102,892 research outputs found

    Refinement Types as Higher Order Dependency Pairs

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    Refinement types are a well-studied manner of performing in-depth analysis on functional programs. The dependency pair method is a very powerful method used to prove termination of rewrite systems; however its extension to higher order rewrite systems is still the object of active research. We observe that a variant of refinement types allow us to express a form of higher-order dependency pair criterion that only uses information at the type level, and we prove the correctness of this criterion

    Improved Functional Flow and Reachability Analyses Using Indexed Linear Tree Grammars

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    The collecting semantics of a program defines the strongest static property of interest. We study the analysis of the collecting semantics of higher-order functional programs, cast as left-linear term rewriting systems. The analysis generalises functional flow analysis and the reachability problem for term rewriting systems, which are both undecidable. We present an algorithm that uses indexed linear tree grammars (ILTGs) both to describe the input set and compute the set that approximates the collecting semantics. ILTGs are equi-expressive with pushdown tree automata, and so, strictly more expressive than regular tree grammars. Our result can be seen as a refinement of Jones and Andersen\u27s procedure, which uses regular tree grammars. The main technical innovation of our algorithm is the use of indices to capture (sets of) substitutions, thus enabling a more precise binding analysis than afforded by regular grammars. We give a simple proof of termination and soundness, and demonstrate that our method is more accurate than other approaches to functional flow and reachability analyses in the literature

    Size-Change Termination as a Contract

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    Termination is an important but undecidable program property, which has led to a large body of work on static methods for conservatively predicting or enforcing termination. One such method is the size-change termination approach of Lee, Jones, and Ben-Amram, which operates in two phases: (1) abstract programs into "size-change graphs," and (2) check these graphs for the size-change property: the existence of paths that lead to infinite decreasing sequences. We transpose these two phases with an operational semantics that accounts for the run-time enforcement of the size-change property, postponing (or entirely avoiding) program abstraction. This choice has two key consequences: (1) size-change termination can be checked at run-time and (2) termination can be rephrased as a safety property analyzed using existing methods for systematic abstraction. We formulate run-time size-change checks as contracts in the style of Findler and Felleisen. The result compliments existing contracts that enforce partial correctness specifications to obtain contracts for total correctness. Our approach combines the robustness of the size-change principle for termination with the precise information available at run-time. It has tunable overhead and can check for nontermination without the conservativeness necessary in static checking. To obtain a sound and computable termination analysis, we apply existing abstract interpretation techniques directly to the operational semantics, avoiding the need for custom abstractions for termination. The resulting analyzer is competitive with with existing, purpose-built analyzers

    Refinement types as higher order dependency pairs

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    International audienceRefinement types are a well-studied manner of performing in-depth analysis on functional programs. The dependency pair method is a very powerful method used to prove termination of rewrite systems; however its extension to higher-order rewrite systems is still the subject of active research. We observe that a variant of refinement types allows us to express a form of higher-order dependency pair method: from the rewrite system labeled with typing information, we build a type-level approximated dependency graph, and describe a type level embedding-order. We describe a syntactic termination criterion involving the graph and the order, and prove our main result: if the graph passes the criterion, then every well-typed term is strongly normalizing.Nous modifions l'approche classique de la terminaison a base de tailles et montrons que le systÚme modifie permet une analyse fine du flot de contrÎle dans un langage d'ordre supérieur. Ceci nous permet de construire un graphe, dit graphe de dépendance approxime, et nous pouvons montrer qu'un critÚre syntaxique sur ce graphe suffit a montrer la terminaison de tout terme bien type

    Towards Static Analysis of Functional Programs using Tree Automata Completion

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    This paper presents the first step of a wider research effort to apply tree automata completion to the static analysis of functional programs. Tree Automata Completion is a family of techniques for computing or approximating the set of terms reachable by a rewriting relation. The completion algorithm we focus on is parameterized by a set E of equations controlling the precision of the approximation and influencing its termination. For completion to be used as a static analysis, the first step is to guarantee its termination. In this work, we thus give a sufficient condition on E and T(F) for completion algorithm to always terminate. In the particular setting of functional programs, this condition can be relaxed into a condition on E and T(C) (terms built on the set of constructors) that is closer to what is done in the field of static analysis, where abstractions are performed on data.Comment: Proceedings of WRLA'14. 201

    Refinement Type Inference via Horn Constraint Optimization

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    We propose a novel method for inferring refinement types of higher-order functional programs. The main advantage of the proposed method is that it can infer maximally preferred (i.e., Pareto optimal) refinement types with respect to a user-specified preference order. The flexible optimization of refinement types enabled by the proposed method paves the way for interesting applications, such as inferring most-general characterization of inputs for which a given program satisfies (or violates) a given safety (or termination) property. Our method reduces such a type optimization problem to a Horn constraint optimization problem by using a new refinement type system that can flexibly reason about non-determinism in programs. Our method then solves the constraint optimization problem by repeatedly improving a current solution until convergence via template-based invariant generation. We have implemented a prototype inference system based on our method, and obtained promising results in preliminary experiments.Comment: 19 page

    12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

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    This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto
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