On the Topological Complexity of Infinitary Rational Relations

We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [Automates et Th'eorie Descriptive, Ph. D. Thesis, Universit'e Paris 7, March 1992].

💡 Research Summary

The paper investigates the topological complexity of infinitary rational relations, i.e., subsets of Σ^ω × Γ^ω recognized by Büchi transducers. While it is well‑known that every such relation is analytic (Σ₁¹), it was an open question—posed by Simonnet in 1992—whether some infinitary rational relation could be analytic yet not Borel. The authors answer this affirmatively by constructing Σ₁¹‑complete (hence non‑Borel) infinitary rational relations.

The work proceeds as follows. First, the authors recall the definition of a Büchi transducer T = (K, Σ, Γ, Δ, q₀, F) and the notion of a successful computation (infinitely many visits to an accepting state). The relation R(T) consists of all input–output ω‑word pairs produced by successful runs; thus every R(T) is analytic.

To obtain non‑Borel examples, the paper exploits the well‑studied correspondence between infinite binary trees and ω‑words. For a finite alphabet Σ with at least two letters, the space of Σ‑labelled infinite binary trees T_ω^Σ is homeomorphic to the Cantor space Σ^ω. For any ω‑language L ⊆ Σ^ω, the set Path(L) of trees that possess at least one branch whose label sequence belongs to L is defined. It is known (Niwinski 1985, Simonnet 1993) that if L is Π₀²‑complete (or of higher Borel complexity) then Path(L) is Σ₁¹‑complete and therefore non‑Borel.

The authors construct a continuous coding h : T_ω^Σ → ((Σ∪{A}) × (Σ∪{A}))^ω, where A is a fresh symbol. The coding interleaves the labels of nodes level by level: odd levels are written into the first component σ₁ in reverse lexicographic order, even levels into the second component σ₂ in normal lexicographic order, with the separator A placed between successive levels. This mapping is continuous.

Next, they define a rational relation R ⊆ ((Σ∪{A}) × (Σ∪{A}))^ω by a Büchi transducer. A pair (y₁, y₂) belongs to R iff it has the specific pattern

 y₁ = x(1)·u₁·A·v₂·x(3)·u₃·A·v₄·…,  y₂ = v₁·x(2)·u₂·A·v₃·x(4)·u₄·A·…

where each x(i)∈Σ, the finite words u_i, v_i satisfy |v_i| = 2|u_i| or 2|u_i|+1, and the infinite word x = x(1)x(2)… belongs to the chosen language L. The length condition on v_i determines whether the next branch step goes left or right, thereby encoding a specific branch of the tree. Consequently, h(t)∈R iff the tree t has a branch whose label sequence lies in L, i.e., iff t∈Path(L). Hence Path(L)=h⁻¹(R).

Since h is continuous and Path(L) is Σ₁¹‑complete, R must also be Σ₁¹‑complete. Moreover, the authors exhibit an explicit Büchi transducer for the concrete case Σ={0,1} and L=(0*·1)^ω, a classic Π₀²‑complete ω‑regular language. The transducer’s states and transitions are described, showing that R is indeed an infinitary rational relation.

Thus the paper establishes the existence of Σ₁¹‑complete infinitary rational relations, providing the first known non‑Borel examples in this class. This result bridges descriptive set theory and the theory of infinite automata, indicating that infinitary rational relations can attain the highest analytic complexity. The authors suggest further investigation into higher projective levels, nondeterministic transducers, and the impact on decision problems for infinite-state systems.