Wadge Degrees of Infinitary Rational Relations

We show that, from the topological point of view, 2-tape B"uchi automata have the same accepting power as Turing machines equipped with a B"uchi acceptance condition. The Borel and the Wadge hierarchies of the class RAT_omega of infinitary rational relations accepted by 2-tape B"uchi automata are equal to the Borel and the Wadge hierarchies of omega-languages accepted by real-time B"uchi 1-counter automata or by B"uchi Turing machines. In particular, for every non-null recursive ordinal $\alpha$, there exist some $\Sigma^0_\alpha$-complete and some $\Pi^0_\alpha$-complete infinitary rational relations. And the supremum of the set of Borel ranks of infinitary rational relations is an ordinal $\gamma^1_2$ which is strictly greater than the first non-recursive ordinal $\omega_1^{CK}$. This very surprising result gives answers to questions of Simonnet (1992) and of Lescow and Thomas (1988,1994).

💡 Research Summary

The paper investigates the topological complexity of infinitary rational relations—subsets of Σ^ω × Γ^ω accepted by two‑tape Büchi automata (2‑BA). A 2‑BA reads two infinite tapes synchronously, consuming finite blocks from the input tape and producing finite blocks on the output tape; a run is successful if it visits an accepting state infinitely often, and the set of all (input, output) pairs generated by successful runs forms the infinitary rational relation R(T).

The authors compare the expressive power of 2‑BA with that of real‑time Büchi 1‑counter automata (r‑BCL(1)ω) and Büchi Turing machines. They recall the Borel hierarchy (Σ⁰_α, Π⁰_α) and the Wadge hierarchy (reduction by continuous functions), and note that any ω‑language accepted by a Büchi Turing machine is analytic (Σ¹_1).

The central technical contribution is a pair of simulations. First, given any Büchi Turing machine M, they construct a 2‑BA T that encodes the whole configuration of M (state, head position, tape contents) on its two tapes. Each step of M is simulated by a finite sequence of transitions of T, preserving the Büchi acceptance condition. Consequently, the projection of R(T) onto the first component coincides with L(M). Second, for any 2‑BA T they build a real‑time Büchi 1‑counter automaton A that reads a merged interleaving of the two tapes (e.g., (a₁,b₁)(a₂,b₂)… ) and uses its counter to keep the two streams synchronized. A accepts exactly the pairs in R(T).

These simulations show that the class RAT_ω of infinitary rational relations accepted by 2‑BA has exactly the same Borel and Wadge hierarchies as the class of ω‑languages accepted by r‑BCL(1)ω or by Büchi Turing machines. In particular:

• For every non‑null recursive ordinal α, there exist Σ⁰_α‑complete and Π⁰_α‑complete infinitary rational relations. This extends earlier results that only exhibited Σ⁰_3‑ and Π⁰_3‑complete examples.
• The supremum of the Borel ranks of infinitary rational relations is the ordinal γ¹_2, which is strictly larger than the Church‑Kleene ordinal ω₁^{CK}. This mirrors the known fact that the Borel ranks of Σ¹_1‑sets (analytic sets) reach γ¹_2.

Thus 2‑tape Büchi automata are topologically as powerful as Büchi Turing machines; they can recognize relations of any analytic complexity, and their Wadge degrees fill the entire Wadge hierarchy of Borel sets. The paper answers the open questions posed by Simonnet (1992) and by Lescow & Thomas (1988, 1994) concerning the topological classification of infinitary rational relations.

Beyond the theoretical classification, the results have several implications. Since 2‑BA can simulate Turing machines, decision problems about infinitary rational relations inherit the full undecidability of ω‑language problems. The Wadge completeness results provide a fine‑grained tool for comparing the complexity of different relations via continuous reductions, which can be useful in the analysis of infinite transducers, infinite games, and verification of systems that manipulate infinite streams.

The authors conclude by noting open directions: whether every ordinal below γ¹_2 actually occurs as the Borel rank of some infinitary rational relation, and how the hierarchy behaves under additional constraints (e.g., deterministic 2‑BA, or restrictions on the form of the output). Extending the analysis to higher projective levels or to other acceptance conditions (Muller, parity) also remains an attractive avenue for future research.