Most Complex Deterministic Union-Free Regular Languages

Janusz A Brzozowski,Sylvie Davies

doi:10.1007/978-3-319-94631-3_4

Abstract

A regular language L is union-free if it can be represented by a regular expression without the union operation. A union-free language is deterministic if it can be accepted by a deterministic one-cycle-free-path finite automaton; this is an automaton which has one final state and exactly one cycle-free path from any state to the final state. Jiraskova and Masopust proved that the state complexities of the basic operations reversal, star, product, and boolean operations in deterministic union-free languages are exactly the same as those in the class of all regular languages. To prove that the bounds are met they used five types of automata, involving eight types of transformations of the set of states of the automata. We show that for each \(n \geqslant 3\) there exists one ternary witness of state complexity n that meets the bound for reversal and product. Moreover, the restrictions of this witness to binary alphabets meet the bounds for star and boolean operations. We also show that the tight upper bounds on the state complexity of binary operations that take arguments over different alphabets are the same as those for arbitrary regular languages. Furthermore, we prove that the maximal syntactic semigroup of a union-free language has \(n^n\) elements, as in the case of regular languages, and that the maximal state complexities of atoms of union-free languages are the same as those for regular languages. Finally, we prove that there exists a most complex union-free language that meets the bounds for all these complexity measures. Altogether this proves that the complexity measures above cannot distinguish union-free languages from regular languages.

Highlights

IntroductionFormal definitions are postponed until Section 2
In 2001, Crvenković, Dolinka and Ésik [11] studied equations satisfied by union-free regular languages, and proved that the class of these languages cannot be axiomatized by a finite set of equations
The state complexity of a regularity-preserving binary operation ◦ on regular languages is the maximal value of κ(L′ ◦ L), epxressed as a function of two parameters m and n, where L′ varies over all regular languages of complexity at most m and L varies over all regular languages of complexity at most n

Summary

Introduction

Formal definitions are postponed until Section 2. In 2001, Crvenković, Dolinka and Ésik [11] studied equations satisfied by union-free regular languages, and proved that the class of these languages cannot be axiomatized by a finite set of equations. This is known to be true for the class of all regular languages. In 2012 Jirásková and Nagy [16] proved that the class of finite unions of deterministic union-free languages is a proper subclass of the class of regular languages They showed that every deterministic union-free language is accepted by a special kind of a one-cycle-free-path DFA called a balloon DFA. A summary of the properties of union-free languages was presented in 2017 in [13]

Preliminaries

Main Results

Conclusions