Relational Properties of Domains?

Andrew M. Pittsy

Cambridge University Computer Laboratory,

Cambridge, England

Abstract

New tools are presented for reasoning about properties of recursively defined domains. We work within a general, category-theoretic framework for various notions of `relation' on domains and for actions of domain constructors on relations. Freyd's analysis of recursive types in terms of a property of mixed initiality/finality is transferred to a corresponding property of invariant relations. The existence of invariant relations is proved under completeness assumptions about the notion of relation. We show how this leads to simpler proofs of the computational adequacy of denotational semantics for functional programming languages with user-declared datatypes. We show how the initiality/finality property of invariant relations can be specialized to yield an induction principle for admissible subsets of recursively defined domains, generalizing the principle of structural induction for inductively defined sets. We also show how the initiality/finality property gives rise to the co-induction principle studied by the author (in Cambridge Univ. Computer Laboratory Tech. Rept. No. 252), by which equalities between elements of recursively defined domains may be proved via an appropriate notion of `bisimulation'.

?Cambridge University Computer Laboratory Technical Report No. 321, December 1993 yResearch supported by UK SERC grant GR/G53279 and EC ESPRIT Basic Research project CLICS-II. Email: [email protected].