Complexity of Winning Strategies for A° Games

For a AiJ game played on w, we show that the winning player has a winning strategy that is recursive in E! , where Ex is the total type-2 object that embodies operation sf . 0. Introduction The complexity of winning strategies for certain definable games played on co can, quite often, be expressed in terms of recursion in (appropriate) higher types. The simplest and the earliest known result is for n° games. It is a well-known result in descriptive set theory that if player II wins a n^ game then she has a winning strategy that is A| recursive. Since the A{ sets are precisely the sets that are recursive in 2E, the Kleene's type-2 object that embodies countable u (cf. [5]), player II has a winning strategy that is recursive in 2E. For Z° games, there is an analogous result. Solovay has shown that a set is &L\ if and only if it is Ij-Ind, where 0 denotes the game quantifier (see [10, 7C.10]). But the S{-Ind sets are precisely the sets that are semi-recursive in E*, by a result of Aczel [1], where E* is the partial type-2 object that embodies operation j/ (see also [5]). Thus ©X" Is tne class °f sets mat are semirecursive in E* . The Third Periodicity Theorem of Moschovakis [10] coupled with the above-mentioned fact shows that if player I wins a Z° game then he has a winning strategy that is recursive in E* . More recently John [7] showed that the complexity of winning strategies for 1°, games is related to recursion in Kolmogorov's operator R. Quite naturally, one would like to have similar results for definable games that are in the higher levels of the arithmetical hierarchy. Such results, it appears, are quite difficult to obtain. In this short note we show that for A^ games the result of Solovay-Aczel can be improved, i.e., we show that for Aj games the winning strategy can be chosen to be recursive in Ei , the (total) type-2 object that embodies operation j/ . Although this result may not be of much significance, we feel that the proof, which is an adaptation of the techniques developed by Burgess in [4], could be useful in analysing A° games for n > 3 . Received by the editors December 20, 1990 and, in revised form, May 10, 1991. 1991 Mathematics Subject Classification. Primary 03D65, 03E15. ©1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page