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13 \title{\uppercase{\textbf{\large{Algorithmic Verification of Reactive Systems}}}}
14 %\title{\textsc{\large{Algorithmic Verification of Reactive Systems}}}
15 %\author{\textsc{\small{Eugen Sawin}}\\
16 \author{\uppercase{\footnotesize{Eugen Sawin}}\\
17 \em{\footnotesize{University of Freiburg, Germany}}}
18 \date{\textsc{\hfill}}
24 %\renewcommand\abstractname{Abstract}
26 Over the past two decades, temporal logic has become a very basic tool for spec-
27 ifying properties of reactive systems. For finite-state systems, it is possible to use
28 techniques based on B\"uchi automata to verify if a system meets its specifications.
29 This is done by synthesizing an automaton which generates all possible models of
30 the given specification and then verifying if the given system refines this most gen-
31 eral automaton. In these notes, we present a self-contained introduction to the basic
32 techniques used for this automated verification. We also describe some recent space-
33 efficient techniques which work on-the-fly.
37 \section{Introduction}
38 Program verification is a fundamental area of study in computer science. The broad goal
39 is to verify whether a program is ``correct''--i.e., whether the implementation of a program
40 meets its specification. This is, in general, too ambitious a goal and several assumptions
41 have to be made in order to render the problem tractable. In these lectures, we will focus
42 on the verification of finite-state reactive programs. For specifying properties of programs,
43 we use linear time temporal logic.
45 What is a reactive program? The general pattern along which a conventional program
46 executes is the following: it accepts an input, performs some computation, and yields an
47 output. Thus, such a program can be viewed as an abstract function from an input domain
48 to an output domain whose behaviour consists of a transformation from initial states to
51 In contrast, a reactive program is not expected to terminate. As the name suggests, such
52 systems “react” to their environment on a continuous basis, responding appropriately to
53 each input. Examples of such systems include operating systems, schedulers, discrete-event
54 controllers etc. (Often, reactive systems are complex distributed programs, so concurrency
55 also has to be taken into account. We will not stress on this aspect in these lectures—we
56 take the view that a run of a distributed system can be represented as a sequence, by
57 arbitrarily interleaving concurrent actions.)
59 To specify the behaviour of a reactive system, we need a mechanism for talking about
60 the way the system evolves along potentially infinite computations. Temporal logic
61 has become a well-established formalism for this purpose. Many varieties of temporal logic
62 have been defined in the past twenty years—we focus on propositional linear time temporal
65 There is an intimate connection between models of LTL formulas and languages of
66 infinite words—the models of an LTL formula constitute an ω-regular language over an
67 appropriate alphabet. As a result, the satisfiability problem for LTL reduces to checking
68 for emptiness of ω-regular languages. This connection was first explicitly pointed out in.
70 \section{$\omega$-regular Languages}
71 \section{B\"uchi Automata}
73 \section{Linear Temporal Logic}
75 \section{Model Checking}
76 \begin{thebibliography}{99}
77 \bibitem{ref:ltl&büchi}
78 \textsc{Madhavan Mukund.}
79 {\em Linear-Time Temporal Logic and B\"uchi Automata}.
80 Winter School on Logic and Computer Science, Indian Statistical Institute, Calcutta, 1997.
82 \bibitem{ref:handbook}
83 \textsc{Patrick Blackburn, Frank Wolter and Johan Van Benthem.}
84 {\em Handbook of Modal Logic (Studies in Logic and Practical Reasoning)}.
85 3rd Edition, Elsevier, Amsterdam, 2007.