Abstract. This tutorial presents an overview of model checking for both discrete and continuous-time Markov chains (DTMCs and CTMCs). Model checking algorithms are given for verifying DTMCs and CTMCs against specifications written in probabilistic extensions of temporal logic, including quantitative properties with rewards. Example properties include the probability that a fault occurs and the expected number of faults in a given time period. We also describe the practical application of stochastic model checking with the probabilistic model checker PRISM by outlining the main features supported by PRISM and three real-world case studies: a probabilistic security protocol, dynamic power management and a biological pathway. 1

Abstract. We compare the definability of total functionals over the reals in two functional-programming approaches to exact real-number datatype of real numbers; and the intensional approach, in which one encodes real numbers using ordinary datatypes. We show that the type hierarchies coincide up to second-order types, and we relate this fact to an analogous comparison of type hierarchies over the external and internal real numbers in Dana Scott’s category of equilogical spaces. We do not know whether similar coincidences hold at third-order types. However, we relate this question to a purely topological conjecture about the Kleene-Kreisel continuous functionals over the natural numbers. Finally, although it is known that, in the extensional approach, parallel primitives are necessary for programming total first-order functions, we demonstrate that, in the intensional approach, such primitives are not needed for second-order types and below. 1

We study the algebraic structure of the monoid of binary channels and show that it is dually isomorphic to the interval domain over the unit interval with the operation from [3]. We show that the capacity of a binary channel is Scott continuous as a map on the interval domain and that its restriction to any maximally commutative submonoid of binary channels is an order isomorphism onto the unit interval. These results allows us to solve an important open problem in the analysis of covert channels: a provably correct method for injecting noise into a covert channel which will reduce its capacity to any level desired in such a way that the practitioner is free to insert the noise at any point in the system.

. We re-interpret the category of relations according to views, pairs of dcpos hP; T i such that T embeds as a set into the set of maximal elements of P . A qualitative view hM; 2i renders the modal transition systems of K. Larsen and B. Thomsen as a partial view, R : X \Theta Y ! P , of relations, R : X \Theta Y ! 2. A quantitative view represents relations as fuzzy (T is the unit interval UI) or interval-valued (P is the interval domain I) relations. We specify functors mediating between these categories to provide soundness of these interpretations. As for probability theory, we propose the view hI; UIi to embed the set of probability measures into the set of maximal elements of a space of partial probability measures. It is hoped that this provides a foundation for reasoning probabilistically about systems with inherent uncertainty, or vagueness, such as the probabilistic specifications of B. Jonsson and K. Larsen. 1 Motivation and outline 1.1 Motivation The work presented here ...

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