On Infinitary Rational Relations and Borel Sets

TL;DR

This paper explores the complexity of infinitary rational relations, demonstrating that some are complete Borel sets at certain levels, revealing intricate topological properties and answering longstanding questions in the field.

Contribution

It establishes the existence of infinitary rational relations with specific Borel complexity levels, advancing understanding of their topological classification.

Findings

Existence of Sigma^0_3-complete infinitary rational relations

Existence of Pi^0_3-complete infinitary rational relations

Some relations are Delta^0_4-sets but not at lower Borel levels

Abstract

We prove in this paper that there exists some infinitary rational relations which are Sigma^0_3-complete Borel sets and some others which are Pi^0_3-complete. This implies that there exists some infinitary rational relations which are Delta^0_4-sets but not (Sigma^0_3U Pi^0_3)-sets. These results give additional answers to questions of Simonnet and of Lescow and Thomas.