[Paper Review] Trees over Infinite Structures and Path Logics with Synchronization

Alex Spelten, Wolfgang Thomas+1 more|Nov 14, 2011

Formal Methods in Verification4 citations

TL;DR

This paper establishes decidability of the chain logic theory for weak tree iterations of infinite relational structures enriched with the equal-level relation, under mild assumptions on the underlying logic. It shows that model-checking remains decidable when quantification over siblings is allowed via a decidable logic, but undecidability arises when extending to strong tree iteration, even for simple structures like the binary tree with equal levels.

ABSTRACT

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.

Motivation & Objective

• To investigate the model-checking problem for infinite tree structures built from infinite relational structures.
• To determine whether decidability holds for path logics with synchronization (via the equal-level relation) in weak tree iterations.
• To separate the logic for tree paths from the logic for the underlying structure’s properties.
• To identify the limits of generalization by showing undecidability in stronger frameworks, such as strong tree iteration.
• To extend prior results on chain logic and infinite alphabets by incorporating sibling quantification via a decidable base logic.

Proposed method

• The paper constructs weak tree iterations $\mathcal{M}^\#$ of an infinite relational structure $\mathcal{M}$, where nodes are sequences from $M^*$ and edges are defined by prefix extension.
• It enriches $\mathcal{M}^\#$ with the equal-level relation $E$, which holds between nodes at the same depth, enabling synchronization across levels.
• It introduces a logic $\mathcal{L}$ over $\mathcal{M}$ whose theory is decidable, and allows quantification over siblings (children of a common parent) using $\mathcal{L}$-formulas.
• It defines a chain logic theory over $\mathcal{M}^\#_E$ that combines path quantifiers (for chains) with $\mathcal{L}$-quantifiers restricted to sibling sets.
• It uses definable coding techniques to simulate existential quantification over finite sets in the tree by quantification over antichains at a fixed level, reducing to first-order logic on levels.
• It proves undecidability for strong tree iteration $\mathcal{M}^*$ by reducing the undecidable weak MSO theory of the binary tree with $E$ to the chain logic theory of $\mathcal{M}^*_E$.

Experimental results

Research questions

• RQ1 Is the chain logic theory decidable for weak tree iterations $\mathcal{M}^\#_E$ when the underlying logic $\mathcal{L}$ over $\mathcal{M}$ is decidable?
• RQ2 Can sibling quantification via a decidable logic $\mathcal{L}$ be integrated into the chain logic framework over $\mathcal{M}^\#_E$ while preserving decidability?
• RQ3 Does the decidability result extend to strong tree iteration $\mathcal{M}^*_E$, as it does for MSO in the classical case?
• RQ4 What is the role of the equal-level relation $E$ in enabling synchronization and its impact on decidability?
• RQ5 Can the framework be generalized to automata over infinite alphabets using $\mathcal{M}$-valued inputs and $\mathcal{L}$-formulas for transitions?

Key findings

• The chain logic theory of $\mathcal{M}^\#_E$ is decidable if the $\mathcal{L}$-theory of $\mathcal{M}$ is decidable, even when allowing $\mathcal{L}$-quantifiers over siblings of any node.
• The decidability result holds under mild assumptions on $\mathcal{L}$, including first-order logic, weak monadic second-order logic, and transitive closure logic.
• The equal-level relation $E$ enables synchronization across tree levels and is essential for modeling path-based properties with coordination.
• The paper demonstrates undecidability for the chain logic theory of $\mathcal{M}^*_E$ when $\mathcal{M}$ is the binary successor structure, showing that strong iteration breaks decidability.
• The undecidability is proven by coding finite sets of nodes before level $L$ into antichains at level $L$, reducing the weak MSO theory of the binary tree with $E$ to the chain logic theory.
• The framework supports a generalized notion of Büchi automata over infinite alphabets, with decidable nonemptiness if the $\mathcal{L}$-theory of $\mathcal{M}$ is decidable.