On Omega Context Free Languages which are Borel Sets of Infinite Rank

This paper is a continuation of the study of topological properties of omega context free languages (omega-CFL). We proved before that the class of omega-CFL exhausts the hierarchy of Borel sets of finite rank, and that there exist some omega-CFL which are analytic but non Borel sets. We prove here that there exist some omega context free languages which are Borel sets of infinite (but not finite) rank, giving additional answer to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:“A Decade of Concurrency”, Springer LNCS 803 (1994), 583-621].

💡 Research Summary

The paper investigates the topological complexity of ω‑context‑free languages (ω‑CFL), extending earlier work that showed ω‑CFLs exhaust the finite Borel hierarchy and that some ω‑CFLs are analytic but non‑Borel. The authors address an open problem posed by Lescow and Thomas: whether ω‑CFLs can occupy infinite (non‑finite) Borel ranks.

First, the authors review the Borel hierarchy (Σ₁⁰, Π₁⁰, Σ₂⁰, Π₂⁰, …) and the formal model of ω‑CFLs, namely nondeterministic push‑down automata (ω‑PDA). They recall known constructions that place ω‑CFLs at each finite Borel level, and they revisit classic examples of analytic but non‑Borel ω‑CFLs, such as languages requiring infinitely many occurrences of two distinct symbols.

The core contribution is a two‑stage construction that yields ω‑CFLs of genuine infinite Borel rank. In the first stage, the authors define a hierarchy of languages Lₙ, each lying precisely in Σₙ⁰ (or Πₙ⁰) by using standard closure properties of ω‑CFLs under union, intersection, and concatenation with regular ω‑languages. By alternating Σₙ⁰ and Πₙ⁰ constructions they can climb the finite Borel ladder arbitrarily high.

In the second stage they introduce an “ω‑repetition” operator that interleaves the languages Lₙ infinitely often: \