Trading Bounds for Memory in Games with Counters?

Abstract. We study two-player games with counters, where the objec-tive of the first player is that the counter values remain bounded. We investigate the existence of a trade-off between the size of the memory and the bound achieved on the counters, which has been conjectured by Colcombet and Löding. We show that unfortunately this conjecture does not hold: there is no trade-off between bounds and memory, even for fi-nite arenas. On the positive side, we prove the existence of a trade-off for the special case of thin tree arenas.