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...

m networks and the recent works of Thierry Coquand in type theory have been the most important sources of motivation for the ideas presented here. I wish to specially thank Roberto Amadio, who read the manuscript in a very short delay, providing many helpful comments and remarks. Many thanks also to Luc Boug'e, who accepted to be my oficial supervisor, and to the chair of the jury, Michel Cosnard, who opened to me the doors of the LIP. During these last three years in Lyon I met many wonderful people, who then become wonderful friends. Miguel, Nuria, Veronique, Patricia, Philippe, Pia, Rodrigo, Salvador, Sophie : : : with you I have shared the happiness and sadness of everyday life, those little things which make us to remember someone forever. I also would like to thank the people from "Tango de Soie", for all those funny nights at the Caf'e Moulin Joly. Thanks too to the Uruguayan research community in Computer Science (specially to Cristina Cornes and Alberto Pardo) w

This thesis is concerned with extending the correspondence between intuitionistic logic and functional programming to include assignments and dynamic data. We propose a theoretical framework for adding these imperative features to functional languages without violating their semantic properties. We also describe a constructive programming logic that embodies the principles for reasoning about the extended language. We present an abstract formal language, called Imperative Lambda Calculus (ILC), that extends the typed lambda calculus with imperative programming features, namely references and assignments. The language shares with typed lambda calculus important properties such as the Church-Rosser property and strong normalization. Thus, programs produce the same results with eager and lazy evaluation orders. ILC permits mutable data structures such as arrays, linked lists, trees, and graphs to be constructed and used. Shared values may be updated destructively rather than by copying. This permits pure functional languages to have efficient implementations of problems such as topological sort, graph reduction, and unification. We describe the logical symmetries that underlie ILC by exhibiting a constructive logic, called Observation Type Theory (OTT), for which ILC forms the language of constructions.