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 \title{\textbf{Verification of Reactive Systems}}

 \author{Eugen Sawin\\   

   University of Freiburg, Germany\\\\

   \texttt{sawine@informatik.uni-freiburg.de}}

 \date{May, 2011}

 \begin{document}

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 \begin{abstract}

 Over the past two decades, temporal logic has become a very basic tool for spec-

 ifying properties of reactive systems. For finite-state systems, it is possible to use

 techniques based on B\"uchi automata to verify if a system meets its specifications.

 This is done by synthesizing an automaton which generates all possible models of

 the given specification and then verifying if the given system refines this most gen-

 eral automaton. In these notes, we present a self-contained introduction to the basic

 techniques used for this automated verification. We also describe some recent space-

 efficient techniques which work on-the-fly.

 \end{abstract}

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 \section{Introduction}

 Program verification is a fundamental area of study in computer science. The broad goal

 is to verify whether a program is “correct”—i.e., whether the implementation of a program

 meets its specification. This is, in general, too ambitious a goal and several assumptions

 have to be made in order to render the problem tractable. In these lectures, we will focus

 on the verification of finite-state reactive programs. For specifying properties of programs,

 we use linear time temporal logic.

 What is a reactive program? The general pattern along which a conventional program

 executes is the following: it accepts an input, performs some computation, and yields an

 output. Thus, such a program can be viewed as an abstract function from an input domain

 to an output domain whose behaviour consists of a transformation from initial states to

 final states.

 In contrast, a reactive program is not expected to terminate. As the name suggests, such

 systems “react” to their environment on a continuous basis, responding appropriately to

 each input. Examples of such systems include operating systems, schedulers, discrete-event

 controllers etc. (Often, reactive systems are complex distributed programs, so concurrency

 also has to be taken into account. We will not stress on this aspect in these lectures—we

 take the view that a run of a distributed system can be represented as a sequence, by

 arbitrarily interleaving concurrent actions.)

 To specify the behaviour of a reactive system, we need a mechanism for talking about

 the way the system evolves along potentially infinite computations. Temporal logic 

 has become a well-established formalism for this purpose. Many varieties of temporal logic

 have been defined in the past twenty years—we focus on propositional linear time temporal

 logic (LTL).

 There is an intimate connection between models of LTL formulas and languages of

 infinite words—the models of an LTL formula constitute an ω-regular language over an

 appropriate alphabet. As a result, the satisfiability problem for LTL reduces to checking

 for emptiness of ω-regular languages. This connection was first explicitly pointed out in.

 \section{$\omega$-regular Languages}

 \section{Linear Temporal Logic}

 \section{B\"uchi Automata}

 Automata are good.

 \section{Model Checking}

 \begin{thebibliography}{99}

 \bibitem{ref:ltl&büchi} Madhavan Mukund. {\em Linear-Time Temporal Logic and B\"uchi Automata}.

 Winter School on Logic and Computer Science, Indian Statistical Institute, Calcutta, 1997.

 \bibitem{ref:handbook} 

 Patrick Blackburn, Frank Wolter and Johan Van Benthem. 

 {\em Handbook of Modal Logic (Studies in Logic and Practical Reasoning)}.

 3rd Edition, Elsevier, Amsterdam, 2007.

 \end{thebibliography}

 \end{document}