Parameterized Complexity Of Representing Models Of MSO Formulas

We introduce Tree Decision Diagrams (TDD) as a model for Boolean functions that generalizes OBDD. They can be seen as a restriction of structured d-DNNF; that is, d-DNNF that respect a vtree $T$. We show that TDDs enjoy the same tractability properties as OBDD, such as model counting, enumeration, conditioning, and apply, and are more succinct. In particular, we show that CNF formulas of treewidth $k$ can be represented by TDDs of FPT size, which is known to be impossible for OBDD. We study the complexity of compiling CNF formulas into deterministic TDDs via bottom-up compilation and relate the complexity of this approach with the notion of factor width introduced by Bova and Szeider.

In Isabelle/HOL, we apply the deep-and-shallow embedding methodology of our prior work to monadic second-order logic (MSO). Three embeddings are developed side by side: a deep embedding (an inductive datatype with an explicit satisfaction relation); a maximal-shallow embedding that translates the connectives and quantifiers directly into HOL, carrying the interpretation and both assignments explicitly; and a minimal-shallow embedding -- a locale that fixes those parameters, collapsing the formula type to bool. The enabling new ingredient is a two-sorted substitution apparatus -- capture-avoiding substitution, renaming, and a substitution lemma per namespace -- in which each binder is transparent for the other; faithfulness of all three embeddings is mechanised and largely automated. Our central contribution is a fully mechanised two-sorted downward Loewenheim-Skolem theorem: the minimal embedding recovers deep validity relative to the (countable) assignment ranges, and this range-relative reading is shown to coincide with the general (Henkin-style) reading of MSO, whereas the standard reading validates strictly more formulas, witnessed by comprehension. Both readings are nonetheless recovered from the minimal embedding, differing only in the admitted interpretations: all of them for the general reading, only the elementary substructures of the full model for the standard. We exercise the embeddings on classical MSO landmarks: the Boolean-closure and graph schemata hold under the full second-order domain yet fail in the minimal embedding, making the dichotomy concrete, while reachability and 2-colorability are refuted throughout.

Weighted first-order model counting (WFOMC) is a central task in lifted probabilistic inference: It asks for the weighted sum of all models of a first-order sentence over a finite domain. A long line of work has identified domain-liftable fragments of first-order logic, that is, syntactic classes for which WFOMC can be solved in time polynomial in the domain size. Among them, the two-variable fragment with counting quantifiers, $\mathbf{C}^2$, is one of the most expressive known liftable fragments. Existing algorithms for $\mathbf{C}^2$, however, establish tractability through multi-stage reductions that eliminate counting quantifiers via cardinality constraints, which introduces substantial practical overhead as the domain size grows. In this paper, we introduce IncrementalWFOMC3, a lifted algorithm for WFOMC on $\mathbf{C}^2$ and its modulo counting extension, $\mathbf{C}^2_{\text{mod}}$. Instead of relying on reduction techniques, IncrementalWFOMC3 operates directly on a Scott normal form that retains counting quantifiers throughout inference. This direct treatment yields two main results. First, we derive a tighter data-complexity bound for WFOMC in $\mathbf{C}^2$, reducing the degree of the polynomial from quadratic to linear in the counting parameters. Second, we prove that $\mathbf{C}^2_{\text{mod}}$ is domain-liftable, extending tractability from $\mathbf{C}^2$ to a richer fragment with native modulo counting support. Finally, our empirical evaluation shows that IncrementalWFOMC3 delivers orders-of-magnitude runtime improvements and better scalability than both existing WFOMC algorithms and state-of-the-art propositional model counters.