Least fixpoints as meanings of recursive definitions.

The variety of power domain constructions proposed in the literature is put into a general algebraic framework. Power constructions are considered algebras on a higher level: for every ground domain, there is a power domain whose algebraic structure is specified by means of axioms concerning the algebraic properties of the basic operations empty set, union, singleton, and extension of functions. A host of derived operations is introduced and investigated algebraically. Every power construction is shown to be equipped with a characteristic semiring such that the resulting power domains become semiring modules. Power homomorphisms are introduced as a means to relate different power constructions. They also allow to define the notion of initial and final constructions for a fixed characteristic semiring. Such initial and final constructions are shown to exist for every semiring, and their basic properties are derived. Finally, the known power constructions are put into the general framewo...

. Two lower bag domain constructions are introduced: the initial construction which gives free lower monoids, and the final construction which is defined explicitly in terms of second order functions. The latter is analyzed closely. For sober dcpo's, the elements of the final lower bag domains can be described concretely as bags. For continuous domains, initial and final lower bag domains coincide. They are continuous again and can be described via a basis which is constructed from a basis of the argument domain. The lower bag domain construction preserves algebraicity and the properties I and M, but does not preserve bounded completeness, property L, or bifiniteness. 1 Introduction Power domain constructions [13, 15, 16] were introduced to describe the denotational semantics of non-deterministic programming languages. A power domain construction is a domain constructor P , which maps domains to domains, together with some families of continuous operations. If X is the semantic domain ...

We propose assertion-consistency (AC) semi-lattices as suitable orders for the analysis of partial models. Such orders express semantic entailment, multiple-viewpoint and multiple-valued analysis, maintain internal consistency of reasoning, and subsume finite De Morgan lattices. We classify those orders that are finite and distributive and apply them to design an efficient algorithm for multiple-viewpoint checking, where checks are delegated to single-viewpoint models — efficiently driven by the order structure. Instrumentations of this algorithm enable the detection and location of inconsistencies across viewpoint boundaries. To validate the approach, we investigate multiple-valued models and their compositional property semantics over a finite distributive AC lattice. We prove that this semantics is computed by our algorithm above whenever the primes of the AC lattice determine ‘projected’ single viewpoints and the order between primes is preserved as refinements of single-viewpoint models. As a case study, we discuss a multiple-valued notion of state-machines with first-order logic plus transitive closure. 1

An R-module M is observable iff all its elements can be distinguished by observing them by means of linear morphisms from M to R. We show that free observable R-modules can be explicitly described as the cores of the final power domains with characteristic semiring R. Then, the general theory is applied to the cases of the lower and the upper semiring. All lower modules are observable, whereas there are non-observable upper modules. Accordingly, all known lower power constructions coincide, whereas there are at least three different upper power constructions. We show that they coincide for continuous ground domains, but differ on more general domains. 1 Introduction A power domain construction maps every domain X into a so-called power domain over X whose points represent sets of points of the ground domain. Power domain constructions were originally proposed to model the semantics of non-deterministic programming languages [Plo76, Smy78, HP79, Mai85]. Other motivations are the sema...

Abstract valuations on a topological space X are functions that map open sets to 0, 1, or one value in between. We define a space of abstract valuations which for a continuous dcpo X is homeomorphic to the Plotkin power domain of X , and for a Hausdorff space X yields the Vietoris hyperspace of X . Thus we obtain a novel concrete representation of the Plotkin power domain. This representation is more similar to the standard representation of the probabilistic power domain than the previously known ones.

. A continuous predicate on a domain, or more generally a topological space, can be concretely described as an open or closed set, or less obviously, as the set of all predicates consistent with it. Generalizing this scenario to quantitative predicates, we obtain under certain wellunderstood hypotheses an isomorphism between continuous functions on points and supremum preserving functions on open sets, both with values in a fixed lattice. The functions on open sets provide a topological foundation for possibility theories in Artificial Intelligence, revealing formal analogies of quantitative predicates with continuous valuations. Three applications of this duality demonstrate its usefulness: we prove a universal property for the space of quantitative predicates, we characterize its inf-irreducible elements, and we show that bicontinuous lattices and Scott-continuous maps form a cartesian closed category. 1 Introduction It is well-known that a predicate p on a set X, i.e. a function p ...

We propose a platform for the specification and analysis of systems. This platform contain models, their refinement and abstraction, and a temporal logic semantics; rendering a sound framework for property validation and refutation. The platform is parametric in a domain of view, an abstraction of a construction based on the Plotkin power domain. For each domain of view E, the resulting platform P[E] contains partial, incomplete systems and complete systems -- the actual implementations. Complete systems correspond to the platform that has as parameter a domain D that is, as a set, isomorphic to the maximal elements of E. If one restricts P[E] to implementations, but retains the temporal logic semantics, re nement, and abstraction relations, one recovers the platform P[D]. This foundation recasts existing work on modal transition systems, presents fuzzy systems, and ponders on the nature of probabilistic platforms. For domains of view E that are determined by a linearly ordered, co...

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...

We show that, in a fairly general setting including higher-types, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a non-deterministic program. The other two involve existential quantification and integration. We also perform first steps towards the semi-decidability of similar tests under the simultaneous presence of non-deterministic and probabilistic choice. Keywords: Non-deterministic and probabilistic computation, higher-type computability theory and exhaustible sets, may and must testing, operational and denotational semantics, powerdomains. 1