On Infinitary Rational Relations and Borel Sets

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.

💡 Research Summary

The paper investigates the descriptive set‑theoretic complexity of infinitary rational relations, i.e., binary relations on infinite words that are recognized by nondeterministic Büchi transducers. After recalling the definition of such relations and the basics of the Borel hierarchy, the authors focus on the third level of this hierarchy, Σ⁰₃ and Π⁰₃, which consist of countable unions of Π⁰₂ sets and countable intersections of Σ⁰₂ sets, respectively. While it was previously known that many infinitary rational relations lie low in the hierarchy (often Σ⁰₂ or Π⁰₂), no examples of Σ⁰₃‑complete or Π⁰₃‑complete relations had been exhibited.

To fill this gap, the authors construct two specific relations, R₁ and R₂. For R₁ they start from a classic Σ⁰₃‑complete set S of ω‑words—namely the set of infinite words that contain infinitely many occurrences of a fixed regular pattern. They define a continuous reduction f that maps any ω‑word x to a pair (x, y) such that the Büchi transducer accepting R₁ reads x, simulates the detection of the pattern, and outputs y in a way that the transducer visits accepting states infinitely often exactly when x ∈ S. Because f is continuous and S is Σ⁰₃‑complete, R₁ is Σ⁰₃‑complete.

For Π⁰₃‑completeness they consider the complement of S, a Π⁰₃‑complete set. The transducer for R₂ is designed to verify that no regular pattern occurs infinitely often. It does this by maintaining a finite set of counters and only entering a Büchi accepting loop when all counters stabilize, which corresponds precisely to the Π⁰₃ condition. A continuous reduction g from the Π⁰₃‑complete set to R₂ shows that R₂ is Π⁰₃‑complete.

Both relations are recognized by Büchi transducers, hence they are Borel sets. Moreover, because Σ⁰₃‑complete and Π⁰₃‑complete sets lie in Δ⁰₄, the constructed relations belong to Δ⁰₄ but not to Σ⁰₃ ∪ Π⁰₃. This demonstrates that infinitary rational relations can reach the fourth level of the Borel hierarchy, answering an open question posed by Simonnet about the maximal Borel complexity of such relations. It also resolves a problem raised by Lescow and Thomas concerning the existence of Δ⁰₄ relations that are not in Σ⁰₃ or Π⁰₃.

The paper concludes with a discussion of possible extensions: exploring higher levels (Σ⁰₅, Π⁰₅), comparing nondeterministic and deterministic transducers, and investigating the impact of these complexity results on automata‑theoretic verification and model‑checking frameworks. The results enrich the understanding of the interplay between automata on infinite words and descriptive set theory.