# Most Complex Non-Returning Regular Languages

## TL;DR

This paper investigates the state complexity of non-returning regular languages, providing tight bounds, witness constructions, and analyzing their algebraic properties to identify the most complex such languages.

## Contribution

It introduces new witnesses for complexity bounds, analyzes their properties, and establishes the existence of a most complex non-returning language meeting all bounds.

## Key findings

- Existence of ternary witnesses meeting reversal bounds for all sizes.
- Binary restrictions of witnesses meet bounds for product, star, and boolean operations.
- Maximal syntactic semigroup size is (n-1)^n with at least n2 generators.

## Abstract

A regular language $L$ is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jir\'askov\'a derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each $n\ge 4$ there exists a ternary witness of state complexity $n$ that meets the bound for reversal and that at least three letters are needed to meet this bound. Moreover, the restrictions of this witness to binary alphabets meet the bounds for product, star, and boolean operations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has $(n-1)^n$ elements and requires at least $\binom{n}{2}$ generators. We find the maximal state complexities of atoms of non-returning languages. Finally, we show that there exists a most complex non-returning language that meets the bounds for all these complexity measures.