Complexity lower bounds for succinct binary structures of bounded clique-width with restrictions

We present a Rice-like complexity lower bound for any MSO-definable problem on binary structures succinctly encoded by circuits. This work extends the framework recently developed as a counterpoint to Courcelle’s theorem for graphs encoded by circuits, in two interplaying directions: (1) by allowing multiple binary relations, and (2) by restricting the interpretation of new symbols. Depending on the pair of an MSO problem $ψ$ and an MSO restriction $χ$, the problem is proven to be NP-hard or coNP-hard or P-hard, as long as $ψ$ is non-trivial on structures satisfying $χ$ with bounded clique-width. Indeed, there are P-complete problems (for logspace reductions) in our extended context. Finally, we strengthen a previous result on the necessity to parameterize the notion of non-triviality, hence supporting the choice of clique-width.

💡 Research Summary

This paper investigates the computational complexity of model‑checking problems expressed in monadic second‑order logic (MSO) on binary structures that are succinctly encoded by Boolean circuits. Building on earlier work that established “Rice‑like” lower bounds for MSO‑definable problems on graphs encoded by circuits, the authors extend the framework in two significant directions. First, they allow multiple binary relations (a set R₂) rather than a single edge relation, thereby covering richer structures such as those arising in automata networks. Second, they introduce a restriction formula χ, also an MSO sentence, which constrains the admissible interpretations of the additional symbols (for example, enforcing a total order, determinism, or strong connectivity).

The central decision problem studied is ψ‑under‑χ‑dynamics: given a circuit that encodes an R₂‑graph, decide whether the MSO property ψ holds under the promise that the structure satisfies χ. The authors define several notions of non‑triviality relative to bounded clique‑width. A formula ψ is called cw‑non‑trivial if there exist infinitely many models and infinitely many counter‑models whose clique‑width is bounded by some constant k. When a restriction χ is present, ψ is cw‑non‑trivial under χ if both models and counter‑models satisfying χ exist in infinite families of bounded clique‑width. They also introduce the concepts of union‑stability (χ’s model class closed under disjoint union) and cw‑size‑independence (for arbitrarily large sizes n there are both a model and a counter‑model of size n satisfying χ).

The main results are as follows.

1. Theorem 2 (NP‑ or coNP‑hardness). If ψ is cw‑non‑trivial under a union‑stable restriction χ, then ψ‑under‑χ‑dynamics is either NP‑hard or coNP‑hard (under polynomial‑time many‑one reductions). The proof adapts the “MSO‑saturating graphs” construction from earlier work, using the union‑stability of χ to glue together arbitrarily many copies of suitable gadgets while preserving the promise.

2. Theorem 3 (P‑hardness). When the pair (ψ, χ) is cw‑size‑independent (i.e., there are arbitrarily large graphs of bounded clique‑width that are both models and counter‑models of ψ while satisfying χ), ψ‑under‑χ‑dynamics becomes P‑hard under log‑space reductions. This covers cases where χ is not union‑stable; a classic example is χ stating that the graph is either a clique or an edgeless graph, and ψ testing “the graph is a clique”. In this setting the problem reduces to evaluating the encoding circuit on a single pair of vertices, which is known to be P‑complete.

3. Theorem 4 (Strong non‑triviality requirement). The authors show that merely having infinitely many models and counter‑models is insufficient for the lower bounds. They construct a first‑order formula ψ that is non‑trivial on planar graphs of maximum degree 4, yet ψ‑dynamics cannot be NP‑ or coNP‑hard unless SAT can be solved in polynomial time on a robust set of instance sizes (as defined by Fortnow and Santhanam). This demonstrates that the bounded‑clique‑width condition is essentially optimal: extending the results to broader graph classes such as planar bounded‑degree graphs would contradict widely believed complexity assumptions.

Technical contributions include two pumping lemmas that generate infinite families of suitable graphs respecting the (ψ, χ) constraints, a detailed method for translating these graphs into succinct circuit encodings, and a careful analysis of the interaction between clique‑width, MSO quantifier rank, and the restriction χ. The paper also discusses the relationship between MSO types of bounded‑clique‑width graphs and the feasibility of algorithmic model‑checking, showing that the type information can be captured by circuits of polynomial size.

In the final sections, the authors illustrate the applicability of their framework to several natural problems on automata networks, such as reachability, limit‑cycle detection, and dynamics of deterministic or injective systems. They argue that any non‑trivial MSO property of such systems, when encoded succinctly, inherits the hardness results established earlier.

Overall, the work provides a comprehensive theory that unifies and extends previous hardness results for succinctly represented structures. By allowing multiple binary relations and incorporating MSO‑expressible restrictions, it delineates precisely when an MSO‑definable property becomes computationally intractable (NP‑hard, coNP‑hard, or P‑hard) on succinct encodings. The necessity of bounded clique‑width is highlighted through a robust lower‑bound argument, reinforcing the role of this graph parameter as the critical barrier between tractable and intractable MSO model‑checking in the succinct setting.