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The state complexity, respectively, nondeterministic state complexity of aregular language $L$ is the number of states of the minimal deterministic,respectively, of a minimal nondeterministic finite automaton for $L$. Some ofthe most studied state complexity questions deal with size comparisons ofnondeterministic finite automata of differing degree of ambiguity. Moregenerally, if for a regular language we compare the size of description by afinite automaton and by a more powerful language definition mechanism, such asa context-free grammar, we encounter non-recursive trade-offs. Operationalstate complexity studies the state complexity of the language resulting from aregularity preserving operation as a function of the complexity of the argumentlanguages. Determining the state complexity of combined operations is generallychallenging and for general combinations of operations that includeintersection and marked concatenation it is uncomputable.

Descriptional Complexity of Finite Automata -- Selected Highlights