Tiered Objects

We investigate the foundations of reasoning over infinite data structures by means of set-theoretical structures arising in the sheaf-theoretic semantics of higher-order intuitionistic logic. Our approach focuses on a natural notion of tiering involving an operation of restriction of elements to levels forming a complete Heyting algebra. We relate these tiered objects to final coalgebras and initial algebras of a wide class of endofunctors of the category of sets, and study their order and convergence properties. As a sample application, we derive a general proof principle for tiered objects

Construction of a cms on a given cpo

In dealing with denotational semantics of programming languages partial orders resp. metric spaces have been used with great benefit in order to provide a meaning to recursive and repetitive constructs. This paper presents two methods to define a metric on a subset M of a cpo D such that M is a complete metric spaces and the metric semantics on M coincides with the cpo semantics on D when the same semantic operators are used. The first method is to add a 'length' on a cpo which means a function ρ : D → IN 0 ∪{∞} of increasing power. The second is based on the ideas of [9] and uses pseudo rank orderings, i.e. monotone sequences of monotone functions ϖn : D → D. We show that SFP domains can be characterized as special kinds of rank orderded cpo's. We also discuss the connection between the Lawson topology and the topology induced by the metric

How to interpret and establish consistency results for semantics of concurrent programming languages

It is meaningful that a language is provided with several semantic descriptions: e.g. one which serves the needs of the implementor, another one that is suitable for specification and yet another one that will be used to explain the language to the user. In this case one has to guarantee that the various semantics are 'consistent'. The attempt of this paper is to clarify the notion 'consistency' and to present a general framework and theorems for consistency results

Metric Completion versus Ideal Completion

AbstractComplete partial orders have been used for a long time for defining semantics of programming languages. In the context of concurrency de Bakker and Zucker (1982) proposed a metric setting for handling concurrency, recursion and nontermination, which has proved to be very successful in many applications. Starting with a semantic domain D for ‘finite behaviour’ we investigate the relation between the ideal completion Idl(D) and the metric completion which are both suitable to model recursion and infinite behaviour. We also consider the properties of semantic operators