Algebraic and Combinatorial Tools for State Complexity : Application to the Star-Xor Problem

TL;DR

This paper introduces algebraic and combinatorial methods, including monsters and modifiers, to analyze the state complexity of the star of symmetrical differences, providing a new approach and a minimal witness example.

Contribution

It presents a novel theoretical framework using monsters and modifiers to determine state complexity, simplifying the process and producing a small alphabet witness.

Findings

Established a method to compute state complexity using monsters and modifiers

Provided a minimal alphabet witness for the star-Xor problem

Demonstrated the effectiveness of algebraic tools in automata theory

Abstract

We investigate the state complexity of the star of symmetrical differences using modifiers and monsters. 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. These recent theoretical concepts allow one to find easily the desired state complexity. We then exhibit a witness with a constant size alphabet.