Characterising Probabilistic Processes Logically

In this paper we work on (bi)simulation semantics of processes that exhibit both nondeterministic and probabilistic behaviour. We propose a probabilistic extension of the modal mu-calculus and show how to derive characteristic formulae for various simulation-like preorders over finite-state processes without divergence. In addition, we show that even without the fixpoint operators this probabilistic mu-calculus can be used to characterise these behavioural relations in the sense that two states are equivalent if and only if they satisfy the same set of formulae.Comment: 18 page

Multiset Bisimulations as a Common Framework for Ordinary and Probabilistic Bisimulations

Our concrete objective is to present both ordinary bisimulations and probabilistic bisimulations in a common coalgebraic framework based on multiset bisimulations. For that we show how to relate the underlying powerset and probabilistic distributions functors with the multiset functor by means of adequate natural transformations. This leads us to the general topic that we investigate in the paper: a natural transformation from a functor F to another G transforms F-bisimulations into G-bisimulations but, in general, it is not possible to express G-bisimulations in terms of F-bisimulations. However, they can be characterized by considering Hughes and Jacobs’ notion of simulation, taking as the order on the functor F the equivalence induced by the epi-mono decomposition of the natural transformation relating F and G. We also consider the case of alternating probabilistic systems where non-deterministic and probabilistic choices are mixed, although only in a partial way, and extend all these results to categorical simulations

Coalgebraic Representation Theory of Fractals

We develop a \emph{representation theory} in which a point of a fractal specified by \emph{metric} means (by a variant of an \emph{iterated function system, IFS}) is represented by a suitable equivalence class of infinite streams of symbols. The framework is categorical: symbolic representatives carry a final coalgebra; an IFS-like metric specification of a fractal is an algebra for the same functor. Relating the two there canonically arises a \emph{representation map}, much like in America and Rutten's use of metric enrichment in denotational semantics. A distinctive feature of our framework is that the canonical representation map is bijective. In the technical development, \emph{gluing} of shapes in a fractal specification is a major challenge. On the metric side we introduce the notion of \emph{injective IFS} to be used in place of conventional IFSs. On the symbolic side we employ Leinster's presheaf framework that uniformly addresses necessary identification of streams---such as $.0111\dotsc=.1000\dotsc$ in the binary expansion of real numbers. Our leading example is the unit interval $\mathbb{I} = [0,1]$