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Algebraic reasoning in lambda calculi

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Kazmierczak, EA (1991) Algebraic reasoning in lambda calculi. PhD thesis, University of Tasmania.

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Abstract

This dissertation examines some aspects of the relationship between λ calculus
and universal algebra. Motivated by the desire to understand the implementation
of abstract data types in functional languages we use λ algebras, or models of
the pure λβ calculus, as a universe in which models for specifications are to be
constructed. Given an arbitrary λ 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, λΣE , is given for reasoning about these models. λΣE is
a simply typed λ 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 λΣE is proved (6.1.5). Finally, a theorem relating
the equational theory to λΣE , namely that for every model of the equational
theory there exists a unique model of λΣE .

Item Type: Thesis (PhD)
Keywords: Lambda calculus
Copyright Holders: The Author
Copyright Information:

Copyright 1991 the Author - The University is continuing to endeavour to trace the copyright
owner(s) and in the meantime this item has been reproduced here in good faith. We
would be pleased to hear from the copyright owner(s).

Additional Information:

Includes bibliographical references (leaves 133-142). Thesis (Ph.D.)--University of Tasmania, 1993

Date Deposited: 19 Dec 2014 02:24
Last Modified: 23 May 2017 00:20
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