${\rm P}\neq{\rm NP}\cap\textrm{co-}{\rm NP}$ for infinite time Turing machines

[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 $NP$ intersect $coNP$. Furthermore, $NP\cap coNP$ is exactly the class of hyperarithmetic sets. For the more general classes, we establish that $P^+ = (NP^+\cap coNP^+) = (NP \capĀ  coNP)$, though $P^{++}$ is properly contained in $NP^{++}\cap coNP^{++}$. Within any contiguous block of infinite clockable ordinals, we show that $P_\alpha$ is not equal to $NP_\alpha\cap coNP_\alpha$, but if $\beta$ begins a gap in the clockable ordinals, then $P_\beta = NP_\beta\cap coNP_\beta$. Finally, we establish that $P^f$ is not equal to $NP^f \capĀ  coNP^f$ for most functions $f$ from the reals to the ordinals, although we provide examples where $P^f = NP^f \cap coNP^f$ and $P^f$ is not equal to $NP^f$.

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