Descriptional Complexity of Finite Automata -- Selected Highlights

TL;DR

This paper reviews key results on the complexity of finite automata, including state complexity, nondeterministic complexity, and the challenges in computing combined operation complexities.

Contribution

It summarizes significant findings in automata descriptional complexity, highlighting non-recursive trade-offs and uncomputability issues in combined operations.

Findings

State complexity measures the minimal number of states in automata for a language.

Non-recursive trade-offs occur when comparing automata with different description mechanisms.

Determining state complexity for combined operations like intersection is generally uncomputable.

Abstract

The state complexity, respectively, nondeterministic state complexity of a regular language $L$ is the number of states of the minimal deterministic, respectively, of a minimal nondeterministic finite automaton for $L$. Some of the most studied state complexity questions deal with size comparisons of nondeterministic finite automata of differing degree of ambiguity. More generally, if for a regular language we compare the size of description by a finite automaton and by a more powerful language definition mechanism, such as a context-free grammar, we encounter non-recursive trade-offs. Operational state complexity studies the state complexity of the language resulting from a regularity preserving operation as a function of the complexity of the argument languages. Determining the state complexity of combined operations is generally challenging and for general combinations of operations that include intersection and marked concatenation it is uncomputable.