Pedro Sánchez Terraf

Nondeterministic Labeled Markov Processes: Bisimulations and Logical Characterization

We extend the theory of labeled Markov processeswith internal nondeterminism, a fundamental conceptfor the further development of a process theory withabstraction on nondeterministic continuous probabilisticsystems. We define nondeterministic labeled Markovprocesses (NLMP) and provide both a state basedbisimulation and an event based bisimulation. We showthe relation between them, including that the largeststate bisimulation is also an event bisimulation. Wealso introduce a variation of the Hennessy-Milnerlogic that characterizes event bisimulation and that issound w.r.t. the state base bisimulation for arbitraryNLMP. This logic, however, is infinitary as it containsa denumerable ∨. We then introduce a finitary sublogicthat characterize both state and event bisimulation forimage finite NLMP whose underlying measure spaceis also analytic. Hence, in this setting, all notions ofbisimulation we deal with turn out to be equal.

Unprovability of the logical characterization of bisimulation

We quickly review labelled Markov processes (LMP) and provide a counterexample showing that in general measurable spaces, event bisimilarity and state bisimilarity differ in LMP. This shows that the Hennessy-Milner logic proposed by Desharnais does not characterize state bisimulation in non-analytic measurable spaces. Furthermore we show that, under current foundations of Mathematics, such logical characterization is unprovable for spaces that are projections of a coanalytic set. Underlying this construction there is a proof that stationary Markov processes over general measurable spaces do not have semi-pullbacks.

Bisimilarity is not Borel

We prove that the relation of bisimilarity between countable labelled transition systems (LTS) is Σ 1 1 -complete (hence not Borel), by reducing the set of non-well orders over the natural numbers continuously to it. This has an impact on the theory of probabilistic and non-deterministic processes over uncountable spaces, since logical characterizations of bisimilarity (as, for instance, those based on the unique structure theorem for analytic spaces) require a countable logic whose formulas have measurable semantics. Our reduction shows that such a logic does not exist in the case of image-infinite processes.

Stochastic non-determinism and effectivity functions

This paper investigates stochastic nondeterminism on continuous state spaces by relating nondeterministic kernels and stochastic effectivity functions to each other. Nondeterministic kernels are functions assigning each state a set o subprobability measures, and effectivity functions assign to each state an upper-closed set of subsets of measures. Both concepts are generalizations of Markov kernels used for defining two different models: Nondeterministic labelled Markov processes and stochastic game models, respectively. We show that an effectivity function that maps into principal filters is given by an image-countable nondeterministic kernel, and that image-finite kernels give rise to effectivity functions. We define state bisimilarity for the latter, considering its connection to morphisms. We provide a logical characterization of bisimilarity in the finitary case. A generalization of congruences (event bisimulations) to effectivity functions and its relation to the categorical presentation of bisimulation are also studied.

A Theory for the Semantics of Stochastic and Non-deterministic Continuous Systems

The description of complex systems involving physical or biological components usually requires to model complex continuous behavior induced by variables such as time, distance, speed, temperature, alkalinity of a solution, etc. Often, such variables can be quantified probabilistically to better understand the behavior of the complex systems. For example, the arrival time of events may be considered a Poisson process or the weight of an individual may be assumed to be distributed according to a log-normal distribution. However, it is also common that the uncertainty on how these variables behave makes us prefer to leave out the choice of a particular probability and rather model it as a purely non-deterministic decision, as it is the case when a system is intended to be deployed in a variety of very different computer or network architectures. Therefore, the semantics of these systems needs to be represented by a variant of probabilistic automata that involves continuous domains on the state space and the transition relation. In this paper, we provide a survey on the theory of such kind of models. We present the theory of the so-called labeled Markov processes LMP and its extension with internal non-determinism NLMP. We show that in these complex domains, the bisimulation relation can be understood in different manners. We show the relation between the different bisimulations and try to understand their expressiveness through examples. We also study variants of Hennessy-Milner logic that provides logical characterizations of some of these bisimulations.

The lattice of congruences of a finite line frame

Let

Directly Indecomposables in Semidegenerate Varieties of Connected po-Groupoids

\mathbf{F}=\left\langle F,R\right\rangle

Bisimulations for non-deterministic labelled markov processes

be a finite Kripke frame. A congruence of

Varieties with definable factor congruences

\mathbf{F}

Bisimulations for Nondeterministic Labeled Markov Processes

is a bisimulation of