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I am trying to solve some questions on partial decidability of languages and I am getting confused in how to construct proper arguments through the idea of Universal Turing Machine.

I am not posting my questions here, so I would like some examples showing a certain language is partially decidable with a proof.

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A language $L$ is partially decidable if there is a turing machine $M$ such that

  • $M$ accepts on input $w$ if $w \in L$
  • $M$ rejects or runs forever on input $w$ if $w \not\in L$.

For example, the language $L = \{\mathrm{enc}(T) : \mathrm{enc}(T) \text{ is the encoding of a TM } T \text{ that halts on } \epsilon \}$ is partially decidable, since one could define a TM $M$ as follows:

 def M(w): 
    if w == enc(T):
        while True:
           simulate T(epsilon) for one step
           if T has halted: 
               accept
    else:
        reject

one would then make a case distinction for every word $w$, so if

  • $w \in L$, then $w = \mathrm{enc}(T)$ for a TM $T$ that halts on $\epsilon$. Therefore, if started with $w$, $M$ enters the if-part. Since $T$ accepts $\epsilon$ after a finite number of steps, the $T$ will halt eventually and $M$ will accept as well.
  • $w \not\in L$. If $w$ is not a valid encoding of a TM, then $M$ will reject after the first step. On the other hand, if $w = \mathrm{enc}(T)$, then $T$ must be a TM that doesn't halt on $\epsilon$. In this case, it is easy to see that the while loop never stops and $M$ will run forever.
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