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I'm exploring the decidability characteristics of a particular language formed by the intersection of two languages, specifically in the context of the Halting Problem. The languages are defined as follows:

  1. $\text{Special Halting Problem Language} K: K := \{ w \in \{0, 1\}^* \mid w\#w \in H \}$, where $H$ is the Halting Problem language defined as: $H := \{ w\#x \mid w, x \in \{0, 1\}^* \text{ and } M_w \text{ halts for the input } x \}$
  2. Arbitrary Language $B$: A language that is not further defined for this question.

The language in question, $A$, is the intersection of these two: $A := K \cap B$

My query pertains to the decidability status of $A$. I understand that if $B$ is undecidable, it doesn't necessarily imply that $A$ will also be undecidable, since the intersection of two undecidable languages can have different decidability properties.

Given this context, is the language $A$ decidable, semi-decidable, or undecidable? I'm looking for insights or theorems that could be applied to determine the decidability of $A$, considering the nature of $K$ and the potential characteristics of $B$.

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    $\begingroup$ The question is not clear. In the definition of $H$, what is $M_w$? Is $B$ arbitrary? What do you mean by "A language that is not further defined for this question"? If $B$ is the set of all words then $A = K$. If $B$ is empty, then $A = \emptyset$. The intersection can be anything, I don't see what you are trying to achieve. $\endgroup$ Commented Jan 6 at 15:07
  • $\begingroup$ Please edit your post to clarify what is the definition of $K$ and $B$. There seems to be some kind of typo in your definition of "Special Halting Problem", where words or symbols appear to be missing. Please proof-read and fix the mistakes. Please define $M_w$ and $H$. $\endgroup$
    – D.W.
    Commented Jan 6 at 21:59

1 Answer 1

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There is no general answer.

$B$ could be decidable or undecidable, and $A$ could too, independently of $B$. This table contains different values of $B$ for all cases.

$B$ decidable $B$ undecidable
$A$ decidable ($A = \emptyset$) $B = \emptyset$ $B = H$
$A$ undecidable ($A = K$) $B = \{0, 1, \#\}^*$ $B = K$
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