[bibtex key=DeolalikarHamkinsSchindler2005:NPcoNP]

Extending results of Schindler [math.LO/0106087] and also my paper with Philip Welch, we establish in the context of infinite time Turing machines that 𝑃 is properly contained in 𝑁⁢𝑃 intersect 𝑐⁢𝑜⁢𝑁⁢𝑃. Furthermore, 𝑁⁢𝑃 ∩𝑐⁢𝑜⁢𝑁⁢𝑃 is exactly the class of hyperarithmetic sets. For the more general classes, we establish that 𝑃+ =(𝑁⁢𝑃+ ∩𝑐⁢𝑜⁢𝑁⁢𝑃+) =(𝑁⁢𝑃 ∩𝑐⁢𝑜⁢𝑁⁢𝑃), though 𝑃++ is properly contained in 𝑁⁢𝑃++ ∩𝑐⁢𝑜⁢𝑁⁢𝑃++. Within any contiguous block of infinite clockable ordinals, we show that 𝑃𝛼 is not equal to 𝑁⁢𝑃𝛼 ∩𝑐⁢𝑜⁢𝑁⁢𝑃𝛼, but if 𝛽 begins a gap in the clockable ordinals, then 𝑃𝛽 =𝑁⁢𝑃𝛽 ∩𝑐⁢𝑜⁢𝑁⁢𝑃𝛽. Finally, we establish that 𝑃𝑓 is not equal to 𝑁⁢𝑃𝑓 ∩𝑐⁢𝑜⁢𝑁⁢𝑃𝑓 for most functions 𝑓 from the reals to the ordinals, although we provide examples where 𝑃𝑓 =𝑁⁢𝑃𝑓 ∩𝑐⁢𝑜⁢𝑁⁢𝑃𝑓 and 𝑃𝑓 is not equal to 𝑁⁢𝑃𝑓.