Skip to main content
SpringerLink
Log in

The Cost of Exactness in Quantitative Reachability

  • Chapter
  • First Online:
Models, Algorithms, Logics and Tools

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10460))

Abstract

In the analysis of reactive systems a quantitative objective assigns a real value to every trace of the system. The value decision problem for a quantitative objective requires a trace whose value is at least a given threshold, and the exact value decision problem requires a trace whose value is exactly the threshold. We compare the computational complexity of the value and exact value decision problems for classical quantitative objectives, such as sum, discounted sum, energy, and mean-payoff for two standard models of reactive systems, namely, graphs and graph games.

This research was supported in part by the Austrian Science Fund (FWF) under grants S11402-N23 and S11407-N23 (RiSE/SHiNE), and Z211-N23 (Wittgenstein Award), ERC Start grant (279307: Graph Games), Vienna Science and Technology Fund (WWTF) through project ICT15-003.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Andersson, D.: An improved algorithm for discounted payoff games. In: Proceedings of 11th ESSLLI Student Session, pp. 91–98 (2006)

    Google Scholar 

  2. Blondin, M., Finkel, A., Göller, S., Haase, C., McKenzie, P.: Reachability in two-dimensional vector addition systems with states is PSPACE-complete. In: Proceedings of LICS: Symposium on Logic in Computer Science, pp. 32–43. IEEE Computer Society (2015)

    Google Scholar 

  3. Boker, U., Henzinger, T.A., Otop, J.: The target discounted-sum problem. In: Proceeings of LICS: Symposium on Logic in Computer Science, pp. 750–761. IEEE Computer Society (2015)

    Google Scholar 

  4. Bonet, B., Geffner, H.: Solving POMDPs: RTDP-Bel vs. point-based algorithms. In: Proceedings of IJCAI: International Joint Conference on Artificial Intelligence, pp. 1641–1646 (2009)

    Google Scholar 

  5. Bouyer, P., Fahrenberg, U., Larsen, K.G., Markey, N., Srba, J.: Infinite runs in weighted timed automata with energy constraints. In: Cassez, F., Jard, C. (eds.) FORMATS 2008. LNCS, vol. 5215, pp. 33–47. Springer, Heidelberg (2008). doi:10.1007/978-3-540-85778-5_4

    Chapter  Google Scholar 

  6. Brázdil, T., Jančar, P., Kučera, A.: Reachability games on extended vector addition systems with states. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 478–489. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14162-1_40

    Chapter  Google Scholar 

  7. Brihaye, T., Geeraerts, G., Haddad, A., Monmege, B.: To reach or not to reach? efficient algorithms for total-payoff games. In: Proceedings of CONCUR: Concurrency Theory. LIPIcs, vol. 42, pp. 297–310. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)

    Google Scholar 

  8. Brim, L., Chaloupka, J., Doyen, L., Gentilini, R., Raskin, J.-F.: Faster algorithms for mean-payoff games. Formal Methods Syst. Des. 38(2), 97–118 (2011)

    Article  MATH  Google Scholar 

  9. Chakrabarti, A., de Alfaro, L., Henzinger, T.A., Stoelinga, M.: Resource interfaces. In: Alur, R., Lee, I. (eds.) EMSOFT 2003. LNCS, vol. 2855, pp. 117–133. Springer, Heidelberg (2003). doi:10.1007/978-3-540-45212-6_9

    Chapter  Google Scholar 

  10. Chatterjee, K., Chmelik, M.: Indefinite-horizon reachability in Goal-DEC-POMDPs. In: Proceedings of ICAPS: International Conference on Automated Planning and Scheduling, pp. 88–96. AAAI Press (2016)

    Google Scholar 

  11. Chatterjee, K., Chmelik, M., Gupta, R., Kanodia, A.: Optimal cost almost-sure reachability in POMDPs. Artif. Intell. 234, 26–48 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chatterjee, K., Forejt, V., Wojtczak, D.: Multi-objective discounted reward verification in graphs and MDPs. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR 2013. LNCS, vol. 8312, pp. 228–242. Springer, Heidelberg (2013). doi:10.1007/978-3-642-45221-5_17

    Chapter  Google Scholar 

  13. Chatterjee, K., Majumdar, R., Henzinger, T.A.: Markov decision processes with multiple objectives. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 325–336. Springer, Heidelberg (2006). doi:10.1007/11672142_26

    Chapter  Google Scholar 

  14. Chatterjee, K., Velner, Y.: Hyperplane Separation technique for multidimensional mean-payoff games. In: D’Argenio, P.R., Melgratti, H. (eds.) CONCUR 2013. LNCS, vol. 8052, pp. 500–515. Springer, Heidelberg (2013). doi:10.1007/978-3-642-40184-8_35

    Chapter  Google Scholar 

  15. Chen, T., Forejt, V., Kwiatkowska, M., Simaitis, A., Trivedi, A., Ummels, M.: Playing stochastic games precisely. In: Koutny, M., Ulidowski, I. (eds.) CONCUR 2012. LNCS, vol. 7454, pp. 348–363. Springer, Heidelberg (2012). doi:10.1007/978-3-642-32940-1_25

    Chapter  Google Scholar 

  16. Ehrenfeucht, A., Mycielski, J.: Positional strategies for mean payoff games. Int. J. Game Theory 8(2), 109–113 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. Filar, J., Vrieze, K.: Competitive Markov Decision Processes. Springer-Verlag, Heidelberg (1997)

    MATH  Google Scholar 

  18. Filiot, E., Gentilini, R., Raskin, J.-F.: Quantitative languages defined by functional automata. Logical Methods Comput. Sci. 11(3) (2015)

    Google Scholar 

  19. Gurvich, V.A., Karzanov, A.V., Khachiyan, L.G.: Cyclic games and an algorithm to find minimax cycle means in directed graphs. USSR Comput. Math. Math. Phy. 28(5), 85–91 (1988)

    Article  MATH  Google Scholar 

  20. Haase, C., Kreutzer, S., Ouaknine, J., Worrell, J.: Reachability in succinct and parametric one-counter automata. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 369–383. Springer, Heidelberg (2009). doi:10.1007/978-3-642-04081-8_25

    Chapter  Google Scholar 

  21. Hansen, T.D., Miltersen, P.B., Zwick, U.: Strategy iteration is strongly polynomial for 2-player turn-based stochastic games with a constant discount factor. J. ACM 60(1), 1:1–1:16 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hunter, P.: Reachability in succinct one-counter games. In: Bojańczyk, M., Lasota, S., Potapov, I. (eds.) RP 2015. LNCS, vol. 9328, pp. 37–49. Springer, Cham (2015). doi:10.1007/978-3-319-24537-9_5

    Chapter  Google Scholar 

  23. Hunter, P., Raskin, J.-F.: Quantitative games with interval objectives. In: Proceedings of FSTTCS, vol. 29 of LIPIcs, pp. 365–377. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2014)

    Google Scholar 

  24. Jurdziński, M., Lazić, R., Schmitz, S.: Fixed-dimensional energy games are in pseudo-polynomial time. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 260–272. Springer, Heidelberg (2015). doi:10.1007/978-3-662-47666-6_21

    Chapter  Google Scholar 

  25. Karp, R.M.: A characterization of the minimum cycle mean in a digraph. Discrete Math. 23, 309–311 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nykänen, M., Ukkonen, E.: The exact path length problem. J. Algorithms 42(1), 41–53 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Puterman, M.L.: Markov Decision Processes. Wiley, Hoboken (1994)

    Book  MATH  Google Scholar 

  28. Reichert, J.: On the complexity of counter reachability games. In: Abdulla, P.A., Potapov, I. (eds.) RP 2013. LNCS, vol. 8169, pp. 196–208. Springer, Heidelberg (2013). doi:10.1007/978-3-642-41036-9_18

    Chapter  Google Scholar 

  29. Velner, Y., Chatterjee, K., Doyen, L., Henzinger, T.A., Rabinovich, A.M., Raskin, J.-F.: The complexity of multi-mean-payoff and multi-energy games. Inf. Comput. 241, 177–196 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theor. Comput. Sci. 158(1&2), 343–359 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IST Austria, Klosterneuburg, Austria

    Krishnendu Chatterjee & Thomas A. Henzinger

  2. CNRS & LSV, ENS Paris-Saclay, Cachan, France

    Laurent Doyen

Authors
  1. Krishnendu Chatterjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Laurent Doyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas A. Henzinger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laurent Doyen .

Editor information

Editors and Affiliations

  1. Reykjavik University, Reykjavik, Iceland

    Luca Aceto

  2. Aalborg University, Aalborg, Denmark

    Giorgio Bacci

  3. Aalborg University, Aalborg, Denmark

    Giovanni Bacci

  4. Reykjavík University, Reykjavik, Iceland

    Anna Ingólfsdóttir

  5. Inria Campus Beaulieu, Rennes, France

    Axel Legay

  6. Aalborg University, Aalborg, Denmark

    Radu Mardare

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Chatterjee, K., Doyen, L., Henzinger, T.A. (2017). The Cost of Exactness in Quantitative Reachability. In: Aceto, L., Bacci, G., Bacci, G., Ingólfsdóttir, A., Legay, A., Mardare, R. (eds) Models, Algorithms, Logics and Tools. Lecture Notes in Computer Science(), vol 10460. Springer, Cham. https://doi.org/10.1007/978-3-319-63121-9_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-63121-9_18

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-63120-2

  • Online ISBN: 978-3-319-63121-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation