The largest Cartesian closed category of domains, considered constructively

DIETER SPREEN
2005 Mathematical Structures in Computer Science  
A conjecture of Smyth [10] is discussed which says that if D and [D → D] are effectively algebraic directedcomplete partial orders with least element (cpo's), then D is an effectively strongly algebraic cpo, where it was left open what exactly is meant by an effectively algebraic and an effectively strongly algebraic cpo. First, notions of an effectively strongly algebraic cpo and an effective SFP object are introduced. The effective SFP objects are just the constructive (computable) objects in
more » ... the effectively given category [9] of indexed ωalgebraic cpo's. Theorem Every effective SFP object is an effectively strongly algebraic cpo, and vice versa. Moreover, this equivalence holds effectively. This shows that the given notion of an effective SFP object is stable. In effectivity considerations of ωalgebraic cpo's it is usual to require that the partial order be decidable on the compact elements. Here, we use a stronger assumption. Theorem If D is an indexed ω-algebraic cpo that has a computable completeness test and [D → D] is an ω-algebraic cpo, then D is an effective SFP object. An ω-algebraic cpo has a computable completeness test, if there is a procedure which decides for any two finite sets X and Y of compact cpo elements whether X is a complete set of upper bounds of Y . It is an open question whether this requirement can be weakened in the above result. Corollary The category of effective SFP objects and continuous maps is the largest Cartesian closed full subcategory of the category of ω-algebraic cpo's that have a computable completeness test. Next, it is studied whether this result also holds in a constructive framework (or, to be more precise, in the framework of recursive mathematics), where one considers categories with constructive domains as objects, that is, domains consisting only of the constructive (computable) elements of an indexed ω-algebraic cpo, and computable maps as morphisms. The notions of a weakly indexed full subcategory and of being constructively Cartesian closed are introduced. The effectivity requirements in these definitions are very weak. Theorem The category of constructive SFP domains is the largest constructively Cartesian closed weakly indexed full subcategory of the category of constructive domains that have a computable completeness test. Constructive (effective) versions of domain-theoretic results are very important both for the foundations of computer science as well as for applications, since programming languages specify computable maps and computable (effectively given) data structures. Moreover, effective versions of classical results are good approximations to what can be proved constructively and such results have turned out to be at the heart of computer science, at least under the viewpoint of developing correct programs. In this respect the results of the paper are relevant to the workshop.
doi:10.1017/s0960129504004591 fatcat:vp4xagskqzhqzhemvfwih46msy