Linearity in the non-deterministic call-by-value setting

We consider the non-deterministic extension of the call-by-value lambda calculus, which corresponds to the additive fragment of the linear-algebraic lambda-calculus. We define a fine-grained type system, capturing the right linearity present in such formalisms. After proving the subject reduction and the strong normalisation properties, we propose a translation of this calculus into the System F with pairs, which corresponds to a non linear fragment of linear logic. The translation provides a deeper understanding of the linearity in our setting.Comment: 15 pages. To appear in WoLLIC 201

Complexity Bounds for Ordinal-Based Termination

`What more than its truth do we know if we have a proof of a theorem in a given formal system?' We examine Kreisel's question in the particular context of program termination proofs, with an eye to deriving complexity bounds on program running times. Our main tool for this are length function theorems, which provide complexity bounds on the use of well quasi orders. We illustrate how to prove such theorems in the simple yet until now untreated case of ordinals. We show how to apply this new theorem to derive complexity bounds on programs when they are proven to terminate thanks to a ranking function into some ordinal.Comment: Invited talk at the 8th International Workshop on Reachability Problems (RP 2014, 22-24 September 2014, Oxford

Confluence of Non-Left-Linear TRSs via Relative Termination

We present a confluence criterion for term rewrite systems by relaxing termination requirements of Knuth and Bendix' confluence criterion, using joinability of extended critical pairs. Because computation of extended critical pairs requires equational unification, which is undecidable, we give a sufficient condition for testing joinability automatically.Logic for Programming, Artificial Intelligence, and Reasoning. Proceedings of the 18th International Conference, LPAR-18, Mérida, Venezuela, March 11-15, 2012

Iterative Lexicographic Path Orders

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Coq formalization of the higher-order recursive path ordering

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Reduction Strategies for Left-Linear Term Rewriting Systems

Abstract. Huet and Lévy (1979) showed that needed reduction is a normalizing strategy for orthogonal (i.e., left-linear and non-overlapping) term rewriting systems. In order to obtain a decidable needed reduction strategy, they proposed the notion of strongly sequential approximation. Extending their seminal work, several better decidable approximations of left-linear term rewriting systems, for example, NV approximation, shal-low approximation, growing approximation, etc., have been investigated in the literature. In all of these works, orthogonality is required to guar-antee approximated decidable needed reductions are actually normaliz-ing strategies. This paper extends these decidable normalizing strategies to left-linear overlapping term rewriting systems. The key idea is the balanced weak Church-Rosser property. We prove that approximated external reduction is a computable normalizing strategy for the class of left-linear term rewriting systems in which every critical pair can be joined with root balanced reductions. This class includes all weakly or-thogonal left-normal systems, for example, combinatory logic CL with the overlapping rules pred · (succ · x) → x and succ · (pred · x) → x, for which leftmost-outermost reduction is a computable normalizing strategy.

Matching power

www.loria.fr/{˜cirstea,˜ckirchne,˜lliquori} Abstract. In this paper we give a simple and uniform presentation of the rewriting calculus, also called Rho Calculus. In addition to its simplicity, this formulation explicitly allows us to encode complex structures such as lists, sets, and objects. We provide extensive examples of the calculus, and we focus on its ability to represent some object oriented calculi, namely the Lambda Calculus of Objects of Fisher, Honsell, and Mitchell, and the Object Calculus of Abadi and Cardelli. Furthermore, the calculus allows us to get object oriented constructions unreachable in other calculi. In summa, we intend to show that because of its matching ability, the Rho Calculus represents a lingua franca to naturally encode many paradigms of computations. This enlightens the capabilities of the rewriting calculus based language ELAN to be used as a logical as well as powerful semantical framework.