Interpreting j-conversion as an expansion rule in the simply-typed -calculus maintains the confluence of reduction in a richer type structure. This use of expansions is supported by categorical models of reduction, where fi-contraction, as the local counit, and j-expansion, as the local unit, are linked by local triangle laws. The latter form reduction loops, but strong normalisation (to the long fij-normal forms) can be recovered by "cutting" the loops.

This thesis concerns rewriting in the typed -calculus. Traditional categorical models of typed -calculus use concepts such as functor, adjunction and algebra to model type constructors and their associated introduction and elimination rules, with the natural categorical equations inherent in these structures providing an equational theory for -terms. One then seeks a rewrite relation which, by transforming terms into canonical forms, provides a decision procedure for this equational theory. Unfortunately the rewrite relations which have been proposed, apart from for the most simple of calculi, either generate the full equational theory but contain no decision procedure, or contain a decision procedure but only for a subtheory of that required. Our proposal is to unify the semantics and reduction theory of the typed -calculus by generalising the notion of model from categorical structures based on term equality to categorical structures based on term reduction. This is accomplished via...

) Delia Kesner CNRS and LRI, B at 490, Universit e Paris-Sud - 91405 Orsay Cedex, France. e-mail:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and non-extensional #-calculi with explicit substitutions, where extensionality is interpreted by #-expansion. For that, we propose a general scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our general scheme makes it possible to treat at the same time many well-known calculi such as ## , ## # and ## , or some other new calculi that we propose in this paper. We also show for those calculi not fitting in the general scheme that can be translated to another one fitting the scheme, such as #s , how to reason about confluence properties of their extensional and non-extensional versions. 1 Introduction The #-calculus is a convenient framework to study functional programming, where the evaluation process is modeled by #-reduction. The...

Confluence of orthogonal rewriting systems can be proved using the Finite Developments Theorem. We present, in a general setting, several adaptations of this proof method for obtaining confluence of `not quite' orthogonal systems. 1. Introduction Rewriting as studied here is based on the analogy: rewriting = substitution + rules. This analogy is useful since it enables a clearcut distinction between the `designer' defined substition process, i.e. management of resources, and the `user' defined rewrite rules, of rewriting systems. For example, application of the `user' defined term rewriting rule 2 \Theta x ! x + x to the term 2 \Theta 3 gives rise to the duplication of the term 3 in the result 3 + 3. How this duplication is actually performed (for example, using sharing) depends on the `designer's' implementation of substitution. This decomposition has been shown useful in [OR94, Oos94] in the case of first-order term rewriting systems (TRSs, [DJ90, Klo92]) and higher-order term r...

<F NaN><F NaN> A A ×B<F NaN><F NaN> oo # 1<F NaN><F NaN> // # 2 B That is: h = #f, g# = ## 1 # h, # 2 # h# Case C A<F NaN><F NaN> // in 1<F NaN><F NaN> << f y y y y y y y y y y y y y y y y y y y A +B<F NaN><F NaN> OO [f,g] h<F NaN><F NaN> OO B<F NaN><F NaN> oo in 2<F NaN><F NaN> bb g E E E E E E E E E E E E E E E E E E E That is: h = [f, g] = [h # in 1 , h # in 2 ] Roberto Di Cosmo Glasgow, September 96 4 From equations to rewriting Two choices to orient