Maximal Syntactic Complexity of Regular Languages Implies Maximal Quotient Complexities of Atoms

TL;DR

This paper explores the relationship between syntactic complexity and atom quotient complexities in regular languages, establishing that maximal syntactic complexity implies maximal atom complexities, but not vice versa.

Contribution

It demonstrates that maximal syntactic complexity ensures maximal atom quotient complexities, providing a deeper understanding of the structure of regular languages.

Findings

Maximal syntactic complexity implies maximal atom quotient complexities.

Regular languages with maximal syntactic complexity have 2^n atoms.

The converse that maximal atom complexity implies maximal syntactic complexity is false.

Abstract

We relate two measures of complexity of regular languages. The first is syntactic complexity, that is, the cardinality of the syntactic semigroup of the language. That semigroup is isomorphic to the semigroup of transformations of states induced by non-empty words in the minimal deterministic finite automaton accepting the language. If the language has n left quotients (its minimal automaton has n states), then its syntactic complexity is at most n^n and this bound is tight. The second measure consists of the quotient (state) complexities of the atoms of the language, where atoms are non-empty intersections of complemented and uncomplemented quotients. A regular language has at most 2^n atoms and this bound is tight. The maximal quotient complexity of any atom with r complemented quotients is 2^n-1, if r=0 or r=n, and 1+\sum_{k=1}^{r} \sum_{h=k+1}^{k+n-r} \binom{h}{n} \binom{k}{h}, otherwise. We prove that if a language has maximal syntactic complexity, then it has 2^n atoms and each atom has maximal quotient complexity, but the converse is false.