State complexity of the star of a Boolean operation

TL;DR

This paper introduces a general method using monsters and modifiers to compute the state complexity of the star of Boolean operations, providing exact results for several cases and unifying the approach.

Contribution

It develops a unified strategy employing monsters and modifiers to determine the state complexity of the star of Boolean operations, including new exact results.

Findings

Recovered state complexity of star of intersection and union

Derived exact state complexity of star of symmetrical difference

Unified approach for various Boolean operations

Abstract

Monsters and modifiers are two concepts recently developed in the state complexity theory. A monster is an automaton in which every function from states to states is represented by at least one letter. A modifier is a set of functions allowing one to transform a set of automata into one automaton. The paper describes a general strategy that can be used to compute the state complexity of many operations. We illustrate it on the problem of the star of a Boolean operation. After applying modifiers on monsters, the states of the resulting automata are assimilated to combinatorial objects: the tableaux. We investigate the combinatorics of these tableaux in order to deduce the state complexity. Specifically, we recover the state complexity of star of intersection and star of union, and we also give the exact state complexity of star of symmetrical difference. We thus harmonize the search strategy for the state complexity of star of any Boolean operations.