User blog comment:SuperJedi224/Something I've been putting together/@comment-1605058-20141113190447

As I mentioned, you have specified pretty much no detail on how to resolve the recursive definitions. Here I have few questions:

All of these things you left undecided when it comes to your function.
 * In what order are rules resolved? For example, suppose we have rules F(0,x)=1, F(x,0)=2. What is the value of F(0,0)?
 * What is the assumed set of numbers in question? For example, suppose we have a single rule, F(2*x)=x. Can we resolve F(3) as 1.5, or is it undefined? Similarly, does P(Sx)=x have P(0)=-1?
 * If in the above we only concern natural numbers, how does the hypothetical compiler know if its input has a desired form? And what if there is more than one such form? For example, if we have F(x*x-x*5+6)=x (where subtraction can be defined using other formula) and we have F(2), is it 1 or 4?
 * Can we have in recursive definitions multiple variables directly related? For example, is the following definition valid? F(2)=2, F(x+y)=F(y) It defines a unique function, which is constantly 2, yet it's not really recursive.
 * If numbers in question are natural, compiler can resolve existence of expression of input, and multiple variables are allowed in definitions as before, then it can solve, in theory, Hilbert's tenth problem, thus making X an uncomputable function.