Semantics Of Functional Programming Languages Research Articles

Computability and totality in domains

We survey the main results on computability and totality in Scott–Eršov-domains as well as their applications to the theory of functionals of higher types and the semantics of functional programming languages. A new density theorem is proved and applied to show the equivalence of the hereditarily computable total continuous functionals with the hereditarily effective operations over a large class of base types.

Relational Properties of Domains

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 ofinvariantrelations. 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. [Theoret. Comput. Sci.124, 195–219 (1994)], by which equalities between elements of recursively defined domains may be proved via an appropriate notion of “bisimulation.”