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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}
