0 Ju l 2 00 3 P 6 = NP ∩ co - NP for Infinite Time Turing Machines

Extending results of Schindler [Sch] and Hamkins and Welch [HW03], we establish in the context of infinite time Turing machines that P is properly contained in NP∩ co-NP. Furthermore, NP∩ co-NP is exactly the class of hyperarithmetic sets. For the more general classes, we establish that P = NP ∩ co-NP = NP∩ co-NP, though P is properly contained in NP ∩ co-NP. Within any contiguous block of infinite clockable ordinals, we show that Pα 6= NPα ∩ co-NPα, but if β begins a gap in the clockable ordinals, then Pβ = NPβ ∩ co-NPβ . Finally, we establish that P f 6= NP ∩ co-NP for most functions f : R → ord, although we provide examples where P f = NP ∩ co-NP and P f 6= NP .