Advanced Topics on State Complexity of Combined Operations

State complexity is a fundamental topic in formal languages and automata theory. The study of state complexity is also strongly motivated by applications of finite automata in software engineering, programming languages, natural language and speech processing and other practical areas. Since many of these applications use automata of large sizes, it is important to know the number of states of the automata. In this thesis, we firstly discuss the state complexities of individual operations on regular languages, including union, intersection, star, catenation, reversal and so on. The state complexity of an operation on unary languages is usually different from that of the same operation on languages over a larger alphabet. Both kinds of state complexities are reviewed in the thesis. Secondly, we study the exact state complexities of twelve combined operations on regular languages. The state complexities of most of these combined operations are not equal to the compositions of the state complexities of the individual operations which make up these combined operations. We also explore the reason for this difference. Finally, we introduce the concept of estimation and approximation of state complexity. We show close estimates and approximations of the state complexities of six combined operations on regular languages which are good enough to use in practice