Novel loop structures and the evolution of mathematical algorithms

In this paper, we analyze the capability of Genetic Programming (GP) to synthesize non-trivial, non-approximative, and deterministic mathematical algorithms with integer-valued results. Such algorithms usually involve loop structures. We raise the question which representation for loops would be most efficient. We define five tree-based program representations which realize the concept of loops in different ways, including two novel methods which use the convergence of variable values as implicit stopping criteria. Based on experiments on four problems under three fitness functions (error sum, hit rate, constant 1) we find that GP can statistically significantly outperform random walks. Still, evolving said algorithms seems to be hard for GP and the success rates are not high. Furthermore, we found that none of the program representations could consistently outperform the others, but the two novel methods with indirect stopping criteria are utilized to a much higher degree than the other three loop instructions.