We consider a programming language that can manipulate both classical and quantum information. Our is type-safe designed for variational programming, which hybrid classical-quantum computational paradigm. The subsystem of the Probabilistic FixPoint Calculus (PFPC), lambda calculus with mixed-variance recursive types, term recursion probabilistic choice. first-order linear type system two subsys...

In this paper, we introduce the notion of super finitely separating functions which gives a characterization of RB-domains. Then we prove that FS-domains and RB-domains are equivalent in some special cases by the following three claims: a dcpo is an RB-domain if and only if there exists an approximate identity for it consisting of super finitely separating functions; a consistent join-semilatti...

Normann proved that the domains of the game model of PCF (the domains of sequential functionals) need not be dcpos. Sazonov has defined natural domains for a theory of such incomplete domains. This paper further develops that theory. It defines lub-rules that infer natural lubs from existing natural lubs, and lub-rule classes that describe axiom systems like that of natural domains. There is a ...

This paper is about a generalization of Scott’s domain theory in such a way that its definitions and theorems become meaningful in quasimetric spaces. The generalization is achieved by a change of logic: the fundamental concepts of original domain theory (order, way-below relation, Scott-open sets, continuous maps, etc.) are interpreted as predicates that are valued in an arbitrary completely d...

Continuous valuations have been proposed by several authors as a way of modeling probabilistic non-determinism in programming language semantics. Let (X;O) be a topological space. A quasi-simple valuation on X is the sup of a directed family of simple valuations. We show that quasisimple valuations are exactly those valuations that extend to continuous valuations to the Alexandroff topology on ...

This paper takes a fresh look at the topic of trace semantics in the theory of coalgebras. The first development of coalgebraic trace semantics used final coalgebras in Kleisli categories, stemming from an initial algebra in the underlying category. This approach requires some non-trivial assumptions, like dcpo enrichment, which do not always hold, even in cases where one can reasonably speak o...

In this paper we show that three major classes of topological spaces are domain-representable, i.e., homeomorphic to the space of maximal elements of some domain (=continuous dcpo) with the relative Scott topology. The three classes are: T3 subcompact spaces, strongly α-favorable spaces with a Gδ-diagonal or with a base of countable order, and complete quasi-developable T3-spaces. It follows th...

In (Simpson 2003) A. Simpson introduced the category PreDom of topological predomains as a framework for denotational semantics containing also most classical spaces, namely all countably based T0 spaces. Countably based T0 spaces are isomorphic to subspaces of Pω where the latter is endowed with the Scott topology. A qcb space (as introduced in (Menni and Simpson 2002)) is a T0 space which app...

The power set of a domain D can be seen as the function space D ! 2 of all functions in the pointwise order. One computational version thereof is the lifted lower powerdomain, which we can think of as the lattice of Scott-closed subsets or, alternatively, as the order dual of the function space D ! 2]. Therefore, logic and the ordinary power set, as well as its information-theoretic version, re...

We prove that the sequential functionals of some fixed types at type level 2, taking finite sequences of unary functions as arguments, do form a directed complete partial ordering. This gives a full characterisation of for which types the partially ordered set of sequential functionals has this property. As a tool, we prove a normal form theorem for the finite sequential functionals of the type...