A probabilistic powerdomain of evaluations
Proceedings of the Fourth Annual Symposium on Logic in computer science
Totally bounded spaces and compact ordered spaces as domains of computation
Topology and category theory in computer science
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An extensional treatment of lazy data flow deadlock
Selected papers of the workshop on Topology and completion in semantics
Towards Computing Distances Between Programs via Scott Domains
LFCS '97 Proceedings of the 4th International Symposium on Logical Foundations of Computer Science
Fundamenta Informaticae
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The notion of a semivaluation has been introduced in [Sch98]. Aside from providing a novel concept generalizing valuations on lattices to the context of semilattices, semivaluations shed new light on the notion of a "partial metric" well known from theoretical computer science (e.g. [Mat94], [Mat95], [O'N97], [BS97] and [BSh97]). As discussed in [Sch98], the characterization of partial metrics in terms of semivaluations is non-trivial and involves the solution of an open problem of the Survey Paper "Non-symmetric Topology" (Problem 7 of [Kün93]) for the class of quasi-uniform semilattices. We recall from [Sch98] that the traditional domain theoretic examples, including the well known class of totally bounded Scott domains (e.g. [Smy91]), all correspond to semivaluation spaces. As such it is possible to study Quantitative Domain Theory (e.g. [FSW96]) in this simplified context, similar to the study of metric lattices (uniform lattices) in the more basic context of valuation spaces (cf. [Bir84] and also [Web91]). Hence the notion of a semivaluation is of sufficient interest to merit an independent study. The main purpose of this short note is to provide a basic introduction to the notion of a semivaluation independent of the domain theoretic considerations of [Sch98] and to discuss a recently obtained characterization of valuations in terms of semivaluations.