Complexity and Cartesian Genetic Programming

Genetic Programming (GP) [1] often uses a tree form of a graph to represent solutions. An extension to this representation, Automatically Defined Functions (ADFs) [1] is to allow the ability to express modules. In [2] 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. Cartesian Genetic Programming (CGP) [3] is a relative new type of representation used in Evolutionary Computation (EC), and differs from the tree based representation in that outputs from previous computations can be reused. This is achieved by representing programs as directed acyclic graphs (DAGs), rather than as trees. Thus computations from subtrees can be reused to reduce the complexity of a function. We prove an analogous result to that in [2]; the complexity of a function using a (Cartesian Program) CP representation is independent of the terminal set (up to an additive constant), provided the different terminal sets can both be simulated. This is essentially due to the fact that if a representation can express Automatic Reused Outputs [3], then it can effectively define its own terminals at a constant cost.