Complete Axiomatization for the Bisimilarity Distance on Markov Chains

In this paper we propose a complete axiomatization of the bisimilarity distance of Desharnais et al. for the class of finite labelled Markov chains. Our axiomatization is given in the style of a quantitative extension of equational logic recently proposed by Mardare, Panangaden, and Plotkin (LICS\u2716) that uses equality relations t =_e s indexed by rationals, expressing that "t is approximately equal to s up to an error e". Notably, our quantitative deductive system extends in a natural way the equational system for probabilistic bisimilarity given by Stark and Smolka by introducing an axiom for dealing with the Kantorovich distance between probability distributions

A complete quantitative deduction system for the bisimilarity distance on Markov chains

In this paper we propose a complete axiomatization of the bisimilarity distance of Desharnais et al. for the class of finite labelled Markov chains. Our axiomatization is given in the style of a quantitative extension of equational logic recently proposed by Mardare, Panangaden, and Plotkin (LICS 2016) that uses equality relations t ≡ ε s indexed by rationals, expressing that “t is approximately equal to s up to an error ε”. Notably, our quantitative deduction system extends in a natural way the equational system for probabilistic bisimilarity given by Stark and Smolka by introducing an axiom for dealing with the Kantorovich distance between probability distributions. The axiomatization is then used to propose a metric extension of a Kleene’s style representation theorem for finite labelled Markov chains, that was proposed (in a more general coalgebraic fashion) by Silva et al. (Inf. Comput. 2011)

Complete axiomatization for the total variation distance of Markov chains

We propose a complete axiomatization for the total variation distance of finite labelled Markov chains. Our axiomatization is given in the form of a quantitative deduction system, a framework recently proposed by Mardare, Panangaden, and Plotkin (LICS 2016) to extend classical equational deduction systems by means of inferences of equality relations t≡εs indexed by rationals, expressing that “t is approximately equal to s up to an error ε”. Notably, the quantitative equational system is obtained by extending our previous axiomatization (CONCUR 2016) for the probabilistic bisimilarity distance with a distributivity axiom for the prefix operator over the probabilistic choice inspired by Rabinovich's (MFPS 1983). Finally, we propose a metric extension to the Kleene-style representation theorem for finite labelled Markov chains w.r.t. trace equivalence due to Silva and Sokolova (MFPS 2011)

Fractals from Regular Behaviours

We are interested in connections between the theory of fractal sets obtained as attractors of iterated function systems and process calculi. To this end, we reinterpret Milner's expressions for processes as contraction operators on a complete metric space. When the space is, for example, the plane, the denotations of fixed point terms correspond to familiar fractal sets. We give a sound and complete axiomatization of fractal equivalence, the congruence on terms consisting of pairs that construct identical self-similar sets in all interpretations. We further make connections to labelled Markov chains and to invariant measures. In all of this work, we use important results from process calculi. For example, we use Rabinovich's completeness theorem for trace equivalence in our own completeness theorem. In addition to our results, we also raise many questions related to both fractals and process calculi

Principles of Markov automata

A substantial amount of today's engineering problems revolve around systems that are concurrent and stochastic by their nature. Solution approaches attacking these problems often rely on the availability of formal mathematical models that reflect such systems as comprehensively as possible. In this thesis, we develop a compositional model, Markov automata, that integrates concurrency, and probabilistic and timed stochastic behaviour. This is achieved by blending two well-studied constituent models, probabilistic automata and interactive Markov chains. A range of strong and weak bisimilarity notions are introduced and evaluated as candidate relations for a natural behavioural equivalence between systems. Among them, weak distribution bisimilarity stands out as a natural notion being more oblivious to the probabilistic branching structure than prior notions. We discuss compositionality, axiomatizations, decision and minimization algorithms, state-based characterizations and normal forms for weak distribution bisimilarity. In addition, we detail how Markov automata and weak distribution bisimilarity can be employed as a semantic basis for generalized stochastic Petri nets, in such a way that known shortcomings of their classical semantics are ironed out in their entirety.Ein beträchtlicher Teil gegenwärtiger ingenieurwissenschafter Probleme erstreckt sich auf Sys- teme, die ihrer Natur nach sowohl stochastisch als auch nebenläufig sind. Lösungsansätze fußen hierbei häufig auf der Verfügbarkeit formaler mathematischer Modelle, die es erlauben, die Spez- ifika jener Systeme möglichst erschöpfend zu erfassen. In dieser Dissertation entwickeln wir ein kompositionelles Modell namens Markov-Automaten, das Nebenläufigkeit mit probabilistis- chen und stochastischen Prozessen integriert. Dies wird durch die Verschmelzung der zweier bekannter Modellklassen erreicht, und zwar die der probabilistischen Automaten und die der interaktiven Markovketten. Wir entwickeln dabei ein Spektrum verschiedener, starker und schwacher Bisimulationsrelationen und beurteilen sie im Hinblick auf ihre Eignung als natür- liche Verhaltensäquivalenz zwischen Systemen. Die schwache Wahrscheinlichkeitsverteilungs- bisimulation sticht dabei als natürliche Wahl hervor, da sie die probabilistische Verzwei- gungsstruktur treffender abstrahiert als bisher bekannte Bisimulationsrelationen. Wir betra- chten des Weiteren Kompositionalitätseigenschaften, Axiomatisierungen, Entscheidungs- und Minimierungsalgorithmen, sowie zustandsbasierte Charakterisierungen und Normalformen für die schwache Wahrscheinlichkeitsverteilungsbisimulation. Abschließend legen wir dar, dass Markov-Automaten und die schwacheWahrscheinlichkeitsverteilungsbisimulation als Grundlage für eine verbesserte Semantik von verallgemeinerten stochastischen Petrinetzen dienen kann, welche bekannte Mängel der klassischen Semantik vollständig behebt

Behavioural Preorders on Stochastic Systems - Logical, Topological, and Computational Aspects

Computer systems can be found everywhere: in space, in our homes, in our cars, in our pockets, and sometimes even in our own bodies. For concerns of safety, economy, and convenience, it is important that such systems work correctly. However, it is a notoriously difficult task to ensure that the software running on computers behaves correctly. One approach to ease this task is that of model checking, where a model of the system is made using some mathematical formalism. Requirements expressed in a formal language can then be verified against the model in order to give guarantees that the model satisfies the requirements. For many computer systems, time is an important factor. As such, we need our formalisms and requirement languages to be able to incorporate real time. We therefore develop formalisms and algorithms that allow us to compare and express properties about real-time systems. We first introduce a logical formalism for reasoning about upper and lower bounds on time, and study the properties of this formalism, including axiomatisation and algorithms for checking when a formula is satisfied. We then consider the question of when a system is faster than another system. We show that this is a difficult question which can not be answered in general, but we identify special cases where this question can be answered. We also show that under this notion of faster-than, a local increase in speed may lead to a global decrease in speed, and we take step towards avoiding this. Finally, we consider how to compare the real-time behaviour of systems not just qualitatively, but also quantitatively. Thus, we are interested in knowing how much one system is faster or slower than another system. This is done by introducing a distance between systems. We show how to compute this distance and that it behaves well with respect to certain properties.Comment: PhD dissertation from Aalborg Universit

Computing Branching Distances Using Quantitative Games

We lay out a general method for computing branching distances between labeled transition systems. We translate the quantitative games used for defining these distances to other, path-building games which are amenable to methods from the theory of quantitative games. We then show for all common types of branching distances how the resulting path-building games can be solved. In the end, we achieve a method which can be used to compute all branching distances in the linear-time--branching-time spectrum