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P is RE, but is the complement of the class of languages decidable in polynomial time also recursively enumerable?

If both are RE then this makes P recursive?

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co-$\mathsf{P} = \mathsf{P}$. To see this, pick your favorite problem $A$ in co-$\mathsf{P}$, let $T$ be a Turing machine for the complement of $A$ (in $\mathsf{P}$) and construct a new Turing Machine $T'$ that simulates $T$, accepts if $T$ rejects, and rejects if $T$ accepts.

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  • $\begingroup$ Thanks! what about the second question? As a consequence of your answer can I give a lexicographic order / index for the languages in P (or machines for languages in P)? This is something i could not do if P were only in RE and not in R. $\endgroup$ – DeeDee May 23 at 14:38
  • $\begingroup$ If both $C$ and and co-$C$ are in RE, then they are also both recursive. In your particular case $P$ is recursive. $\endgroup$ – Steven May 23 at 14:39

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