Maximal Syntactic Complexity of Regular Languages Implies Maximal Quotient Complexities of Atoms
Year: 2013
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Brzozowski, Janusz, Davies, Gareth
Institution: University of Waterloo
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Award: G3764228066
OpenAlex ID: W1609937380