[Solved]: Is the language of TMs that decide some language Turing-recognizable?

Problem Detail: 

Is the language

$\qquad L=\{ \langle \text{M} \rangle \; | \; \text{M is a Turing machine that decides some language} \}$

a Turing-recognizable language? I think it's not, as, even if I am able to tell somehow that a Turing machine halts for some input there are still infinite strings to check for. Similarly I think that this problem is not even co-recognizable. Am I right? If yes is there a more precise proof ?

Asked By : sashas

Answered By : Yuval Filmus

This language is usually known as TOT, the language of machines computing total functions. It is $\Pi_2$-complete, and in particular is neither recognizable nor co-recognizable.

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Question Source : http://cs.stackexchange.com/questions/48777