[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Multi-weighted Automata and MSO Logic

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

Weighted automata are non-deterministic automata where the transitions are equipped with weights. They can model quantitative aspects of systems like costs or energy consumption. The value of a run can be computed, for example, as the maximum, average, or discounted sum of transition weights. In multi-weighted automata, transitions carry several weights and can model, for example, the ratio between rewards and costs, or the efficiency of use of a primary resource under some upper bound constraint on a secondary resource. Here, we introduce a general model for multi-weighted automata as well as a multi-weighted MSO logic. In our main results, we show that this multi-weighted MSO logic and multi-weighted auto-mata are expressively equivalent both for finite and infinite words. The translation process is effective, leading to decidability results for our multi-weighted MSO logic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Andersson, D.: Improved combinatorial algorithms for discounted payoff games. Master’s thesis Up-psala University, Department of Information Technology (2006)

  2. Bauer, S., Juhl, L., Larsen, K., Legay, A., Srba, J.: A logic for accumulated-weight reasoning on multiweighted modal automata Im: TASE 2012, pp. 77–84. IEEE (2012)

  3. Berstel, J., Reutenauer, C.: Rational Series and Their Languages. EATCS Monographs on Theoretical Computer Science, vol. 12. Springer (1988)

  4. Bloem, R., Greimel, K., Henzinger, T.A., Jobstmann, B.: Synthesizing robust systems, pp. 85–92 (2009)

  5. Bollig, B., Gastin, P.: Weighted versus probabilistic logics. In: DLT 2009.LNCS, vol. 5583, pp. 18–38. Springer, Heidelberg (2009)

    Google Scholar 

  6. Bollig, B., Gastin, P., Monmege, B., Zeitoun, M.: Logical characterization of weighted pebble walking automata. In: CSL/LICS’14. ACM Press (2014)

  7. Bouyer, P., Brinksma, E., Larsen, K.G.: Staying alive as cheaply as possible. In: HSCC 2004. LNCS, vol. 2993, pp. 203–218. Springer, Heidelberg (2004)

    Google Scholar 

  8. Bouyer, P., Brinksma, E., Larsen, K.G.: Optimal infinite scheduling for multi-priced timed automata. Formal Methods in System Design 32, 3–23 (2008)

    Article  MATH  Google Scholar 

  9. Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik und Grundl. Math 6, 66–92 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carton, O., Michel, M.: Unambiguous Büchi automata. In: LATIN 2000. LNCS, vol. 1776, pp. 407–416. Springer, Heidelberg (2000)

    Google Scholar 

  11. Chatterjee, K., Doyen, L., Henzinger, T.A.: Quantitative languages. In: CSL 2008. LNCS, vol. 5213, pp. 385–400. Springer, Heidelberg (2008)

    Google Scholar 

  12. Chatterjee, K., Doyen, L., Henzinger, T.A.: Expressiveness and closure properties for quantitative languages. Logical Methods in Comp. Sci, vol. 6 (2010)

  13. Droste, M., Gastin, P.: Weighted automata and weighted logics. Theoret. Comp. Sci 380(1-2), 69–86 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Droste, M., Gastin, P.: Weighted automata and weighted logics. In: Droste, M., Kuich, W., Vogler, H. (eds.) [15] chapter 5

  15. In: Droste, M, Kuich, W, Vogler, H (eds.) : Handbook of Weighted Automata. EATCS Monographs on Theoretical Computer Science. Springer (2009)

  16. Droste, M., Kuske, D.: Skew and infinitary formal power series, vol. 2719, pp. 426–438. Springer (2003)

  17. Droste, M., Meinecke, I.: Weighted automata and weighted MSO logics for average and long-time behaviors. Inf. Comput 220-221, 44–59 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Droste, M., Perevoshchikov, V.: Multi-weighted automata and MSO logic. In: CSR 2013. LNCS, vol. 7913, pp. 418–430. Springer, Heidelberg (2013)

    Google Scholar 

  19. Droste, M., Perevoshchikov, V.: A Nivat theorem for weighted timed automata and relative distance logic, vol. 8573, pp. 171–182. Springer (2009)

  20. Droste, M., Vogler, H.: Weighted automata and multi-valued logics over arbitrary bounded lattices. Theoret. Comp. Science 418, 14–36 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Eilenberg, S.: Automata, Languages and Machines, volume A. Academic Press, New York (1974)

    MATH  Google Scholar 

  22. Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Amer. Math. Soc 98, 21–51 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fahrenberg, U., Juhl, L., Larsen, K.G., Srba, J.: Energy games in multiweighted automata. In: ICTAC 2011. LNCS, vol. 6916, pp. 95–115. Springer (2011)

  24. Fahrenberg, U., Larsen, K.G.: Discount-optimal infinite runs in priced timed automata. Electr. Notes Theor. Comput. Sci 239, 179–191 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fahrenberg, U., Larsen, K.G.: Discounting in time. Electr. Notes Theor. Comput. Sci 253, 25–31 (2009)

    Article  Google Scholar 

  26. Fahrenberg, U., Larsen, K.G., Thrane, C.R.: Model-based verification and analysis for real-time systems. In: Software and Systems Safety - Specification and Verification. NATO Science for Peace and Security Series - D: Information and Communication Securita, vol. 30, pp. 231–259. IOS Press (2011)

  27. Filiot, E., Gentilini, R., Raskin, J.-F.: Quantitative languages defined by functional automata. In: CONCUR 2012. LNCS, vol. 7454, pp. 132–146. Springer (2012)

  28. Fränzle, M., Swaminathan, M. In: Ouaknine, J., Vaandrager, F.W (eds.) : Revisiting decidability and optimum reachability for multi-priced timed automata, vol. 5813, pp. 149–163. Springer, Heidelberg (2009)

  29. Kuich, W.: Semirings and formal power series: their relevance to formal languages and automata. In: Handbook of Formal Languages, vol. 1, pp. 609–677. Springer (1997)

  30. Kuich, W., Salomaa, A.: Semirings, Automata and Languages. EATCS Monographs on Theoretical Computer Science, vol. 5. Springer (1986)

  31. Larsen, K.G., Rasmussen, J.I.: Optimal conditional reachability for multi-priced timed automata. In: FOSSACS 2005. LNCS, vol. 3441, pp. 234–249. Springer (2005)

  32. Megiddo, N.: Combinatorial optimization with rational objective functions. In: STOC 1978. ACM, pp. 1–12 (1978)

  33. Puri, A., Tripakis, S.: Algorithms for the multi-constrained routing problem, vol. 2368, pp. 338–347. Springer (2002)

  34. Quaas, K.: MSO Logics for Weighted Timed Automata. Formal Methods in System Design, vol. 38 (2011)

  35. Sakarovitch, J.: Rational and recognisable power series. In: Droste, M., Kuich, W., Vogler, H. (eds.) [15] chapter 4

  36. Salomaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Text and Monographs in Computer Science. Springer (1978)

Download references

Acknowledgments

The authors would like to thank the anonymous referees for helpful suggestions and comments. They would like to thank Paul Gastin for the valuable discussion and interesting ideas which helped to improve the quality of this paper.

Author information

Authors and Affiliations

  1. Universität Leipzig, Institut für Informatik, 04109, Leipzig, Germany

    Manfred Droste & Vitaly Perevoshchikov

Authors
  1. Manfred Droste
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vitaly Perevoshchikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vitaly Perevoshchikov.

Additional information

A preliminary version of this paper appeared in the proceedings of the 8th International Computer Science Symposium in Russia (CSR 2013) [18]

Part of this work was done while M.D. held a “Catedra de excelencia” from Universitat Rovira i Virgili, Tarragona, Spain. He would like to thank his colleagues for their hospitality.

Supported by DFG Graduiertenkolleg 1763 (QuantLA)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Droste, M., Perevoshchikov, V. Multi-weighted Automata and MSO Logic. Theory Comput Syst 59, 231–261 (2016). https://doi.org/10.1007/s00224-015-9658-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-015-9658-9

Keywords