On Robot Games of Degree Two

  • Vesa Halava
  • Reino Niskanen
  • Igor Potapov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8977)


Robot Game is a two player vector addition game played in integer lattice \(\mathbb {Z}^n\). In a degree \(k\) case both players have \(k\) vectors and in each turn the vector chosen by a player is added to the current configuration vector of the game. One of the players, called Attacker, tries to play the game from the initial configuration to the origin while the other player, Defender, tries to avoid origin. The decision problem is to decide whether or not Attacker has a winning strategy. We prove that the problem is decidable in polynomial time for the degree two games in any dimension \(n\).


Automata and concurrency Reachability games Vector addition game Decidability Winning strategy 


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  1. 1.
    Abdulla, P.A., Bouajjani, A., d’Orso, J.: Deciding monotonic games. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 1–14. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  2. 2.
    Arul, A., Reichert, J.: The complexity of robot games on the integer line. In: Proceedings of QAPL 2013. EPTCS, vol. 117, pp. 132–148 (2013)Google Scholar
  3. 3.
    Bradley, G.H.: Algorithms for hermite and smith normal matrices and linear dio-phantine equations. Math. Comp., Amer. Math. Soc. 25, 897–907 (1971)CrossRefzbMATHGoogle Scholar
  4. 4.
    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) CrossRefGoogle Scholar
  5. 5.
    Chatterjee, K., Fijalkow, N.: Infinite-state games with finitary conditions. In: Proceedings of CSL 2013. LIPIcs, vol. 23, pp. 181–196 (2013)Google Scholar
  6. 6.
    Doyen, L., Rabinovich, A.: Robot games. Tech. Rep. LSV-13-02, LSV, ENS Cachan (2013)Google Scholar
  7. 7.
    Haase, C., Halfon, S.: Integer vector addition systems with states. In: Ouaknine, J., Potapov, I., Worrell, J. (eds.) RP 2014. LNCS, vol. 8762, pp. 112–124. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  8. 8.
    Halava, V., Harju, T., Niskanen, R., Potapov, I.: Weighted automata on infinite words in the context of attacker-defender games. Tech. Rep. 1118, TUCS (2014)Google Scholar
  9. 9.
    Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. J. ACM 47(2), 312–360 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Bull. Amer. Math. Soc. 74(5), 1025–1029 (1968)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. J. ACM 49(5), 672–713 (2002)CrossRefMathSciNetGoogle Scholar
  12. 12.
    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) CrossRefGoogle Scholar
  13. 13.
    Walukiewicz, I.: Pushdown processes: Games and model-checking. Inf. Comput. 164(2), 234–263 (2001)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  2. 2.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK

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