Reactive Systems

Abstract

Formal methods is the term used to describe the specification and verification of software and software systems using mathematical logic. Various methodologies have been developed and incorporated into software tools. An important subclass is distributed systems. There are many books that look at particular methodologies for such systems, e.g. CSP, process algebra. This book offers a more balanced introduction for graduate students that describes the various approaches, their strengths and weaknesses, and when they are best used. Milner's CCS and its operational semantics are introduced, together with notions of behavioural equivalence based on bisimulation techniques and with variants of Hennessy-Milner modal logics. Later in the book, the presented theories are extended to take timing issues into account. The book has arisen from various courses taught in Iceland and Denmark and is designed to give students a broad introduction to the area, with exercises throughout.

A Propositional Dynamic Logic for Instantial Neighborhood Semantics

“…Therefore, in the special case of a composite action of the type π 1 ; π 2 executed at some state w, the action of the agent in the computation π 2 at a later state v resulting as the outcome of the computation π 1 might not be determined uniquely by the state v of the system. It may also depend on other parameters, which might change during the execution of π 1 . This accounts for the extra non-determinism involved in our sequential composition operator, where an action of type π 1 ; π 2 need not specify a unique response to each outcome of the first action.…”

“…That is, each action by the agent is followed by a response from the environment, which is not uniquely determined. This is in contrast with reactive systems, where the behaviour of the system is non-deterministic but completely determined by the actions of the agent [1]. Many different logics for open systems have been proposed, perhaps the most well known being the alternating-time temporal logic ATL introduced by Alur et al in [2].…”

Compositionality for quantitative specifications

“…Using property (2), we see that a 1 a 2 must be defined when u 1 = u 2 and [l 1 , r 1 ] ∩ [l 2 , r 2 ] = ∅, and by (2), if a 1 a 2 = a 3 is defined, then u 3 = u 1 and u 3 = u 2 , and…”

“…This is also sometimes called Hennessy-Milner logic with maximal fixed points and represented using equation systems in Hennessy-Milner logic with variables, see [2,46]. We will use this representation below.…”

On the origins of bisimulation and coinduction

“…For instance, on LTSs in which labels have a structure, which may be useful when processes may exchange values in communications; or on LTSs equipped with a special action to represent movements internal to processes, in which case one may wish to abstract from such action in the bisimulation game yielding the so-called weak bisimulations and weak bisimilarity. Examples of these kinds may be found, e.g., in [Milner 1989;Sangiorgi and Walker 2001;Aceto et al 2007;Sangiorgi et al 2007]. Also, we do not discuss in this paper enhancements of the bisimulation proof method, intended to relieve the amount of work needed to prove bisimilarity results, such as bisimulation up-to techniques; see, e.g., [Milner 1989;Sangiorgi 1998;Pous 2007].…”