Operational Complexity in Subregular Classes

Abstract

The state complexity of a regular operation is a function that assigns the maximal state complexity of the language resulting from the operation to the sizes of deterministic finite automata recognizing the operands of the operation. We study the state complexity of intersection, union, concatenation, star, and reversal on the classes of combinational, singleton, finitely generated left ideal, symmetric definite, star, comet, two-sided comet, ordered, star-free, and power-separating languages. We get the exact state complexities in all cases. The complexity of all operations on combinational languages is given by a constant function. The state complexity of the considered operations on singleton languages is $\min \{m,n\}$, $m+n-3$, $m+n-2$, $n-1$, and n, respectively, and on finitely generated left ideals, it is $mn-2$, $mn-2$, $m+n-1$, $n+1$, and $2^{n-2}+2$. The state complexity of concatenation, star, and reversal is $m2^{n}-2^{n-1}-m+1$, n, $2^n$ and $m2^{n-1}-2^{n-2}+1$, $n+1$, $2^{n-1}+1$ for star and symmetric definite languages, respectively. We also show that the complexity of reversal on ordered and power-separating languages is $2^n-1$, which proves that the lower bound for star-free languages given by [Brzozowski, Liu, Int. J. Found. Comput. Sci. 23, 1261–1276, 2012] is tight. In all the remaining cases, the complexity is the same as for regular languages. Except for reversal on finitely generated left ideals and ordered languages, all our witnesses are described over a fixed alphabet.

Supported by the Slovak Grant Agency for Science (VEGA), contract 2/0096/23 “Automata and Formal Languages: Descriptional and Computational Complexity”.

M. Hospodár—This publication was supported by the Operational Programme Integrated Infrastructure (OPII) for the project 313011BWH2: “InoCHF - Research and development in the field of innovative technologies in the management of patients with CHF”, co-financed by the European Regional Development Fund.