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Categorical Fixed Point Calculus
, 1995
"... A number of latticetheoretic fixed point rules are generalised to category theory and applied to the construction of isomorphisms between list structures. 1 Introduction Category theoreticians view a preordered set as a particular sort of category in which there is at most one arrow between any pa ..."
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A number of latticetheoretic fixed point rules are generalised to category theory and applied to the construction of isomorphisms between list structures. 1 Introduction Category theoreticians view a preordered set as a particular sort of category in which there is at most one arrow between any pair of objects. According to this view, several concepts of lattice theory are instances of concepts of category theory as shown in table 1. An alternative viewpoint, advocated by Lambek [9], is that lattice theory is a valuable source of inspiration for novel results in category theory. Indeed, it is our view that for the purposes of advancing programming methodology category theory may profitably be regarded as "coherently constructive lattice theory 1 ". That is to say, arrows between objects of a category may be seen as "witnesses" to a preordering between the objects. Category theory is thus "constructive" because it is a theory about how to construct such witnesses rather than a theor...
Mathematics of Recursive Program Construction
, 2001
"... A discipline for the design of recursive programs is presented. The core concept ..."
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A discipline for the design of recursive programs is presented. The core concept
Calculating invariants as coreflexive bisimulations
, 2008
"... Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to computer systems engineering. In this paper we reduce the first to a particular case of the second and show how both together pave the way to a theory of coalgebras which regards invariant predicates as ..."
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Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to computer systems engineering. In this paper we reduce the first to a particular case of the second and show how both together pave the way to a theory of coalgebras which regards invariant predicates as types. An outcome of such a theory is a calculus of invariants ’ proof obligation discharge, a fragment of which is presented in the paper. The approach has two main ingredients: one is that of adopting relations as “first class citizens” in a pointfree reasoning style; the other lies on a synergy found between a relational construct, Reynolds ’ relation on functions involved in the abstraction theorem on parametric polymorphism and the coalgebraic account of bisimulation and invariants. In this process, we provide an elegant proof of the equivalence between two different definitions of bisimulation found in coalgebra literature (due to B. Jacobs and Aczel & Mendler, respectively) and their instantiation to the classical ParkMilner definition popular in process algebra.
DatatypeGeneric Reasoning
"... Abstract. Datatypegeneric programs are programs that are parameterised by a datatype. Designing datatypegeneric programs brings new challenges and new opportunities. We review the allegorical foundations of a methodology of designing datatypegeneric programs. The effectiveness of the methodology ..."
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Abstract. Datatypegeneric programs are programs that are parameterised by a datatype. Designing datatypegeneric programs brings new challenges and new opportunities. We review the allegorical foundations of a methodology of designing datatypegeneric programs. The effectiveness of the methodology is demonstrated by an extraordinarily concise proof of the wellfoundedness of a datatypegeneric occursin relation.
DatatypeGeneric Termination Proofs
"... Abstract. Datatypegeneric programs are programs that are parameterised by a datatype. We review the allegorical foundations of a methodology of designing datatypegeneric programs. The notion of Freductivity, where F parametrises a datatype, is reviewed and a number of its properties are presented ..."
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Abstract. Datatypegeneric programs are programs that are parameterised by a datatype. We review the allegorical foundations of a methodology of designing datatypegeneric programs. The notion of Freductivity, where F parametrises a datatype, is reviewed and a number of its properties are presented. The properties are used to give concise, effective proofs of termination of a number of datatypegeneric programming schemas. The paper concludes with a concise proof of the wellfoundedness of a datatypegeneric occursin relation.