Kleene, Rabin, and Scott Are Available

  • Jochen Hoenicke
  • Roland Meyer
  • Ernst-Rüdiger Olderog
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6269)


We are concerned with the availability of systems, defined as the ratio between time of correct functioning and uptime. We propose to model guaranteed availability in terms of regular availability expressions (rae) and availability automata. We prove that the intersection problem of rae is undecidable. We establish a Kleene theorem that shows the equivalence of the formalisms and states precise correspondence of flat rae and simple availability automata. For these automata, we provide an extension of the powerset construction for finite automata due to Rabin and Scott. As a consequence, we can state a complementation algorithm. This enables us to solve the synthesis problem and to reduce model checking of availability properties to reachability.


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  1. Asarin, E., Caspi, P., Maler, O.: Timed regular expressions. Journal of the ACM 49, 172–206 (2002)CrossRefMathSciNetGoogle Scholar
  2. Alur, R., Dill, D.L.: A theory of timed automata. Theoretical Computer Science 126, 183–235 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  3. Beyer, D., Henzinger, T.A., Jhala, R., Majumdar, R.: The software model checker BLAST. STTT 9(5-6), 505–525 (2007)CrossRefGoogle Scholar
  4. Ball, T., Rajamani, S.K.: The SLAM project: debugging system software via static analysis. In: Proc. of POPL, pp. 1–3. ACM, New York (2002)Google Scholar
  5. Chaochen, Z., Hansen, M.R.: Duration Calculus: A Formal Approach to Real-Time Systems. EATCS Monographs. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  6. Cassez, F., Larsen, K.G.: The impressive power of stopwatches. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 138–152. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. Droste, M., Kuich, W., Vogler, H. (eds.): Handbook of Weighted Automata. EATCS Monographs. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  8. de Souza e Silva, E., Gail, H.R.: Calculating availability and performability measures of repairable computer systems using randomization. Journal of the ACM 36(1), 171–193 (1989)zbMATHCrossRefGoogle Scholar
  9. Larsen, K.G., Rasmussen, J.I.: Automata-theoretic techniques for modal logics of programs. Theoretical Computer Science 390, 197–213 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  10. Minsky, M.: Computation: Finite and Infinite Machines. Prentice Hall, Englewood Cliffs (1967)zbMATHGoogle Scholar
  11. Rabin, M.O., Scott, D.S.: Finite automata and their decision problems. IBM Journal of Research 3(2), 115–125 (1959)MathSciNetGoogle Scholar
  12. Rubino, G., Sericola, B.: Interval availability distribution computation. In: Proc. of FTCS, pp. 48–55. IEEE, Los Alamitos (1993)Google Scholar
  13. Seidl, H., Schwentick, T., Muscholl, A.: Numerical document queries. In: Proc. of PODS, pp. 155–166. ACM, New York (2003)Google Scholar
  14. Trivedi, K.S.: Probability and Statistics with Reliability, Queuing, and Computer Science Applications, 2nd edn. Wiley, Chichester (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jochen Hoenicke
    • 1
  • Roland Meyer
    • 2
  • Ernst-Rüdiger Olderog
    • 3
  1. 1.University of Freiburg 
  2. 2.LIAFA, Paris Diderot University & CNRS 
  3. 3.University of Oldenburg 

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