Solution Of Recursive Domain Equations Research Articles

The Bicategory-Theoretic Solution of Recursive Domain Equations

We generalise the traditional approach of Smyth and Plotkin to the solution of recursive domain equations from order-enriched structures to bicategorical ones and thereby develop a bicategorical theory for recursively defined domains in accordance with Axiomatic Domain Theory.

A Convenient Category of Domains

We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ω-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, can be used as the basis for a theory of computability, and provides a model of parametric polymorphism.

A logical model for relational abstract domains

In this article we introduce the notion of Heyting completion in abstract interpretation. We prove that Heyting completion provides a model for Cousot's reduced cardinal power of abstract domains and that it supplies a logical basis to specify relational domains for program analysis and abstract interpretation. We study the algebraic properties of Heyting completion in relation with other well-known domain transformers, like reduced product and disjunctive completion. This provides a uniform algebraic setting where complex abstract domains can be specified by simple logic formulas, or as solutions of recursive abstract domain equations, involving few basic operations for domain construction, all characterized by a clean logical interpretation. We apply our framework to characterize directionality and condensing and in downward closed analysis of (constraint) logic programs.

A characterization of distance between 1-bounded compact ultrametric spaces through a universal space

The category of 1-bounded compact ultrametric spaces (KUMs) and non-distance increasing functions has been extensively used in the semantics of concurrent programming languages. In this paper a universal space U for KUMs is introduced, such that each KUM can be isometrically embedded in it. The space U consists of a suitable subset of the space of functions from [0, 1) to N, endowed with a “prefix-based” ultrametric. U allows to characterize the distance between KUMs introduced in Alessi et al. (1995) in terms of the Hausdorff distance between its compact subsets. As applications, it is proved how to derive the existence of limits for Cauchy towers of spaces without using the classical categorical construction and how to find solutions of recursive domain equations inside ℘ nco(U) .

Using information systems to solve recursive domain equations

This paper aims to make the following main contribution: to show how to use the concrete nature of Scott's information systems to advantage in solving recursive domain equations. The method is based on the substructure relation between information systems. This essentially makes a complete partial order (cpo) of information systems. Standard domain constructions like function space can be made continous on this cpo so the solution of recursive domain equations reduces to the more familiar construction of forming the least fixed-point of a continuous function.

Recursive domain equations for concrete data structures

In the theory of denotational semantics of programming languages, we study event structures which combine features of Scott's information systems and Kahn and Plotkin's concrete data structures and model computational processes. We show that a simple approximation concept for event structures allows us to obtain straightforward solutions of recursive domain equations for event domains. From this, we derive a generalization of corresponding theorems of Kahn and Plotkin respectively Berry and Curien for concrete data structures and concrete domains.

The Category-Theoretic Solution of Recursive Domain Equations

Recursive specifications of domains plays a crucial role in denotational semantics as developed by Scott and Strachey and their followers. The purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place. The idea is to follow the well-known analogy between partial orders and categories, generalizing from least fixed-points of continuous functions over cpos to initial ones of continuous functors over $\omega $-categories. To apply these general ideas we introduce Wand’s ${\bf O}$-categories where the morphism-sets have a partial order structure and which include almost all the categories occurring in semantics. The idea is to find solutions in a derived category of embeddings and we give order-theoretic conditions which are easy to verify and which imply the needed categorical ones. The main tool is a very general form of the limit-colimit coincidence remarked by Scott. In the concluding section we outline how compatibility considerations are to be included in the framework. A future paper will show how Scott’s universal domain method can be included too.