Unambiguous Tree Languages Are Topologically Harder Than Deterministic Ones

TL;DR

This paper demonstrates that unambiguous tree languages can be topologically more complex than deterministic ones, providing examples that surpass known complexity classes in the hierarchy of descriptive set theory.

Contribution

It introduces a specific unambiguous tree language that is topologically harder than any countable boolean combination of analytic and coanalytic sets, advancing understanding of automata complexity.

Findings

Unambiguous parity automaton recognizes a set that is analytic-complete.

Constructed unambiguous language exceeds the complexity of any difference hierarchy of analytic sets.

Shows unambiguous languages can be topologically more complex than deterministic ones.

Abstract

The paper gives an example of a tree language G that is recognised by an unambiguous parity automaton and is analytic-complete as a set in Cantor space. This already shows that the unambiguous languages are topologically more complex than the deterministic ones, that are all coanalytic. Using set G as a building block we construct an unambiguous language that is topologically harder than any countable boolean combination of analytic and coanalytic sets. In particular the language is harder than any set in difference hierarchy of analytic sets considered by O.Finkel and P.Simonnet in the context of nondeterministic automata.