Date of Award
Engineering and Applied Science
This dissertation develops a new theory of finite function complexity. This novel approach is based on the structure of preimage sets generated under repeat application of the inverse. We encode this information in our primary data-structure, an square matrix called the sigma matrix. This matrix allows us to easily encode information about the functional digraph and cycle structure of the associated endofunction. Additionally, the sigma matrix is of interest in its own right. The columns of sigma matrices are integer partitions of the domain size n, the size of the domain is always an eigenvalue of the sigma matrix, and calculation of the sigma matrix is highly efficient -- requiring no direct calculation of inverses. The problem of finding the number of unique sigma matrices on X^X as a function of n, the size of X, gives rise to a novel integer sequence \[CapitalSigma](n). We use the sigma matrix and a natural ordering on these matrices as part of a flexible and informative definition of preimage complexity. We give several examples of preimage complexity measures and examine the partition of all endofunctions induced by these measures. Finally we show how the integer sequence \[CapitalSigma](n) corresponds to the number of complexity classes induced by the discrete preimage complexity function.
Fournier-Eaton, Bradford M., "A Theory of Preimage Complexity: Data-structures, Complexity Measures and Applications to Endofunctions and Associated Digraphs" (2020). University of New Orleans Theses and Dissertations. 2794.