On Polynomial-Size Programs Winning Finite-State Games

Finite-state reactive programs are identiied with nite au-tomata which realize winning strategies in B uchi-Landweber games. The games are speciied by nite \game graphs" equipped with diierent winning conditions (\Rabin condition", \Streett condition" and \Muller con-dition", deened in analogy to the theory of !-automata). We show that for two classes of games with Muller winning condition polynomials are both an upper and a lower bound for the size of winning reactive programs. Also we give a new proof for the existence of no-memory strategies in games with Rabin winning condition, as well as an exponential lower bound for games with Streett winning condition.