Computing overlappings by unification in the deterministic lambda calculus LR with letrec, case, constructors, seq and variable chains

Correctness of program transformations in extended lambda calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach to proving correctness is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules.The method is similar to the computation of critical pairs for the completion of term rewriting systems. We describe an effective unification algorithm to determine all overlaps of transformations with reduction rules for the lambda calculus LR which comprises a recursive let-expressions, constructor applications, case expressions and a seq construct for strict evaluation. The unification algorithm employs many-sorted terms, the equational theory of left-commutativity modeling multi-sets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive let-expressions. As a result the algorithm computes a finite set of overlappings for the reduction rules of the calculus LR that serve as a starting point to the automatization of the analysis of program transformations

Two Decades of Maude

This paper is a tribute to José Meseguer, from the rest of us in the Maude team, reviewing the past, the present, and the future of the language and system with which we have been working for around two decades under his leadership. After reviewing the origins and the language's main features, we present the latest additions to the language and some features currently under development. This paper is not an introduction to Maude, and some familiarity with it and with rewriting logic are indeed assumed.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Interning Ground Terms in XSB

This paper presents an implementation of interning of ground terms in the XSB Tabled Prolog system. This is related to the idea of hash-consing. I describe the concept of interning atoms and discuss the issues around interning ground structured terms, motivating why tabling Prolog systems may change the cost-benefit tradeoffs from those of traditional Prolog systems. I describe the details of the implementation of interning ground terms in the XSB Tabled Prolog System and show some of its performance properties. This implementation achieves the effects of that of Zhou and Have but is tuned for XSB's representations and is arguably simpler.Comment: Proceedings of the 13th International Colloquium on Implementation of Constraint LOgic Programming Systems (CICLOPS 2013), Istanbul, Turkey, August 25, 201

Ein mehrsortiger Resolutionskalkül mit Paramodulation

Der Resolutionskalkül mit Paramodulationsregel wird zu einem mehrsortigen Kalkül erweitert. Als Grundlage für das automatische Beweisen erhält man mit diesem Kalkül einen stark reduzierten Suchraum und einfachere Beweise. Die Vollständigkeit, die Korrektheit und der Sortensatz, der den neuen Kalkül mit seinem einsortigen Gegenstück in Beziehung setzt, werden bewiesen. Ergebnisse über Grundtermersetzungen und Unifikation in einem mehrsortigen Kalkül werden vorgestellt. Die Implementierung eines automatischen Beweisers für den neuen Kalkül wird beschrieben. Die Vorteile der Methode werden anhand ausgewählter Beispiele belegt.The resolution calculus with paramodulationrule is extended to a many-sorted calculus. As a basis for Automated Theorem Proving, this many-sorted calculus leads to a remarkable reduction of the search space and also to simpler proofs. Soundness and completeness of the new calculus and the Sort-Theorem, which relates the many-sorted calculus to its one-sorted counterpart, are shown. In addition results about groundterm rewriting and unification in a many-sorted calculus are obtained. The practical consequences for an implementation of an automated theorem prover based on the many-sorted calculus are described. The advantages of the proposed method is verified by certain examples

Multi-level Contextual Type Theory

Contextual type theory distinguishes between bound variables and meta-variables to write potentially incomplete terms in the presence of binders. It has found good use as a framework for concise explanations of higher-order unification, characterize holes in proofs, and in developing a foundation for programming with higher-order abstract syntax, as embodied by the programming and reasoning environment Beluga. However, to reason about these applications, we need to introduce meta^2-variables to characterize the dependency on meta-variables and bound variables. In other words, we must go beyond a two-level system granting only bound variables and meta-variables. In this paper we generalize contextual type theory to n levels for arbitrary n, so as to obtain a formal system offering bound variables, meta-variables and so on all the way to meta^n-variables. We obtain a uniform account by collapsing all these different kinds of variables into a single notion of variabe indexed by some level k. We give a decidable bi-directional type system which characterizes beta-eta-normal forms together with a generalized substitution operation.Comment: In Proceedings LFMTP 2011, arXiv:1110.668