On Infinite Real Trace Rational Languages of Maximum Topological Complexity

TL;DR

This paper investigates the topological complexity of rational languages of infinite real traces, showing they can be highly complex, including non-Borel and Sigma^1_1-complete sets, thus revealing their maximum possible topological complexity.

Contribution

It proves that all rational languages of infinite real traces are analytic and constructs examples with maximum topological complexity, including non-Borel and Sigma^1_1-complete sets.

Findings

All rational languages of infinite real traces are analytic.

Existence of rational languages that are non-Borel sets.

Some rational languages are Sigma^1_1-complete, indicating maximum topological complexity.

Abstract

We consider the set of infinite real traces, over a dependence alphabet (Gamma, D) with no isolated letter, equipped with the topology induced by the prefix metric. We then prove that all rational languages of infinite real traces are analytic sets and that there exist some rational languages of infinite real traces which are analytic but non Borel sets, and even Sigma^1_1-complete, hence of maximum possible topological complexity.