Reduction Strategies in Lambda Term Normalization and their Effects on Heap Usage

Many automatic theorem-provers rely on rewriting. Using theorems as rewrite rules helps to simplify the subgoals that arise during a proof. LCF is an interactive theorem-prover intended for reasoning about computation. Its implementation of rewriting is presented in detail. LCF provides a family of rewriting functions, and operators to combine them. A succession of functions is described, from pattern matching primitives to the rewriting tool that performs most inferences i...

Substitution plays a prominent role in the foundation and implementation of mathematics and computation. In the lambda calculus, we cannot define alpha congruence without a form of substitution but for substitution and reduction to work, we need to assume a form of alpha congruence (e.g., when we take lambda terms modulo bound variables). Students on a lambda calculus course usually find this confusing. The elegant writings and research of the Curry school have settled this p...

In this article we investigate how a subterm pattern matching algorithm can be exploited to implement efficient term rewriting procedures. From the left-hand sides of the rewrite system we construct a set automaton, which can be used to find all redexes in a term efficiently. We formally describe a procedure that, given a rewrite strategy, interleaves pattern matching steps and rewriting steps and thus smoothly integrates redex discovery and subterm replacement. We then prese...

We present a prototype of an integrated reasoning environment for educational purposes. The presented tool is a fragment of a proof assistant and automated theorem prover. We describe the existing and planned functionality of the theorem prover and especially the functionality of the educational fragment. This currently supports working with terms of the untyped lambda calculus and addresses both undergraduate students and researchers. We show how the tool can be used to supp...

This volume contains selected papers from the proceedings of the First International Workshop on Strategies in Rewriting, Proving, and Programming (IWS 2010), which was held on July 9, 2010, in Edinburgh, UK. Strategies are ubiquitous in programming languages, automated deduction and reasoning systems. In the two communities of Rewriting and Programming on one side, and of Deduction and Proof engines (Provers, Assistants, Solvers) on the other side, workshops have been launch...

This paper presents simple, syntactic strong normalization proofs for the simply-typed lambda-calculus and the polymorphic lambda-calculus (system F) with the full set of logical connectives, and all the permutative reductions. The normalization proofs use translations of terms and types to systems, for which strong normalization property is known.

This paper introduces a new term rewriting system that is similar to the embedded read-back mechanism for interaction nets presented in our previous work, but is easier to follow than in the original setting and thus to analyze its properties. Namely, we verify that it correctly represents the lambda calculus. Further, we show that there is exactly one reduction sequence that starts with any term in our term rewriting system. Finally, we represent the leftmost strategy which ...

Abstract machines for the strong evaluation of lambda-terms (that is, under abstractions) are a mostly neglected topic, despite their use in the implementation of proof assistants and higher-order logic programming languages. This paper introduces a machine for the simplest form of strong evaluation, leftmost-outermost (call-by-name) evaluation to normal form, proving it correct, complete, and bounding its overhead. Such a machine, deemed Strong Milner Abstract Machine, is a ...

Termination is a major question in both logic and computer science. In logic, termination is at the heart of proof theory where it is usually called strong normalization (of cut elimination). In computer science, termination has always been an important issue for showing programs correct. In the early days of logic, strong normalization was usually shown by assigning ordinals to expressions in such a way that eliminating a cut would yield an expression with a smaller ordinal....

This paper is an expository contribution reporting on published work. It focusses on an approach followed in the rewriting community to formalize the concept of strategy. Based on rewriting concepts, several definitions of strategy are reviewed and connected: in order to catch the higher-order nature of strategies, a strategy is defined as a proof term expressed in the rewriting logic or in the rewriting calculus; to address in a coherent way deduction and computation, a stra...