9 research outputs found
A Proof System for Compositional Verification of Probabilistic Concurrent Processes
Abstract. We present a formal proof system for compositional verification of probabilistic concurrent processes. Processes are specified using an SOS-style process algebra with probabilistic operators. Properties are expressed using a probabilistic modal µ-calculus. And the proof system is formulated as a sequent calculus in which sequents are given a quantitative interpretation. A key feature is that the probabilistic scenario is handled by introducing the notion of Markov proof, according to which proof trees contain probabilistic branches and are required to satisfy a condition formulated byinterpretingthemas Markov Decision Processes. We present simple but illustrative examples demonstrating the applicability of the approach to the compositional verification of infinite state processes. Our main result is the soundness of the proof system, which is proved by applying the coupling method from probability theory to the game semantics of the probabilistic modal µ-calculus.
Generalized labelled Markov processes, coalgebraically
Coalgebras of measurable spaces are of interest in probability theory as a formalization of Labelled Markov Processes (LMPs). We discuss some general facts related to the notions of bisimulation and cocongruence on these systems, providing a faithful characterization of bisimulation on LMPs on generic measurable
spaces. This has been used to prove that bisimilarity on single LMPs is an equivalence, without assuming the state space to be analytic. As the second main contribution, we introduce the first specification rule format to define well-behaved composition operators for LMPs. This allows one to define process description languages on LMPs which are always guaranteed to have a fully-abstract semantics
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
Modular Markovian Logic.
We introduce Modular Markovian Logic (MML) for compositional continuous-time and continuous-space Markov processes. MML combines operators specific to stochastic logics with operators reflecting the modular structure of the models, similar to those used by spatial and separation logics. We present a complete Hilbert-style axiomatization for MML, prove the small model property and analyze the relation between stochastic bisimulation and logical equivalence. © 2011 Springer-Verlag
Modular Markovian Logic.
We introduce Modular Markovian Logic (MML) for compositional continuous-time and continuous-space Markov processes. MML combines operators specific to stochastic logics with operators reflecting the modular structure of the models, similar to those used by spatial and separation logics. We present a complete Hilbert-style axiomatization for MML, prove the small model property and analyze the relation between stochastic bisimulation and logical equivalence. © 2011 Springer-Verlag
Modular Markovian logic
Abstract. We introduce Modular Markovian Logic (MML) for compositional continuous-time and continuous-space Markov processes. MML combines operators specific to stochastic logics with operators reflecting the modular structure of the models, similar to those used by spatial and separation logics. We present a complete Hilbert-style axiomatization for MML, prove the small model property and analyze the relation between stochastic bisimulation and logical equivalence.