# Is the "intersection" of the special Halting Problem with a language always undecidable?

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 $$B$$.

• 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. Commented Jan 6 at 15:07
• 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 $H$.
– D.W.
Commented Jan 6 at 21:59

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