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What exactly does Turing Equivalence mean?

First of all you should separate the (math) specifications of $L$ and its "execution" from the input/output functions of $L$. If such input/output is crucial for $L$ then you're trying to ...
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What exactly is the relation between Haskell and category theory?

Category theory is an abstract branch of mathematics and you do not need to learn it before you start writing Haskell code and similarly, you do not need to learn Haskell before you start learning ...
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