Algebraic reasoning in lambda calculi

This dissertation examines some aspects of the relationship between ˜í¬™ calculus and universal algebra. Motivated by the desire to understand the implementation of abstract data types in functional languages we use ˜í¬™ algebras, or models of the pure ˜í¬™˜í‚⧠calculus, as a universe in which models for specifications are to be constructed. Given an arbitrary ˜í¬™ algebra M we construct from it a cartesian closed category with sum objects which we call C(M) . The objects of C(M) are interpreted as types while the arrows can be thought of as effective functions. Unlike set theory where the semantics of an abstract data type is often chosen to be the (isomorphism) class of initial models the simple model of types and functions presented here contains no initial object. This means more care is required when choosing the semantics of an abstract data type. In particular the category C(M) , which is an extension of the cartesian closed monoids of [Koy82, LS86, Bar84] is cartesian closed (4.1.6) and so the basic properties of functional languages such as currying and higher order functions can be modelled. Using the idea of a T-algebra [LS81, SP82] we construct algebras as the greatest fixed points of endofunctors over C(M) (4.2.5). While in general systems of equations do not always exhibit computational properties in many cases they do, especially if a set of Church-Rosser and strongly normalising rewrite rules can be extracted from the equations by a technique such as the Knuth-Bendix completion procedure [KB70]. It is known [KS81] that a Church-Rosser strongly normalising set of rewrite rules is associated with the word problem in the free algebra and if the word problem is solvable then there is a recursive function taking each term in the free algebra to its normal form. We construct models for systems of Church-Rosser, strongly normalising sets of rewrite rules based on this characterisation of the word problem (4.3.14). Secondly a calculus, ˜í¬™˜í¬£E , is given for reasoning about these models. ˜í¬™˜í¬£E is a simply typed ˜í¬™ calculus augmented with the operations and equations of the original specification. It is shown that this calculus is a conservative extension of the usual equational calculus if a strongly normalising and Church-Rosser set of rewrite rules can be generated from the original equations (5.2.10). Thirdly, given a specification and an arbitrary model of that specification in C(M) , the soundness of deduction in ˜í¬™˜í¬£E is proved (6.1.5). Finally, a theorem relating the equational theory to ˜í¬™˜í¬£E , namely that for every model of the equational theory there exists a unique model of ˜í¬™˜í¬£E .