# Complexity of regular bifix-free languages

## TL;DR

This paper investigates the complexity properties of regular bifix-free languages, establishing bounds and characterizations for various operations, and constructing specific language streams that meet these bounds with minimal alphabet size.

## Contribution

It provides the first known universal stream of bifix-free languages meeting all bounds for basic operations, and characterizes state and atom complexities with tight bounds.

## Key findings

- Existence of a universal bifix-free language stream meeting all bounds.
- Characterization of state complexity for product, star, and reversal operations.
- Minimum alphabet sizes required for certain complexity bounds.

## Abstract

We study descriptive complexity properties of the class of regular bifix-free languages, which is the intersection of prefix-free and suffix-free regular languages. We show that there exist a single ternary universal (stream of) bifix-free languages that meet all the bounds for the state complexity basic operations (Boolean operations, product, star, and reversal). This is in contrast with suffix-free languages, where it is known that there does not exist such a stream. Then we present a stream of bifix-free languages that is most complex in terms of all basic operations, syntactic complexity, and the number of atoms and their complexities, which requires a superexponential alphabet. We also complete the previous results by characterizing state complexity of product, star, and reversal, and establishing tight upper bounds for atom complexities of bifix-free languages. We show that to meet the bound for reversal we require at least 3 letters and to meet the bound for atom complexities $n+1$ letters are sufficient and necessary. For the cases of product, star, and reversal we show that there are no gaps (magic numbers) in the interval of possible state complexities of the languages resulted from an operation; in particular, the state complexity of the product $L_m L_n$ is always $m+n-2$, while of the star is either $n-1$ or $n-2$.