Are pure functions always computable functions?

Question

Are pure functions always computable functions?

In computer programming, a Pure function is a function that has the following properties:
(1) the function return values are identical for identical arguments (no variation with local static variables, non-local variables, mutable reference arguments or input streams), and

(2) the function application has no side effects (no mutation of local static variables, non-local variables, mutable reference arguments or input/output streams).

Computable functions are the basic objects of study in computability theory.
Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output.

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Answer 1

You can define the Halting problem as a decision problem, where the function takes in a program and input, and determines whether the program will terminate or not. This is a pure function, but it's well-known that the Halting problem has no algorithm.

Answer 2

It depends what you mean by "pure function". You have to carefully define it.

If "pure function" refers to computer code to compute some function, and if that code is guaranteed to always terminate, then yes, it computes a computable function.

If "pure function" refers to computer code to compute some function, and if that code is not guaranteed to always terminate, then no, it doesn't necessarily compute a computable function.

If "pure function" refers to a mathematical function (rather than an implementation in code), then it might or might not be computable.

When you need a precise definition, Wikipedia is not a reliable source. Terms like "computable function" are a mathematical concept that have a precise definition. To find the exact definition, I recommend studying a good textbook on the subject of computability. You might find discussion of computable functions in many places (including Wikipedia), where people talk informally about the notion of a computable function; there they speak with the understanding that others who have studied the subject can translate that informal discussion into precise mathematics if needed. I am speaking informally here in my answer. It is beyond the scope of an answer on this site to teach you the subject of computability theory; that's self-study you'll need to do on your own.

In my experience, the term "pure function" does not have a standard, accepted definition. Different people may use that phrase in similar but slightly different ways. So if you are working on a subject where the exact definition matters, you may need to define what you mean by "pure function" (which particular definition you are using) to enable clear communication and clear thinking.