finite model theory

This paper investigates the computational properties of first-order logic (FO) and its extensions under the semiring semantics introduced by Grädel and Tannen, focusing on model checking and data complexity. For any commutative positive semiring $K$, we establish the first exact classification of data complexity for FO under $K$-semantics and provide a logical characterization of $ ext{FAC}^0_K $—the class of constant-depth arithmetic circuits over $K$. Our method unifies logical expressiveness and algebraic computation by proving equivalences among FO under semiring semantics, arithmetic circuits over $K$, and generalized Blum–Shub–Smale machines. The results yield a novel interdisciplinary framework bridging logic, database theory, and algebraic computation, substantially advancing the foundational theory of semiring semantics.

Traditional circuit complexity theory is confined to the Boolean domain and struggles to characterize computational power over more general algebraic structures. This work develops a unified algebraic framework that algebraizes Barrington’s branching programs and the model of Idziak et al., enabling a systematic study of language classes recognized by circuits over finite algebras—particularly simple algebras and those belonging to congruence-modular varieties. By integrating universal algebra, congruence-modularity theory, non-uniform automata, and circuit complexity analysis, the paper provides the first complete characterization of the language classes captured by such algebraic circuits. This establishes a bidirectional correspondence between algebraic structure and computational complexity, laying a foundational theoretical basis for algebraic models of computation.

This paper investigates the decidability of the widely applicable query containment problem in first-order logic, focusing on cases admitting structurally simple countermodels—characterized by bounded treewidth, cliquewidth, and a newly introduced width measure, partitionwidth. We introduce the notion of “width-bounded universal model sets” and develop a unified framework grounded in partitionwidth, systematically integrating model-theoretic methods, graph width theory, and existential rule techniques. Partitionwidth is employed as the central width parameter for the first time, subsuming and extending classical decidable classes such as Datalog± and guarded rules. We establish decidability for several classes of homomorphism-closed queries under finite-partitionwidth rule sets. Furthermore, we expose inherent limitations of finite-unification sets and propose principled repairs to restore decidability.

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.

Graph Neural Networks (GNNs) suffer from limited analyzability and verifiability due to the lack of precise formal characterizations of their expressive power and decidability boundaries. Method: We establish, for the first time, an exact expressive equivalence between GNNs and a decidable logic fragment—namely, first-order logic extended with Presburger quantifiers—by integrating model theory, formal semantics of GNNs, and decidability theory. Based on this equivalence, we develop a Presburger arithmetic–based logical characterization framework. Results: We design the first sound and complete decision procedures for key GNN verification tasks—including output range checking and functional equivalence verification. Concurrently, we rigorously prove the undecidability of several static analysis problems, such as generalization guarantee verification. Our work provides both theoretical foundations and practical tools for ensuring GNN reliability.

This work addresses the absence of a unified Nerode-style characterization of regularity applicable to arbitrary output domains. Building on a communication complexity framework, it partitions input strings between two parties, Alice and Bob, who collaboratively compute function values by exchanging a constant number of messages drawn either from the output domain or a finite signal set. This approach relaxes traditional computability assumptions, accommodating non-Boolean output domains and infinite alphabets, and extends—within the nominal sets framework—to languages over atomic words. The study demonstrates that the proposed model aligns with established computational models across diverse output domains, thereby offering a unified conjecture and theoretical foundation for settings previously lacking a Nerode-type characterization.

This work addresses foundational questions in circuit complexity by reconceptualizing its theoretical framework from an information-theoretic perspective: Boolean circuits are treated as compact encodings of truth tables, rather than conventional computational models. We introduce “circuit description complexity”—a novel paradigm inspired by Kolmogorov complexity—that unifies the characterization of circuit size and the intrinsic information content of truth tables. Our framework rigorously reproduces several classical circuit lower bounds (e.g., for AC⁰ and monotone circuits), provides the first information-theoretic explanation for why most Boolean functions admit a unique optimal circuit structure, and reveals structural properties of optimal circuits—including hierarchical organization and sparsity. Methodologically, we integrate tools from descriptive complexity, information theory, and circuit lower-bound techniques. By reframing computation as information compression rather than functional realization, our approach advances the fundamental understanding of computational resources.

This study addresses the problem of systematically constructing logical systems corresponding to program abstractions to support formal reasoning. By associating logical systems with finite abstract domains, the work proposes a general method: for a given abstract lattice, it constructs a logic whose Lindenbaum–Tarski algebra is isomorphic to the abstraction, and derives corresponding axioms and inference rules. This approach establishes, for the first time, a systematic connection between abstract interpretation and proof theory as well as algebraic logic, enabling logical modeling of non-Cartesian abstractions such as octagons. The resulting logical connectives and inference systems preserve the concretization map and, under suitable conditions, satisfy soundness and completeness. The framework naturally extends to Cartesian products, multi-variable settings, and non-Cartesian abstract domains.

This study systematically investigates the computational complexity of superintuitionistic and modal logics, focusing on whether propositional logic polynomially reduces to single- or two-variable fragments, and whether first-order logic reduces to fragments with few (one or two) unary predicate letters and minimal individual variables (two or three). Methodologically, it integrates Kripke semantics, model theory, algorithmic reductions, and complexity classification techniques. The work establishes, for the first time, general sufficient conditions for polynomial-time reducibility of logics to small fragments; constructs multiple irreducibility counterexamples; and determines precise complexity bounds for over ten classes of non-classical logics. It also proves Kripke incompleteness for several first-order systems and extends the Church–Trakhtenbrot theorem to quasi-monadic predicate logic, yielding a novel undecidability characterization.

This study investigates whether semantically universal quantum circuit description languages (QCDLs) can effectively decide the validity of programs. By integrating computability theory, formal language theory, and models of quantum unitary operators, the work presents a formal characterization of QCDLs and establishes, for the first time, that the set of valid programs in any semantically universal QCDL is not semi-decidable. This result underscores a fundamental divergence between quantum and classical programming languages at the level of semantic structure and reveals an intrinsic limitation of quantum circuit description languages: no compiler can reliably recognize all valid programs. Consequently, the findings highlight the unique challenges posed by quantum computation in the realm of formal semantics.