# Domains in a realizability framework

## Abstract

We aim to relate and develop various approaches to a theory of domains in a realizability framework.

In the first part we develop the theory in an arbitrary *partial cartesian closed category* and, as a particular instance, in the category of *partial equivalence relations* (per) over an arbitrary *partial combinatory algebra* (pca). In this way one obtains the category of *separated* pers that may be regarded as a universe of partially ordered sets and monotone maps with strong closure properties.

Pursuing the analogy with domain theory one tries to build a category of directed complete partial orders and continuous maps. We consider two main approaches: one based on *Kleene's pca* of numbers and partial recursive functions and the other based on *Scott's D* _{∞} *λ-model*.

Contents: 0. Introduction, 1. Dominance and Intrinsic Preorder, 2. Separated Partial Equivalence Relations (Σper), 3. N-completeness, 4. Σ-connection and Extensionality, 5. D-completeness and Uniformity, 6. Conclusion.

## Keywords

Full Subcategory Terminal Object Realizability Framework Admissible Family Partial Recursive Function## References

- Abadi M., Plotkin G. [1990] "A per model of polymorphism and recursive types", 5th IEEE-LICS, Philadelphia.Google Scholar
- Amadio R. [1989] "Recursion over realizability structures", TR 1/89 Dipartimento di Informatica, Università di Pisa, (to appear on Info.&Comp.).Google Scholar
- Amadio R., Cardelli L. [1990] "Subtyping Recursive Types", DEC-SRC TR #62, ext. abs. to appear in Proc. ACM-POPL91.Google Scholar
- Bruce K., Longo G. [1990] "A modest model of records, inheritance and bounded quantification", Info&Comp., 87, 1–2, (196–240), Preliminary version in Proc. of IEEE-LICS 88.Google Scholar
- Carboni A., Freyd P., Scedrov A. [1987] "A categorical approach to realizability and polymorphic types" 3rd ACM Symp. on Math. Found of Language semantics, New Orleans.Google Scholar
- Cardelli L., Longo G. [1990] "A semantic basis for Quest", in Proc LISP&FP, Nice.Google Scholar
- Coppo M. [1985] "A completeness theorem for recursively defined types", in Proc. 12th ICALP, Nafplion, Greece 1985, SLNCS 194.Google Scholar
- Curien P-L., Obtulowicz A. [1988] "Partiality, Cartesian Closedness and Toposes", Information & Computation, 80, (50–95).Google Scholar
- Droste M., Göbel R. [1990] "Universal domains in the theory of denotational semantics of programming languages", IEEE-LICS 90, Philadelphia.Google Scholar
- Freyd P., Mulry P., Rosolini G., Scott D. [1990] "Extensional Pers", 5th IEEE-LICS, Philadelphia.Google Scholar
- Gunter C., Jung A. [1990] "Coherence and Consistency in Domains", J.P.Apl.Alg., 63, (49–66).Google Scholar
- Hyland M. [1982] "The effective Topos," in The Brouwer Symposium, (Troelstra, Van Dalen eds.), North-Holland.Google Scholar
- Hyland M. [1988] "A small complete category", APAL, 40, 2, (135–165).Google Scholar
- Longo G., Moggi E. [1984] "Cartesian closed categories of enumerations for effective type-structures", in Symp. on Semantics of Data Types, Kahn&al. (eds.), SLNCS 173.Google Scholar
- Longo G., Moggi E. [1988] "Constructive Natural Deduction and its Modest Interpretation", CMU Report CS-88-131, Lecture delivered at the workshop "The semantics of natural and programming languages" Stanford, March 1987.Google Scholar
- Mac Lane S. [1971] "Categories for the working mathematician", Springer-Verlag, New York.Google Scholar
- McCarthy D. [1984] "Realizability and Recursive mathematics" Ph. D. Thesis, Merton College, Oxford.Google Scholar
- Moggi E. [1988] "Partial morphisms in categories of effective objects", Info.&Comp., 76, (250–277).Google Scholar
- Mulry P. [1981] "Generalized Banach-Mazur functionals in the topos of recursive sets", J. Pure Apl. Algebra, 26, 1, (71–83).Google Scholar
- Myhill J., Shepherdson J. [1955] "Effective operations on partial recursive functions", Zeit. für Math. Logik und Grund. der Math., 1, (310–317).Google Scholar
- Paulin-Mohring C. [1989] "Extraction de programmes dans le calcul des contructions", thèse, Paris VII.Google Scholar
- Phoa W. [1990] "Effective domains and intrinsic structure", 5th IEEE-LICS, Philadelphia.Google Scholar
- Plotkin G. [1980/4] "Domains", lecture notes, C.S. Dept., Edinburgh.Google Scholar
- Robinson E., Rosolini P. [1988] "Categories of partial maps", Info&Co., 79, (95–130).Google Scholar
- Rosolini G. [1986] "Continuity and effectiveness in Topoi" PhD Thesis, Oxford University.Google Scholar
- Scott D. [1976] "Data types as lattices," SIAM Journal of Computing, 5 (pp. 522–587).Google Scholar
- Smyth M. [1977] "Effectively Given Domains", Theoret. Comput. Sci. 5, (255–272).Google Scholar
- Smyth M., Plotkin G. [1982] "The category-theoretic solution of recursive domain equations" SIAM Journal of Computing 11, (pp.761–783).Google Scholar
- Soare R. [1987] "Recursively enumerable sets and degrees", Springer-Verlag.Google Scholar
- Streicher T. [1990] "Independence results for impredicative calculi of dependent types", preprint.Google Scholar
- Troelstra A., van Dalen D. [1988] "Constructivism in Mathematics", 2 vols., North-Holland.Google Scholar
- Wand M. [1979] "Fixed-point Constructions in Order-enriched Categories", Theoret. Comp. Sci. (13–30).Google Scholar