Mixed powerdomains for probability and nondeterminism

Abstract

We consider mixed powerdomains combining ordinary nondeterminism and probabilistic nondeterminism. We characterise them as free algebras for suitable (in)equational theories; we establish functional representation theorems; and we show equivalencies between state transformers and appropriately healthy predicate transformers. The extended nonnegative reals serve as 'truth-values'. As usual with powerdomains, everything comes in three flavours: lower, upper, and order-convex. The powerdomains are suitable convex sets of subprobability valuations, corresponding to resolving nondeterministic choice before probabilistic choice. Algebraically this corresponds to the probabilistic choice operator distributing over the nondeterministic choice operator. (An alternative approach to combining the two forms of nondeterminism would be to resolve probabilistic choice first, arriving at a domain-theoretic version of random sets. However, as we also show, the algebraic approach then runs into difficulties.)Rather than working directly with valuations, we take a domain-theoretic functionalanalytic approach, employing domain-theoretic abstract convex sets called Kegelspitzen; these are equivalent to the abstract probabilistic algebras of Graham and Jones, but are more convenient to work with. So we define power Kegelspitzen, and consider free algebras, functional representations, and predicate transformers. To do so we make use of previous work on domain-theoretic cones (d-cones), with the bridge between the two of them being provided by a free d-cone construction on Kegelspitzen.

Monads for Measurable Queries in Probabilistic Databases

“…Further over to the semantic side, we note that the relevance of bags for probability has recently been emphasised by Jacobs [16,17]. Bags are a form of non-determinism, and the problem for combining nondeterminism and probability is notoriously subtle, although there has been plenty of recent progress [9,19,22,23,14,29]. The particular combination we use here is trouble-free.…”

“…Having established the measurability of the query language, in Section 5 we turn to investigate languages for generating probabilistic databases. For this we turn to the composite of the monads, P • B, which we have already shown to be a monad in [6] (see also [16,19]). As we demonstrate, the language for the monad P • B appears to be ideal for generating probabilistic databases, at least as an intermediate language.…”

“…We recall the distributive law distr between the monads P and B that we provided in an earlier paper [6] (see also [16,19,32]):…”

Quantum Expectation Transformers for Cost Analysis

“…Kegelspitzen [15] are dcpo's (directed complete partial orders) that enjoy a convex structure. We define our quantum expectation transformer by making use of Kegelspitzen, but for simplicity, we define Kegelspitzen using -cpo's (complete partial orders), instead of dcpo's, because the former notion is more familiar to most readers.…”

Differentials and distances in probabilistic coherence spaces

“…• Models based on Scott continuous functions on domains endowed with additional probability related structures. Among these models we can mention Plotkin and Keimel Kegelspitzen [13] (domains equipped with an algebraic convex structure) and ω-quasi Borel spaces [15] (domains equipped with a generalized notion of measurability), this latter semantics, as far as we understand the situation, requiring the use of an adapted probabilistic powerdomain construction.…”