Roland Carl Backhouse

Algebraic and coalgebraic methods in the mathematics of program construction

Ordered Sets and Complete Lattices.- Algebras and Coalgebras.- Galois Connections and Fixed Point Calculus.- Calculating Functional Programs.- Algebra of Program Termination.- Exercises in Coalgebraic Specification.- Algebraic Methods for Optimization Problems.- Temporal Algebra.

Demonic Operators and Monotype Factors

This paper tackles the problem of constructing a compact, point-free proof of the associativity of demonic composition of binary relations and its distributivity through demonic choice. In order to achieve this goal, a definition of demonic composition is proposed in which angelic composition is restricted by means of a so-called ‘monotype factor’. Monotype factors are characterised by a Galois connection similar to the Galois connection between composition and factorisation of binary relations. The identification of such a connection is argued to be highly conducive to the desired compactness of calculation.

Elements of a Relational Theory of Datatypes

The “Boom hierarchy” is a hierarchy of types that begins at the level of trees and includes lists, bags and sets. This hierarchy forms the basis for the calculus of total functions developed by Bird and Meertens, and which has become known as the “Bird-Meertens formalism”.

A calculational approach to mathematical induction

Abstract Several concise formulations of mathematical induction are presented and proved equivalent. The formulations are expressed in variable-free relation algebra and thus are in terms of relations only, without mentioning the related objects. It is shown that the induction principle in this form, when combined with the explicit use of Galois connections, lends itself very well for use in calculational proofs. Two non-trivial examples are presented. The first is a proof of Newmans lemma. The second is a calculation of a condition under which the union of two well-founded relations is well-founded. In both cases the calculations lead to generalisations of the known results. In the case of the latter example, one lemma generalises three different conditions.

Do-it-yourself type theory

This paper provides a tutorial introduction to a constructive theory of types based on, but incorporating some extensions to, that originally developed by Per Martin-Löf. The emphasis is on the relevance of the theory to the construction of computer programs and, in particular, on the formal relationship between program and data structure. Topics discussed include the principle of propositions as types, free types, congruence types, types with information loss and mutually recursive types. Several examples of program development within the theory are also discussed in detail.

Galois Connections and Fixed Point Calculus

Fixed point calculus is about the solution of recursive equations defined by a monotonic endofunction on a partially ordered set. This tutorial presents the basic theory of fixed point calculus together with a number of applications of direct relevance to the construction of computer programs. The tutorial also summarises the theory and application of Galois connections between partially ordered sets. In particular, the intimate relation between Galois connections and fixed point equations is amply demonstrated.

Categorical Fixed Point Calculus

A number of lattice-theoretic fixed point rules are generalised to category theory and applied to the construction of isomorphisms between list structures.

Exercises in quantifier manipulation

The Eindhoven quantifier notation is systematic, unlike standard mathematicial notation. This has the major advantage that calculations with quantified expressions become more straightforward because the calculational rules need be given just once for a great variety of different quantifiers. We demonstrate the ease of calculation with finite quantifications by considering a number of examples. Two are simple warm-up exercises, using boolean equality as the quantifier. Three are taken from books of challenging mathematical problems, and one is a problem concocted by the authors to demonstrate the techniques.

Calculating path algorithms

A calculational derivation is given of two abstract path algorithms. The first is an all-pairs algorithm, two well-known instances of which are Warshalls (reachability) algorithm and Floyds shortest-path algorithm; instances of the second are Dijkstras shortest-path algorithm and breadth-first/depth-first search of a directed graph. The basis for the derivations is the algebra of regular languages.

Induction and Recursion on Datatypes

A new induction principle is introduced. The principle is based on a property of relations, called reductivity, that generalises the property of admitting induction to one relative to a given datatype. The principle is used to characterise a broad class of recursive equations that have a unique solution and is also related to standard techniques for proving termination of programs. Methods based on the structure of the given datatype for constructing reductive relations are discussed.