Motivation: I would like to be able to use functional programming of toys in languages without first-order functions, using natural numbers instead of functions.
A universal function is a function f: N → (N → N), which is equivalent to f: N * N → N, which lists all possible computable functions. In other words, there exists a number k such that f (k) is a quadratic function, there exists a number j such that f (j) is the nth simple function, etc.
To write such a function, you can take any language containing Turing (a programming language compiler, lambda calculus, Turing machines ...) and list all the programs. I would like to allow not only evaluation, but also operations on such functions as addition, composition, currying. For example, given the indices of the two functions f, g, I would like to know what is the index of the function f + g or f composed with g. This would allow "functional toy programming."
What is a good way to write such a code library? I am not looking for a minimalistic Turing Tering that will try to calculate factorial 10, and I don't want to write an advanced compiler. It should have some basic functions, such as adding and the ability to write a loop, but not much more.
Welcome to all high-level languages. Pseudocode, Haskell and Python are preferred. You can assume arbitrary arithmetic of accuracy. Use evalor the like is not permitted.
eval
Explanation: The listed functions will consist of all partial recursive (computable) - this includes functions that do not stop at some inputs. A universal function will hang in these cases; Of course, this is inevitable. See Also: m-recursive functions - http://en.wikipedia.org/wiki/M-recursive_function .
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