Proving Termination Using Recursive Path Orders and SAT Solving

Schneider–Kamp, Peter; Thiemann, René; Annov, Elena; Codish, Michael; Giesl, Jürgen

doi:10.1007/978-3-540-74621-8_18

Cited by 24 publications

(26 citation statements)

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“…This paper extends the preliminary work presented in [9] and [40] substantially. 1 It contains the details for the complete SAT encoding of RPO, including the use of RPO for constraints like (1) and (3), and a formal analysis of the size of the encoding.…”

Section: Introductionsupporting

confidence: 78%

SAT Solving for Termination Proofs with Recursive Path Orders and Dependency Pairs

et al. 2010

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This paper introduces a propositional encoding for recursive path orders (RPO), in connection with dependency pairs. Hence, we capture in a uniform setting all common instances of RPO, i.e., lexicographic path orders (LPO), multiset path orders (MPO), and lexicographic path orders with status (LPOS). This facilitates the application of SAT solvers for termination analysis of term rewrite systems (TRSs).We address four main inter-related issues and show how to encode them as satisfiability problems of propositional formulas that can be efficiently handled by SAT solving: (A) the lexicographic comparison w.r.t. a permutation of the arguments; (B) the multiset extension of a base order; (C) the combined search for a path order together with an argument filter to orient a set of inequalities; and (D) how the choice of the argument filter influences the set of inequalities that have to be oriented (so-called usable rules).We have implemented our contributions in the termination prover AProVE. Extensive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power.

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Section: Introductionsupporting

confidence: 78%

SAT Solving for Termination Proofs with Recursive Path Orders and Dependency Pairs

et al. 2010

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“…We give in particular the first full formalisation in COQ of the powerful subterm criterion [29] for termination of rewriting systems 2 , and we propose an extension (Theorem 4). Another example is our extended notion of matrix interpretations [16].…”

Section: Introductionmentioning

confidence: 99%

A3PAT, an approach for certified automated termination proofs

Contejean

Paskevich

Urbain

et al. 2010

Proceedings of the 2010 ACM SIGPLAN Workshop on Partial Evaluation and Program Manipulation

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Software engineering, automated reasoning, rule-based 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.

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“…In order to provide a fair comparison we focus on a small set of DP processors: the dependency graph processor, the RPO reduction pair implemented by means of the SAT encoding of [Codish et al, 2005;Schneider-Kamp et al, 2007;Codish et al, 2006] extended to account for our notion of goal-directed usable rules, the subterm criterion of [Hirokawa and Middeldorp, 2004] and the narrowing and instantiation graph refinement processors [Arts and Giesl , 2000]. It is our opinion that this set of processors constitutes a realistic tool which can be used to solve a very reasonable class of termination problems.…”

Section: Resultsmentioning

confidence: 99%

“…In particular, narradar provides an RPO based reduction pair implemented by encoding the order restrictions in propositional form and then using the [Yices] SAT solver to find the precedence and argument filtering which orient the rules and pairs. This approach is described in detail in [Codish et al, 2005;Schneider-Kamp et al, 2007;Codish et al, 2006]. narradar extends this approach with two additional constraint in the propositions generated for narrowing problems, which ensure that:…”

Section: Discussionmentioning

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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Proving Termination Using Recursive Path Orders and SAT Solving

Cited by 24 publications

References 23 publications

SAT Solving for Termination Proofs with Recursive Path Orders and Dependency Pairs

SAT Solving for Termination Proofs with Recursive Path Orders and Dependency Pairs

A3PAT, an approach for certified automated termination proofs

Termination of Narrowing: Automated Proofs and Modularity Properties

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