Date of Award

Fall 12-18-2015

Degree Type


Degree Name


Degree Program




Major Professor

Kenneth Holladay

Second Advisor

Ralph Saxton

Third Advisor

Jairo Santanilla


This paper develops a data structure based on preimage sets of functions on a finite set. This structure, called the sigma matrix, is shown to be particularly well-suited for exploring the structural characteristics of recursive functions relevant to investigations of complexity. The matrix is easy to compute by hand, defined for any finite function, reflects intrinsic properties of its generating function, and the map taking functions to sigma matrices admits a simple polynomial-time algorithm . Finally, we develop a flexible measure of preimage complexity using the aforementioned matrix. This measure naturally partitions all functions on a finite set by characteristics inherent in each function's preimage structure.


The University of New Orleans and its agents retain the non-exclusive license to archive and make accessible this dissertation or thesis in whole or in part in all forms of media, now or hereafter known. The author retains all other ownership rights to the copyright of the thesis or dissertation.

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.