Trees over Infinite Structures and Path Logics with Synchronization

We provide decidability and undecidability results on the model-checking problem for infinite tree structures. These tree structures are built from sequences of elements of infinite relational structures. More precisely, we deal with the tree iteration of a relational structure M in the sense of Shelah-Stupp. In contrast to classical results where model-checking is shown decidable for MSO-logic, we show decidability of the tree model-checking problem for logics that allow only path quantifiers and chain quantifiers (where chains are subsets of paths), as they appear in branching time logics; however, at the same time the tree is enriched by the equal-level relation (which holds between vertices u, v if they are on the same tree level). We separate cleanly the tree logic from the logic used for expressing properties of the underlying structure M. We illustrate the scope of the decidability results by showing that two slight extensions of the framework lead to undecidability. In particular, this applies to the (stronger) tree iteration in the sense of Muchnik-Walukiewicz.

💡 Research Summary

The paper investigates the model‑checking problem for infinite tree structures that are generated from an infinite relational structure M by means of the Shelah‑Stupp “weak” tree iteration (denoted M#). The authors enrich the tree with the equal‑level relation E, which holds between two nodes that lie on the same depth, and study logics that are restricted to quantification over paths and over chains (subsets of paths). This restriction corresponds to the expressive power needed in many branching‑time verification formalisms, where one is interested in properties of computation paths rather than arbitrary subsets of the tree.

The main positive result shows that, for any logic L whose L‑theory of M is decidable (e.g., FO, MSO, WMSO, TC, counting extensions), the “chain‑L‑theory” of the enriched weak tree M#E is decidable. The proof proceeds by encoding each chain c as a pair of ω‑words (α,β): α describes the underlying infinite path (a word over M) and β is a binary word marking which positions of α belong to c. An n‑tuple of chains thus becomes a single ω‑word over the alphabet (M × {0,1})ⁿ. This translation reduces chain‑logic formulas to formulas of a hybrid logic called M‑L‑MSO, which is MSO over ω‑words where the atomic predicates are defined by L‑formulas on the letters. The authors then introduce M‑L‑Büchi automata, whose transitions are specified by L‑formulas. Standard closure properties (union, complement, projection) and the emptiness problem for these automata are shown to be decidable whenever the L‑theory of M is decidable. Consequently, model‑checking a chain‑logic formula on M#E reduces to checking emptiness of an appropriate M‑L‑Büchi automaton, establishing decidability.

The paper also explores the limits of this approach. When the “strong” tree iteration M* is considered (obtained from M# by adding the clone predicate C, which connects a node with its copy one level deeper), the situation changes dramatically. For input alphabets of the form Mⁿ with n > 1, the emptiness problem for the corresponding strong M‑L‑Büchi automata becomes undecidable. The authors prove that already for the simple successor structure of the natural numbers (M = (N, succ)), the first‑order theory and even the chain‑theory of M*E are undecidable. The clone predicate introduces a form of horizontal synchronization that cannot be captured by the restricted chain quantification.

A further negative result concerns extending the equal‑level quantification from chains to whole levels. While the chain‑logic of M#E remains decidable, allowing quantification over arbitrary subsets of a level (i.e., full MSO on each level) leads to undecidability, even for the binary alphabet {0,1}. This shows that the decidability frontier is precisely at the point where quantification is limited to chains (or equivalently, to singletons) rather than to arbitrary level‑wide sets.

The paper is organized as follows: Section 2 fixes terminology and defines the weak and strong tree iterations, the equal‑level relation E, and the clone predicate C. Section 3 develops the theory of M‑L‑Büchi automata, establishing their equivalence with M‑L‑MSO and proving closure and decidability results. Section 4 applies these automata to obtain the decidability of the chain‑L‑theory of M#E under the assumption that the L‑theory of M is decidable. Section 5 presents the two undecidability results for M*E and for level‑wide quantification. The conclusion discusses possible extensions and open problems.

In summary, the work delineates a clear boundary for decidable model‑checking on infinite trees built from infinite structures: when one restricts set quantification to chains and keeps synchronization to the equal‑level relation, decidability is preserved and can be realized via a novel automata‑theoretic framework. Introducing stronger synchronization (the clone predicate) or allowing full level quantification immediately crosses into undecidable territory, aligning with known limits of Büchi automata over infinite alphabets.