Monoids and the State Complexity of the Operation root(<i>L</i>)

Abstract

In this thesis, we cover the general topic of state complexity. In particular, we examine the bounds on the state complexity of some different representations of regular languages. As well, we consider the state complexity of the operation root(<i>L</i>). We give quick treatment of the deterministic state complexity bounds for nondeterministic finite automata and regular expressions. This includes an improvement on the worst-case lower bound for a regular expression, relative to its alphabetic length. The focus of this thesis is the study of the increase in state complexity of a regular language <i>L</i> under the operation root(<i>L</i>). This operation requires us to examine the connections between abstract algebra and formal languages. We present results, some original to this thesis, concerning the size of the largest monoid generated by two elements. Also, we give good bounds on the worst-case state complexity of root(<i>L</i>). In turn, these new results concerning root(<i>L</i>) allow us to improve previous bounds given for the state complexity of two-way deterministic finite automata.