Decidable languages are closed under complement. Recognizable languages that are closed under complement are decidable.

Checking the equivalence of two languages is undecidable. ETM reduces to EQTM.

Checking whether the language of a Turing Machine is empty is undecidable. ATM reduces to ETM.

The halting problem is undecidable. ATM reduces to HALTTM.

Checking whether a TM accepts a string is undecidable. Here is how.

Checking whether a context free language is empty is decidable. Here is how.

Checking whether a string belongs to a context free language is decidable. Here is how.

The equivalence of regular languages is decidable. Here is how.

Checking whether a regular language is empty is decidable. Here is how.

Checking whether a string belongs to a regular language is decidable. Here is how.