Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular omega-Language

It was noticed by Harel in [Har86] that “one can define $\Sigma_1^1$-complete versions of the well-known Post Correspondence Problem”. We first give a complete proof of this result, showing that the infinite Post Correspondence Problem in a regular $\omega$-language is $\Sigma_1^1$-complete, hence located beyond the arithmetical hierarchy and highly undecidable. We infer from this result that it is $\Pi_1^1$-complete to determine whether two given infinitary rational relations are disjoint. Then we prove that there is an amazing gap between two decision problems about $\omega$-rational functions realized by finite state B"uchi transducers. Indeed Prieur proved in [Pri01, Pri02] that it is decidable whether a given $\omega$-rational function is continuous, while we show here that it is $\Sigma_1^1$-complete to determine whether a given $\omega$-rational function has at least one point of continuity. Next we prove that it is $\Pi_1^1$-complete to determine whether the continuity set of a given $\omega$-rational function is $\omega$-regular. This gives the exact complexity of two problems which were shown to be undecidable in [CFS08].

💡 Research Summary

The paper investigates the high undecidability of several decision problems concerning infinite Post’s Correspondence Problem (PCP) restricted to regular ω‑languages, infinitary rational relations, and ω‑rational functions realized by Büchi transducers. The authors first revisit the analytical hierarchy, defining Σ¹₁ and Π¹₁ classes via second‑order quantification over functions, and recall a classic Σ¹₁‑complete problem: given a Turing machine M, does M, started on a blank tape, have an infinite computation that visits the initial state infinitely often?

Using this benchmark, they introduce a variant of the infinite PCP, denoted ω‑PCP(Reg), where solutions must belong to a regular ω‑language L(A) accepted by a given Büchi automaton A. An instance consists of two n‑tuples of non‑empty words (x₁,…,xₙ) and (y₁,…,yₙ) together with A. A solution is an infinite index sequence i₁i₂…∈L(A) such that the concatenations of the selected x‑words and y‑words coincide.

The authors prove that ω‑PCP(Reg) is Σ¹₁‑complete. Membership in Σ¹₁ follows by constructing a Büchi Turing machine M_Φ(I) that accepts exactly the set of index sequences satisfying the two constraints; non‑emptiness of its ω‑language can be expressed by an existential second‑order formula, placing the problem in Σ¹₁. For Σ¹₁‑hardness, they reduce the aforementioned Σ¹₁‑complete Turing‑machine problem (P) to ω‑PCP(Reg). The reduction encodes the computation of M into the pairs (x_i, y_i) and designs A so that any infinite run of M that revisits the initial state corresponds precisely to a solution of the constructed ω‑PCP(Reg) instance. This establishes Σ¹₁‑completeness.

Armed with this result, the paper derives three new high‑undecidability results.

1. Disjointness of infinitary rational relations: Given two infinitary rational relations R₁ and R₂ (each recognized by a Büchi transducer), deciding whether R₁∩R₂=∅ is Π¹₁‑complete. The problem is the complement of “∃ω‑word w∈R₁∩R₂”, which is Σ¹₁‑complete by the previous theorem; thus its complement lies in Π¹₁ and is Π¹₁‑hard via a similar reduction.

2. Existence of a continuity point for ω‑rational functions: For an ω‑rational function f (realized by a finite‑state Büchi transducer), determining whether there exists at least one point of continuity is Σ¹₁‑complete. Although Prieur showed that the global continuity of f is decidable, the existence of a single continuity point requires expressing “∃x∈Σ^ω ∀ε>0 ∃δ>0 …”, a Σ¹₁‑formula. The authors reduce ω‑PCP(Reg) to this problem, showing Σ¹₁‑hardness, and the formula itself places it in Σ¹₁.

3. Regularity of the continuity set: Deciding whether the set C(f) of continuity points of a given ω‑rational function f is an ω‑regular language is Π¹₁‑complete. The property “C(f) is ω‑regular” can be written as “∀ω‑word w (w∈C(f) → w∈L)” for some regular L, which is a Π¹₁ statement. By reducing the complement of ω‑PCP(Reg) to this decision problem, Π¹₁‑hardness follows.

Thus the paper pinpoints the exact analytical complexity of several problems that were previously known only to be undecidable. It demonstrates a unified method: encode a Σ¹₁‑complete infinite PCP instance into the structure of the problem at hand, thereby transferring the high undecidability. The results highlight a striking gap between decidable global continuity and Σ¹₁‑complete existence of a continuity point, and they place the regularity of continuity sets at the Π¹₁ level. Overall, the work deepens our understanding of the analytical hierarchy’s relevance to automata theory, rational relations, and infinite‑word transductions.