) CSDPs can be collapsing. This complicates the handling o ..."
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Cited by 13 (2 self)
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Contextsensitive dependency pairs (CSDPs) are currently the most powerful method for automated termination analysis of contextsensitive rewriting. However, compared to DPs for ordinary rewriting, CSDPs suffer from two main drawbacks: (a) CSDPs can be collapsing. This complicates the handling of CSDPs and makes them less powerful in practice. (b) There does not exist a “DP framework” for CSDPs which would allow one to apply them in a flexible and modular way. This paper solves drawback (a) by introducing a new definition of CSDPs. With our definition, CSDPs are always noncollapsing and thus, they can be handled like ordinary DPs. This allows us to solve drawback (b) as well, i.e., we extend the existing DP framework for ordinary DPs to contextsensitive rewriting. We implemented our results in the tool AProVE and successfully evaluated them on a large collection of examples.
Polynomial Interpretations with Negative Coefficients
, 2004
"... Polynomial interpretations are a useful technique for proving termination of term rewrite systems. We show how polynomial interpretations with negative coefficients, like x 1 for a unary function symbol or x y for a binary function symbol, can be used to extend the class of rewrite systems that ca ..."
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Polynomial interpretations are a useful technique for proving termination of term rewrite systems. We show how polynomial interpretations with negative coefficients, like x 1 for a unary function symbol or x y for a binary function symbol, can be used to extend the class of rewrite systems that can be automatically proved terminating.