Formal Methods in System Design

, Volume 38, Issue 3, pp 193–222 | Cite as

MSO logics for weighted timed automata

  • Karin QuaasEmail author


We aim to generalize Büchi’s fundamental theorem on the coincidence of recognizable and MSO-definable languages to a weighted timed setting. For this, we investigate weighted timed automata and show how we can extend Wilke’s relative distance logic with weights taken from an arbitrary semiring. We show that every formula in our logic can effectively be transformed into a weighted timed automaton, and vice versa. The results indicate the robustness of weighted timed automata and may also be used for specification purposes.


Weighted timed automata Weighted MSO logic Real-time systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alur R, Dill DL (1994) A theory of timed automata. Theor Comput Sci 126(2):183–235 CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Alur R, La Torre S, Pappas GJ (2004) Optimal paths in weighted timed automata. Theor Comput Sci 318:297–322 CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Alur R, Madhusudan P (2004) Decision problems for timed automata: A survey. In: Bernardo M, Corradini F (eds) SFM-RT. LNCS, vol 3185. Springer, Berlin, pp 1–24 Google Scholar
  4. 4.
    Behrmann G, Fehnker A, Hune T, Larsen K, Pettersson P, Romijn J, Vaandrager F (2001) Minimum-cost reachability for priced timed automata. In: Di Benedetto MD, Sangiovanni-Vincentelli A (eds) HSCC. LNCS, vol 2034. Springer, Berlin, pp 147–161 Google Scholar
  5. 5.
    Bollig B, Meinecke I (2007) Weighted distributed systems and their logics. In: Artëmov SN, Nerode A (eds) LFCS. LNCS, vol 4514. Springer, Berlin, pp 54–68 Google Scholar
  6. 6.
    Bouyer P, Brihaye T, Bruyère V, Raskin J-F (2007) On the optimal reachability problem on weighted timed automata. Form Methods Syst Des 31(2):135–175 CrossRefzbMATHGoogle Scholar
  7. 7.
    Bouyer P, Brihaye T, Markey N (2006) Improved undecidability results on weighted timed automata. Inf Process Lett 98(5):188–194 CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bouyer P, Larsen KG, Markey N (2008) Model checking one-clock priced timed automata. Log Methods Comput Sci 4:1–28 MathSciNetGoogle Scholar
  9. 9.
    Bouyer P, Markey N (2007) Costs are expensive! In: Raskin J-F, Thiagarajan PS (eds) FORMATS. LNCS, vol 4763. Springer, Berlin, pp 53–68 Google Scholar
  10. 10.
    Büchi JR (1960) On a decision method in restricted second order arithmetics. In: Nagel E et al. (eds) Proc intern congress on logic, methodology and philosophy of sciences. Stanford University Press, Stanford, pp 1–11 Google Scholar
  11. 11.
    Chiplunkar A, Narayanan Krishna S, Jain C (2009) Model checking logic WCTL with multi constrained modalities on one clock priced timed automata. In: Ouaknine J, Vaandrager FW (eds) FORMATS. LNCS, vol 5813. Springer, Berlin, pp 88–102 Google Scholar
  12. 12.
    Droste M, Gastin P (2005) Weighted automata and weighted logics. In: Caires L, Italiano GF, Monteiro L, Palamidessi C, Yung M (eds) ICALP. LNCS, vol 3580. Springer, Berlin, pp 513–525 Google Scholar
  13. 13.
    Droste M, Gastin P (2009) Weighted automata and weighted logics. In: Droste M, Kuich W, Vogler H (eds) Handbook of weighted automata. EATCS monographs in theoretical computer science. Springer, Berlin, pp 175–211 CrossRefGoogle Scholar
  14. 14.
    Droste M, Quaas K (2008) A Kleene-Schützenberger theorem for weighted timed automata. In: Amadio RM (ed) FoSSaCS. LNCS, vol 4962. Springer, Berlin, pp 142–156 Google Scholar
  15. 15.
    Droste M, Rahonis G (2006) Weighted automata and weighted logics on infinite words. In: Ibarra OH, Dang Z (eds) Developments in language theory. LNCS, vol 4036. Springer, Berlin, pp 49–58 CrossRefGoogle Scholar
  16. 16.
    Droste M, Vogler H (2006) Weighted tree automata and weighted logics. Theor Comput Sci 366(3):228–247 CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    D’Souza D (2003) A logical characterisation of event clock automata. Int J Found Comput Sci 14(4):625–640 CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Furia CA, Rossi M (2008) MTL with bounded variability: Decidability and complexity. In: Cassez F, Jard C (eds) FORMATS. LNCS, vol 5215. Springer, Berlin, pp 109–123 Google Scholar
  19. 19.
    Mathissen C (2007) Definable transductions and weighted logics for texts. In: Harju T, Karhumäki J, Lepistö A (eds) Developments in language theory. LNCS, vol 4588. Springer, Berlin, pp 324–336 CrossRefGoogle Scholar
  20. 20.
    Mathissen C (2008) Weighted logics for nested words and algebraic formal power series. In: Aceto L, Damgård I, Goldberg L Ann, Halldórsson MM, Ingólfsdóttir A, Walukiewicz I (eds) ICALP (2). LNCS, vol 5126. Springer, Berlin, pp 221–232 Google Scholar
  21. 21.
    Mäurer I (2006) Weighted picture automata and weighted logics. In: Durand B, Thomas W (eds) STACS. LNCS, vol 3884. Springer, Berlin, pp 313–324 Google Scholar
  22. 22.
    Meinecke I (2006) Weighted logics for traces. In: Grigoriev D, Harrison J, Hirsch EA (eds) CSR. LNCS, vol 3967. Springer, Berlin, pp 235–246 Google Scholar
  23. 23.
    Quaas K (2009) Kleene-Schützenberger and Büchi theorems for weighted timed automata. PhD thesis, Universität Leipzig, Institut für Informatik, Abteilung Automaten und Sprachen Google Scholar
  24. 24.
    Quaas K (2009) Weighted timed MSO logics. In: Diekert V, Nowotka D (eds) DLT 2009, Proceedings. LNCS, vol 5583. Springer, Berlin, pp 419–430 Google Scholar
  25. 25.
    Alur R, Henzinger TA (1990) Real-time logics: complexity and expressiveness. In: Fifth annual IEEE symposium on logic in computer science. IEEE Computer Society Press, Washington, pp 390–401 CrossRefGoogle Scholar
  26. 26.
    Thomas W (1990) Automata on infinite objects. In: van Leeuwen J (ed) Handbook of theoretical computer science, Volume B: Formal models and semantics (B). Elsevier and MIT Press, Amsterdam/Cambridge, pp 133–192 Google Scholar
  27. 27.
    Thomas W (1997) Languages, automata and logic. In: Rozenberg G, Salomaa A (eds) Handbook of formal languages. Springer, Berlin, pp 389–485 Google Scholar
  28. 28.
    Wilke T (1994) Automaten und Logiken zur Beschreibung zeitabhängiger Systeme. PhD thesis, Christian-Albrecht-Universität Kiel Google Scholar
  29. 29.
    Wilke T (1994) Specifying timed state sequences in powerful decidable logics and timed automata. In: Langmaack H, de Roever W-P, Vytopil J (eds) Formal techniques in real-time and fault-tolerant systems, Lübeck, Germany. LNCS, vol 863. Springer, Berlin, pp 694–715 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institut für InformatikUniversität LeipzigLeipzigGermany

Personalised recommendations