Complete Sets of Reductions Modulo A Class of Equational Theories which Generate Infinite Congruence Classes

In this paper we present a generalization of the Knuth-Bendix procedure for generating a complete set of reductions modulo an equational theory. Previous such completion procedures have been restricted to equational theories which generate finite congruence classes. The distinguishing feature of this work is that we are able to generate complete sets of reductions for some equational theories which generate infinite congruence classes. In particular, we are able to handle the class of equational theories which contain the associative, commutative, and identity laws for one or more operators. We first generalize the notion of rewriting modulo an equational theory to include a special form of conditional reduction. We are able to show that this conditional rewriting relation restores the finite termination property which is often lost when rewriting in the presence of infinite congruence classes. We then develop Church-Rosser tests based on the conditional rewriting relation and set forth a completion procedure incorporating these tests. Finally, we describe a computer program which implements the theory and give the results of several experiments using the program

Locally Chain-Parsable Languages

If a context-free language enjoys the local parsability property then, no matter how the source string is segmented, each segment can be parsed in- dependently, and an efficient parallel parsing algorithm becomes possible. The new class of locally chain-parsable languages (LCPL), included in deterministic context-free languages, is here defined by means of the chain-driven automa- ton and characterized by decidable properties of grammar derivations. Such au- tomaton decides to reduce or not a factor in a way purely driven by the terminal characters, thus extending the well-known concept of Input-Driven (ID) (visibly) pushdown machines. LCPL extend and improve the practically relevant operator- precedence languages (Floyd), which are known to strictly include the ID lan- guages, and for which a parallel-parser generator exists. Consistently with the classical results for ID, chain-compatible LCPL are closed under reversal and Boolean operations, and language inclusion is decidable

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Toward a theory of input-driven locally parsable languages

If a context-free language enjoys the local parsability property then, no matter how the source string is segmented, each segment can be parsed independently, and an efficient parallel parsing algorithm becomes possible. The new class of locally chain parsable languages (LCPLs), included in the deterministic context-free language family, is here defined by means of the chain-driven automaton and characterized by decidable properties of grammar derivations. Such automaton decides whether to reduce or not a substring in a way purely driven by the terminal characters, thus extending the well-known concept of input-driven (ID) alias visibly pushdown machines. The LCPL family extends and improves the practically relevant Floyd's operator-precedence (OP) languages which are known to strictly include the ID languages, and for which a parallel-parser generator exists

Modeling, Sharing, and Recursion for Weak Reduction Strategies using Explicit Substitution

We present the lambda sigma^a_w calculus, a formal synthesis of the concepts ofsharing and explicit substitution for weak reduction. We show howlambda sigma^a_w can be used as a foundation of implementations of functionalprogramming languages by modelling the essential ingredients of suchimplementations, namely weak reduction strategies, recursion, spaceleaks, recursive data structures, and parallel evaluation, in a uniform way.First, we give a precise account of the major reduction strategiesused in functional programming and the consequences of choosing lambda-graph-reduction vs. environment-based evaluation. Second, we showhow to add constructors and explicit recursion to give a precise accountof recursive functions and data structures even with respect tospace complexity. Third, we formalize the notion of space leaks in lambda sigma^a_wand use this to define a space leak free calculus; this suggests optimisationsfor call-by-need reduction that prevent space leaking and enablesus to prove that the "trimming" performed by the STG machine doesnot leak space.In summary we give a formal account of several implementationtechniques used by state of the art implementations of functional programminglanguages.Keywords. Implementation of functional programming, lambdacalculus, weak reduction, explicit substitution, sharing, recursion, spaceleaks