presburger mso specification

This paper addresses the precise expressibility characterization of Monadic Second-Order (MSO) and Guarded Second-Order (GSO) logic over finite structures in terms of Datalog programs. To establish expressibility criteria, we introduce an existential pebble game, combined with homomorphism-closure analysis, Constraint Satisfaction Problem (CSP) modeling, and countably categorical structure theory. This yields the first necessary and sufficient conditions for MSO/GSO definability by Datalog. In particular, we prove that every complement-closed, homomorphism-closed GSO class must be a finite union of countably categorical CSPs. We further propose the notion of *canonical Datalog programs*, enabling the construction of width-bounded (l,k)-optimal inference programs for homomorphism-closed MSO/GSO classes—programs that are sound and maximally complete among all sound Datalog programs. Our results establish a tight correspondence between logical expressibility and Datalog’s computational power.

This paper addresses the decidability of monadic second-order logic (MSO) extended over the natural-number order structure ⟨ℕ; <⟩, where the extension includes unary dynamic predicates defined by arbitrary integer linear recurrence sequences (ILRS). To resolve this long-standing problem, the authors introduce *prodisjunctivity*—a novel logical property capturing effective conjunctive separability—as a central tool for characterizing definability of such dynamic predicates. Integrating automata theory, logical semantics, and algebraic analysis of recurrence sequences, they establish the decidability of the extended MSO theory. This result constitutes the first systematic solution to the MSO decidability problem for ILRS-induced predicates. Moreover, it extends the scope of Büchi’s theorem beyond fixed regular predicates and establishes a general new paradigm for higher-order logical reasoning over dynamic structures.

This work addresses the decidability of monadic second-order (MSO) logic over the natural number structure ⟨ℕ; <, P₁,…,P_d⟩, where each P_i is a canonical arithmetic predicate (e.g., powers of k, perfect k-th powers ℕ^k, or the Fibonacci sequence Fib). We develop an interdisciplinary decidability framework integrating symbolic dynamical systems, transcendental number theory (including the Schanuel Conjecture), finite automata theory, and logical analysis. Unconditionally—i.e., without unproven hypotheses—we establish MSO decidability for key structures such as (ℕ; <, Pow2, Fib) and (ℕ; <, Pow2, Pow3, Pow6). We further prove Turing equivalence between the MSO theory of (ℕ; <, Pow2, ℕ²) and that of binary normal numbers. Crucially, our approach uncovers deep connections between combinatorial encoding properties of arithmetic predicates and their representability by finite automata, yielding a systematic methodological advance for decidability research at the interface of logic and number theory.

This work addresses the lack of a unified and scalable theoretical framework for verifying concurrent programs under weak memory models. It proposes a novel approach that leverages monadic second-order logic (MSO) as a meta-theory, integrated with treewidth analysis of graph structures, to establish a uniform framework for verification and robustness checking. The study establishes, for the first time, an intrinsic connection between MSO axiomatizability and bounded treewidth, introduces the new notion of “reads-from robustness,” and proves that models such as TSO are not MSO-axiomatizable due to their unbounded treewidth. Building on these insights, the paper identifies several classes of weak memory models that are amenable to MSO axiomatization and provides either automated verification algorithms or methods for generating robustness counterexamples for them.

This paper investigates the computational complexity of monadic second-order (MSO) logic defining colorings on the infinite two-dimensional grid. **Problem:** It addresses (1) the decidability of whether an MSO formula defines a subshift, and (2) the complexity characterization of the associated language families. **Method:** For the first time, it systematically establishes an exact correspondence between the quantifier alternation depth of MSO formulas and the computational complexity of subshift definability, integrating subshift theory, automata theory, model theory of infinite graphs, and formal language complexity analysis. **Contribution/Results:** For each quantifier alternation class Σₙ and Πₙ, it provides tight complexity classifications—e.g., Σₙᵖ-completeness—for the subshift definability problem; moreover, it delivers complete and optimal upper and lower bounds for the class of languages definable by MSO over ℤ², thereby filling a fundamental gap in understanding the systematic relationship between MSO expressiveness and computational complexity on multidimensional discrete structures.

研究通过扩展计数单点序逻辑并引入最优解谓词，为固定团宽和树宽图上的双层图优化问题提供了固定参数可处理的算法。

This work investigates the impact of unary and binary structure operations—such as disjoint union and Cartesian product—that are definable in first-order logic on the recognizability of classes of finite structures. By leveraging a backward translation theorem and a splitting theorem, it reduces first-order properties of output structures to finitely many first-order properties of input structures, preserving quantifier depth in the quantifier-free case and extending to logical fragments enriched with modulo-counting existential quantifiers. Combining first-order transductions with tree automata techniques over structures of bounded treewidth or cliquewidth, the paper establishes the recognizability of such finite structure classes under these operations and provides effective automata-based decision procedures, thereby forging a novel connection between structural recognizability and automata theory.

本文在Isabelle/HOL中采用深浅嵌入方法处理单子二阶逻辑，通过三种嵌入方式实现，并自动化验证其忠实性。

This work investigates the constant-time testability of first-order logic with modular counting (FOMOD) properties on graph classes of bounded degree and bounded connected component size. The authors introduce a FOMOD Hanf normal form tailored to such graph classes and formulate a number-theoretic “patchability” condition that enables global property inference from constant-sized local samples. By integrating fine-grained local structure analysis with the expressive power of modular counting logic, the approach not only establishes the constant-time testability of FOMOD properties on these graphs but also lays a theoretical foundation for efficiently testing richer counting logics, such as counting monadic second-order logic (CMSO).

This study addresses the efficiency bottlenecks inherent in model checking and compositional reconstruction over graph classes excluding a fixed subgraph. To overcome these limitations, this work proposes subconnected logic, which extends first-order logic by introducing a depth-first forest order. This formalization establishes an equivalence among subgraph exclusion, bounded twin-width, and monadic dependence, thereby enabling efficient model checking and dependence analysis through the theoretical framework of bounded twin-width. The primary contribution lies in deriving polynomial-time algorithms that significantly enhance query processing performance for sparse graph classes.