[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 $P^{+}=(N{P}^{+}\cap coN{P}^{+})=(NP\cap coNP)$, though $P^{++}$ is properly contained in $N{P}^{++}\cap coN{P}^{++}$. Within any contiguous block of infinite clockable ordinals, we show that $P_{\alpha }$ is not equal to $N{P}_{\alpha }\cap coN{P}_{\alpha }$, but if $\beta$ begins a gap in the clockable ordinals, then $P_{\beta }=N{P}_{\beta }\cap coN{P}_{\beta }$. Finally, we establish that $P^f$ is not equal to $N{P}^f\cap coN{P}^f$ for most functions $f$ from the reals to the ordinals, although we provide examples where $P^f=N{P}^f\cap coN{P}^f$ and $P^f$ is not equal to $N{P}^f$.