Theory of recursive functions and effective computability
Theory of recursive functions and effective computability
Genetic programming: on the programming of computers by means of natural selection
Genetic programming: on the programming of computers by means of natural selection
An introduction to Kolmogorov complexity and its applications
An introduction to Kolmogorov complexity and its applications
Neural networks for pattern recognition
Neural networks for pattern recognition
Simultaneous evolution of programs and their control structures
Advances in genetic programming
Evolving recursive programs for tree search
Advances in genetic programming
Evolving recursive functions for the even-parity problem using genetic programming
Advances in genetic programming
Genetic programming: an introduction: on the automatic evolution of computer programs and its applications
Genetic Programming and Evolvable Machines
A Representation for the Adaptive Generation of Simple Sequential Programs
Proceedings of the 1st International Conference on Genetic Algorithms
Evolving Turing Machines for Biosequence Recognition and Analysis
EuroGP '01 Proceedings of the 4th European Conference on Genetic Programming
Algorithm evolution with internal reinforcement for signal understanding
Algorithm evolution with internal reinforcement for signal understanding
Advances in Minimum Description Length: Theory and Applications (Neural Information Processing)
Advances in Minimum Description Length: Theory and Applications (Neural Information Processing)
Modularity in genetic programming
EuroGP'03 Proceedings of the 6th European conference on Genetic programming
Theoretical results in genetic programming: the next ten years?
Genetic Programming and Evolvable Machines
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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. 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) [3], 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 [3] and the proof in [2].