# prove language is decidable

The question is:

The language L contains $$DFAs$$ which can accept languages equal to $$\Sigma$$*

prove this language is decidable.

I'm new to the Decidability topic and I don't know where should I start proving this?

reduce? Rice theorem? or we don't need things like reducing and...I would appreciate it if someone teaches me the way of thinking and proving these questions. thanks in advance.

About reduction: for as long as you can reduce this problem to some other decidable problem, its fine. That is, you can show that $$L\le_p L'$$ for some decidable $$L'$$ and this would ensure that $$L$$ is also decidable.
However, the best way to do this (in my opinion) is to directly build an algorithm that decides the question. If you want a small hint, try to answer the following: how would you decide the language of all $$DFA$$s that accept $$\emptyset$$?
If you have an algorithm which decides "Does DFA $$M$$ accept $$\Sigma^*$$?" you just need to check if the input is a DFA (in whatever reasonable notation you specify), if not, reject; if it is a DFA, apply the above algorithm. This decides your language.
• @nirshahar, the algorithm to see if $M$ accepts $\Sigma^*$. Commented Jun 8, 2021 at 22:45