We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...

The method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive. This develops results of Raney on completely distributive lattices and of Hoofman on continuous Scott domains, and also generalizes Smyth's "R-structures". Various constructions on continuous posets have neat descriptions in terms of these continuous information systems; here we describe Hoffmann-Lawson duality (which could not be done easily with R-structures) and Vietoris power locales. 2 We also use the method to give a partial answer to a question of Johnstone's: in the context of continuous posets, Vietoris algebras are the same as localic semilattices.

An abstract notion of category of information systems or I-category is introduced as a generalisation of Scott's well-known category of information systems. As in the theory of partial orders, I-categories can be complete or !-algebraic, and it is shown that !-algebraic I-categories can be obtained from a certain completion of countable I-categories. The proposed axioms for a complete I-category introduce a global partial order on the morphisms of the category, making them a cpo. An initial algebra theorem for a class of functors continuous on the cpo of morphisms is proved, thus giving canonical solution of domain equations; an effective version of these results for !-algebraic I-categories is also provided. Some basic examples of I-categories representing the categories of sets, Boolean algebras, Scott domains and continuous Scott domains are constructed. 1 Introduction A distinctive feature of information systems representing Scott domains, as expressed in [Sco82, LW84], is that th...

The probabilistic power domain construction of Jones and Plotkin [6, 7] is defined by a construction on dcpo's. We present alternative definitions in terms of information systems `a la Vickers [12], and in terms of locales. On continuous domains, all three definitions coincide. 1 Introduction To model probabilistic and randomized algorithms in the semantic framework of dcpo's and Scott continuous functions, Jones and Plotkin introduce in [6, 7] the probabilistic power domain construction PD . It forms a computational monad in the sense of [8] in the category of dcpo's and continuous functions and various of its subcategories of `domains'. Every probabilistic powerdomain PDX is equipped with a family of binary operations + p indexed by a real number p between 0 and 1 such that A+ p B denotes the result of choosing A with probability p and B with probability 1 \Gamma p. Other applications of PD were found in [1]. The probabilistic powerdomain of the upper power space [10] of a second ...

This paper is a summary of the following six publications: (1) Stable Power Domains [Hec94d] (2) Product Operations in Strong Monads [Hec93b] (3) Power Domains Supporting Recursion and Failure [Hec92] (4) Lower Bag Domains [Hec94a] (5) Probabilistic Domains [Hec94b] (6) Probabilistic Power Domains, Information Systems, and Locales [Hec94c] After a general introduction in Section 0, the main results of these six publications are summarized in Sections 1 through 6. 0 Introduction In this section, we provide a common framework for the summarized papers. In Subsection 0.1, Moggi's approach to specify denotational semantics by means of strong monads is introduced. In Subsection 0.2, we specialize this approach to languages with a binary choice construct. Strong monads can be obtained in at least two ways: as free constructions w.r.t. algebraic theories (Subsection 0.3), and by using second order functions (Subsection 0.4). Finally, formal definitions of those concepts which are used in all...

Girard categories (GC's) were defined in [14] as categorical models for linear logic. It was shown that the Kleisli category of a GC is Cartesian closed.

Continuous DCPOs are shown to be precisely the complete objects (in the sense of the completions of Brummer, Giuli and Herrlich, [2], [1]) in a certain category of generalized information systems (related to those of Vickers, [9]), with naturally de ned morphisms and embeddings.

A small modification of Vickers' definition of continuous information systems allows for a representation of the category of continuous domains (continuous DCPOs) and several other categories (Scott domains, continuous Scott domains, continuous lattices, algebraic lattices, and others) as Kleisli categories of suitable monads.