## A Uniform Approach to Domain Theory in Realizability Models (1996)

Venue: | Mathematical Structures in Computer Science |

Citations: | 19 - 6 self |

### BibTeX

@INPROCEEDINGS{Longley96auniform,

author = {John R. Longley and Alex K. Simpson},

title = {A Uniform Approach to Domain Theory in Realizability Models},

booktitle = {Mathematical Structures in Computer Science},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper we provide a uniform approach to modelling them in categories of modest sets. To do this, we identify appropriate structure for doing "domain theory" in such "realizability models". In Sections 2 and 3 we introduce PCAs and define the associated "realizability" categories of assemblies and modest sets. Next, in Section 4, we prepare for our development of domain theory with an analysis of nontermination. Previous approaches have used (relatively complicated) categorical formulations of partial maps for this purpose. Instead, motivated by the idea that A provides a primitive programming language, we consider a simple notion of "diverging" computation within A itself. This leads to a theory of divergences from which a notion of (computable) partial function is derived together with a lift monad classifying partial functions. The next task is to isolate a subcategory of modest sets with sufficient structure for supporting analogues of the usual domain-theoretic constructions. First, we expect to be able to interpret the standard constructions of total type theory in this category, so it should inherit cartesian-closure, coproducts and the natural numbers from modest sets. Second, it should interact well with the notion of partiality, so it should be closed under application of the lift functor. Third, it should allow the recursive definition of partial functions. This is achieved by obtaining a fixpoint object in the category, as defined in (Crole and Pitts 1992). Finally, although there is in principle no definitive list of requirements on such a category, one would like it to support more complicated constructions such as those required to interpret polymorphic and recursive types. The central part of the paper (Sections 5, 6, 7 and 9) is devoted to establish...