Symmetric Groups and Quotient Complexity of Boolean Operations

TL;DR

This paper investigates the quotient complexity of boolean operations on regular languages with symmetric group transition semigroups, establishing conditions for maximal complexity and linking automata theory with group theory.

Contribution

It provides a complete characterization of when the quotient complexity reaches the product of individual complexities for languages with symmetric group automata, generalizing uniform minimality.

Findings

Maximal quotient complexity is achieved under specific conditions related to group conjugacy.

The paper introduces a generalized notion of uniform minimality for automata products.

A novel connection between boolean operation complexity and group theory is established.

Abstract

The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L' are binary regular languages with quotient complexities m and n, and that the transition semigroups of the minimal deterministic automata accepting L and L' are the symmetric groups S_m and S_n of degrees m and n, respectively. Denote by o any binary boolean operation that is not a constant and not a function of one argument only. For m,n >= 2 with (m,n) not in {(2,2),(3,4),(4,3),(4,4)} we prove that the quotient complexity of LoL' is mn if and only either (a) m is not equal to n or (b) m=n and the bases (ordered pairs of generators) of S_m and S_n are not conjugate. For (m,n)\in {(2,2),(3,4),(4,3),(4,4)} we give examples to show that this need not hold. In proving these results we generalize the notion of uniform minimality to direct products of automata. We also establish a non-trivial connection between complexity of boolean operations and group theory.