# Question regarding notation of a language decidability

$$1.\: A_{DFA} = \{\langle B, w \rangle \mid B \text{ is a } DFA \text{ that accepts input string } w \}$$ $$2.\:A_{DFA} = \{\langle B \rangle \mid B \text{ is a } DFA \text{ that accepts input string } w \}$$

I know how to proof that 1 is decidable by constructing a machine that always halts and accepts whenever $$B$$ accepts, otherwise rejects. what's the difference between 1 and 2. I find that it doesn't make sense to not include $$w$$ inside the encoding brackets but I have seen this notation in other places. are they both the same?

In the first one, $$w$$ is a part of the input.
In the second one, $$w$$ is fixed beforehand, and the language has to depend on what you fix it to be.
• Will that affect the proof that $A_{DFA}$ is decidable. by other means, How is it possible to proof that a language is decidable if $w$ is not part of the input? Apr 11, 2021 at 19:08
• It definitely can only help if $w$ is fixed beforehand and is not a part of the input, but in any case both languages are decidable since we can emulate the DFA until it halts (and it always halts) Apr 11, 2021 at 22:12