Abstract
At CSL 2002, Jerzy Marcinkowsi and Tomasz Truderung presented the notions of positive games and persistent strategies [8]. A strategy is persistent if, given any finite or infinite run played on a game graph, each time the player visits some vertex already encountered, this player repeats the decision made when visiting this vertex for the first time. Such strategies require memory, but once a choice is made, it is made for ever. So, persistent strategies are a weakening of memoryless strategies.
The same authors established a direct relation between positive games and the existence of persistent winning strategies. We give a description of such games by means of their topological complexity. In games played on finite graphs, positive games are unexpectedly simple. On the contrary, infinite game graphs, as well as infinite alphabets, yield positive sets involved in non determined games.
Last, we discuss positive Muller winning conditions. Although they do not help to discriminate between memoryless and LAR winning strategies, they bear a strong topological characterization.
The author sincerely thanks Erich Grädel for numerous remarks and corrections.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Duparc, J.: La forme normale des Boréliens de rangs finis. Th‘ese de Doctorat, Université Denis Diderot Paris VII. Juillet (1995)
Duparc, J.: Wadge hierarchy and Veblen hierarchy: part I Borel sets of finite rank. J. Symbolic Logic 66(1), 56–86 (2001)
Duparc, J.: A Hierarchy of Deterministic Context-Free ω-languages. Theoretical Computer Science 290(3), 1253–1300 (2003)
Duparc, J., Finkel, O., Ressayre, J.-P.: Computer Science and the fine Structure of Borel Sets. Theoretical Computer Science 257(1-2), 85–105 (2001)
Higman, G.: Ordering by divisibility in abstract algebras. Proc. London Math. Soc. 2, 326–336 (1952)
Kechris, A.S.: Classical descriptive set theory. Graduate texts in mathematics, vol. 156. Springer, Heidelberg (1994)
Louveau, A.: Some Results in the Wadge Hierarchy of Borel Sets. Cabal Sem 79-81. Lecture Notes in Mathematics, vol. 1019, pp. 28-55
Marcinkowski, J., Truderung, T.: Optimal Complexity Bounds for Positive LTL Games. In: Bradfield, J.C. (ed.) CSL 2002 and EACSL 2002. LNCS, vol. 2471, pp. 262–275. Springer, Heidelberg (2002)
Martin, D.A.: Borel determinacy. Ann. of Math. 102(2), 363–371 (1975)
Martin, D.A., Steel, J.R.: A proof of projective determinacy. J. Amer. Math. Soc. 2(1), 71–125 (1989)
Nash-Williams, C.S.J.A.: On well-quasi-ordering finite trees. In: Proceedings of the Cambridge Philosophical Society, vol. 59, pp. 833–835 (1963)
Thomas, W.: Automata on Infinite Objects. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, ch. 4, Elsevier Science Publishers B. V, Amsterdam (1990)
Thomas, W.: Languages, automata, and logic. In: Handbook of formal languages, vol. 3, pp. 389–455. Springer, Berlin (1997)
Wadge, W.W.: Degrees of complexity of subsets of the Baire space. Notice A.M.S., A-714 (1972)
Wadge, W.W.: Ph.D. Thesis, Berkeley
Wagner, K.W.: On ω-regular sets. Inform. and Control 43(2), 123–177 (1979)
Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. In: Theoret. Comput. Sci., vol. 200(1-2), pp. 135–183 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Duparc, J. (2003). Positive Games and Persistent Strategies. In: Baaz, M., Makowsky, J.A. (eds) Computer Science Logic. CSL 2003. Lecture Notes in Computer Science, vol 2803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45220-1_17
Download citation
DOI: https://doi.org/10.1007/978-3-540-45220-1_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40801-7
Online ISBN: 978-3-540-45220-1
eBook Packages: Springer Book Archive