Observationally-induced algebras in Domain Theory

Vol. 10(3:18)2014, pp. 1–26 www.lmcs-online.or

A Recipe for State-and-Effect Triangles

In the semantics of programming languages one can view programs as state transformers, or as predicate transformers. Recently the author has introduced state-and-effect triangles which capture this situation categorically, involving an adjunction between state- and predicate-transformers. The current paper exploits a classical result in category theory, part of Jon Beck's monadicity theorem, to systematically construct such a state-and-effect triangle from an adjunction. The power of this construction is illustrated in many examples, covering many monads occurring in program semantics, including (probabilistic) power domains

Location equivalence in a parametric setting

AbstractLocation equivalence has been presented in [5] as a bisimulation-based equivalence able to take into account the spatial distribution of processes.In this work, the parametric approach of [12] is applied to location equivalence. An observation domain for localities is identified and the associated equivalence is shown to coincide with the equivalence introducted in [6,16]. The observation of a computation is a forest (defined up to isomorphism) whose nodes are the events (labeled by observable actions) and where the arcs describe the sublocation relation.We show in the paper that our approach is really parametric. By performing minor changes in the definitions, many equivalences are captured: partial and mixed ordering causal semantics, interleaving, and a variation of location equivalence where the generation ordering is not evidenced. It seems difficult to modify the definitions of [6,16] to obtain the last observation. The equivalence induced by this observation corresponds to the very intuitive assumption that different locations cannot share a common clock, and hence the ordering between events occurring in different places cannot be determined.Thanks to the general results proved in [12] for the parametric approach, all the observation equivalences described in this paper come equipped with sound and complete axiomatizations

Involutive Markov categories and the quantum de Finetti theorem

Markov categories have recently emerged as a powerful high-level framework for probability theory and theoretical statistics. Here we study a quantum version of this concept, called involutive Markov categories. First, we show that these are equivalent to Parzygnat's quantum Markov categories but argue that they are simpler to work with. Our main examples of involutive Markov categories involve C*-algebras (of any dimension) as objects and completely positive unital maps as morphisms in the picture of interest. Second, we prove a quantum de Finetti theorem for both the minimal and the maximal C*-tensor norms, and we develop a categorical description of such quantum de Finetti theorems which amounts to a universal property of state spaces.Comment: 49 page

A Stepwise Planned Approach to the Solution of Hilbert's Sixth Problem. III : Measurements and von Neumann Projection/Collapse Rule

Supmech, the universal mechanics developed in the previous two papers, accommodates both quantum and classical mechanics as subdisciplines (a brief outline is included for completeness); this feature facilitates, in a supmech based treatment of quantum measurements, an unambiguous treatment of the apparatus as a quantum system approximated well by a classical one. Taking explicitly into consideration the fact that observations on the apparatus are made when it has `settled down after the measurement interaction' and are restricted to macroscopically distinguishable pointer readings, the unwanted superpositions of (system + apparatus) states are shown to be suppressed; this provides a genuinely physics based justification for the (traditionally \emph{postulated}) von Neumann projection/collapse rule. The decoherence mechanism brought into play by the stated observational constraints is free from the objections against the traditional decoherence program.Comment: 29 pages; one section and two references added; results unchange