Germain Faure

A ρ-Calculus of Explicit Constraint Application

Theoretical presentations of the @r-calculus often treat the matching constraint computations as an atomic operation although matching constraints are explicitly expressed. Actual implementations have to take a much more realistic view: computations needed in order to find the solutions of a matching equation can be really important in some matching theories and the substitution application usually involves a term traversal. Following the works on explicit substitutions in the @l-calculus, we propose, study and exemplify a @r-calculus with explicit constraint handling, up to the level of substitution applications. The approach is general, allowing the extension to various matching theories. We show that the calculus is powerful enough to deal with errors. We establish the confluence of the calculus and the termination of the explicit constraint handling and application sub-calculus.

A ρ-calculus of explicit constraint application

Theoretical presentations of the rewriting or ρ-calculus often treat the matching constraint computations as an atomic operation although matching constraints are explicitly expressed. Actual implementations have to take a more realistic view: computations needed in order to find the solutions of a matching equation can have an important impact on the (efficiency of the) calculus for some matching theories and the substitution application usually involves a term traversal.Following the works on explicit substitutions in the λ-calculus, we present two versions of the ρ-calculus, one with explicit matching and one with explicit substitutions, together with a version that combines the two and considers efficiency issues and more precisely the composition of substitutions. The approach is general, allowing for potential extensions to various matching theories. We establish the confluence of the calculus and the termination of the explicit constraint handling and application sub-calculus.

Confluence of pattern-based calculi

Different pattern calculi integrate the functional mechanisms from the λ-calculus and the matching capabilities from rewriting. Several approaches are used to obtain the confluence but in practice the proof methods share the same structure and each variation on the way pattern-abstractions are applied needs another proof of confluence. We propose here a generic confluence proof where the way pattern-abstractions are applied is axiomatized. Intuitively, the conditions guarantee that the matching is stable by substitution and by reduction. We show that our approach directly applies to different pattern calculi, namely the lambda calculus with patterns, the pure pattern calculus and the rewriting calculus. We also characterize a class of matching algorithms and consequently of pattern-calculi that are not confluent.

Exceptions in the Rewriting Calculus

In the context of the rewriting calculus, we introduce and study an exception mechanism that allows us to express in a simple way rewriting strategies and that is therefore also useful for expressing theorem proving tactics. The proposed exception mechanism is expressed in a confluent calculus which gives the ability to simply express the semantics of the first tactical and to describe in full details the expression of conditional rewriting.

Term Collections in λ and ρ-calculi

A hose construction, and method of making same, are provided wherein the hose construction comprises a hose having a hose end and a hose clamp assembly fastened to opposite sides of a first arcuate length of the hose end with a second arcuate length of the hose end defining the remaining 360 DEG circumference thereof and with the assembly being movable between an unclamped and a clamped position thereof and in the clamped position employs the second arcuate length of the hose end as clamping means; and, the assembly comprises a first straight member, an L-shaped member, and a simple link connected between the members which cooperate with the members to move the assembly between the unclamped and a clamped positions thereof with optimum simplicity and the assembly has means for locking same in its clamped position.