A theory of classes: proofs and models

Authors:
Barnaby P. Hilken;David E. Rydeheard
Affiliations:
Department of Computer Science, University of Manchester, Oxford Road, Manchester M13 9PL, U.K. E-mail: der@cs.man.ac.uk;Department of Computer Science, University of Manchester, Oxford Road, Manchester M13 9PL, U.K. E-mail: der@cs.man.ac.uk
Venue:
Mathematical Structures in Computer Science
Year:
1997

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Abstract

We investigate the proof structure and models of theories of classes, where classes are ‘collections’ of entities. The theories are weaker than set theories and arise from a study of type classes in programming languages, as well as from comprehension schemata in categories. We introduce two languages of proofs: one a simple type theory and the other involving proof environments for storing and retrieving proofs. The relationship between these languages is defined in terms of a normalisation result for proofs. We use this result to define a categorical semantics for classes and establish its coherence. Finally, we show how the formal systems relate to type classes in programming languages.