Invariance of Function Complexity Under Primitive Recursive Functions
Genetic Programming (GP)  often uses a tree form of a graph to represent solutions. An extension to this representation, Automatically Defined Functions (ADFs)  is to allow the ability to express modules. In  we proved that the complexity of a function is independent of the primitive set (function set and terminal set) if the representation has the ability to express modules. This is essentially due to the fact that if a representation can express modules, then it can effectively define its own primitives at a constant cost. This is reminiscent of the result that the complexity of a bit string is independent of the choice of Universal Turing Machine (UTM) (within an additive constant) , the constant depending on the UTM but not on the function.
The representations typically used in GP are not capable of expressing recursion, however a few researchers have introduced recursion into their representations. These representations are then capable of expressing a wider classes of functions, for example the primitive recursive functions (PRFs). We prove that given two representations which express the PRFs (and only the PRFs), the complexity of a function with respect to either of these representations is invariant within an additive constant. This is in the same vein as the proof of the invariants of Kolmogorov complexity  and the proof in .
KeywordsGenetic Programming Recursive Function Additive Constant Minimum Description Length Total Function
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