Minimal Representations of Natural Numbers Under a Set of Operators

This paper studies the minimal length representation of the natural numbers. Let O be a fixed set of integer-valued functions (primarily hyperoperations). For each n, what is the shortest way of expressing n as a combinations of functions in O to the constant 1? For example, if O contains the two functions Sx (successor of x) and *xy (x times y) then the shortest representation of 12 is *SSS1SS1, with 8 symbols. This is taken to mean that 8 is complexity of 12 under O. We make a study of such minimal representations and complexities in this paper, proving and/or rightly predicting bounds on complexities, discussing some relevant patterns in the complexities and minimal representations of the natural numbers and listing the results gleaned from computational analysis. Computationally, the first 4.5 million natural numbers were probed to verify our mathematically obtained results. Due to the finiteness of the problem, we used the method of exhaustion of possibilities to state some other results as well.