category-theoretic modeling

This work introduces a dynamical mechanism into algebraic artificial chemistry models to unify their algebraic structure with dynamic behavior. To this end, it pioneers the use of category theory as an organizational framework in this domain, constructing a functor that endows λ-calculus-based interaction-combinator algebraic models—such as Fontana and Buss’s AlChemy—with a formal semantics of dynamical evolution. This approach not only generalizes the classical AlChemy framework but also establishes a rigorous correspondence between algebraic structure and system dynamics. By integrating algebraic rigor with expressive dynamical capabilities, the study formulates a novel paradigm for artificial chemistry, laying a theoretical foundation for future formal modeling and analysis.

This paper establishes a deep categorical equivalence between Gödel’s completeness theorem and the compactness theorem in classical first-order logic. The syntactic and semantic foundations of these cornerstone results have long lacked a unified structural account. Method: We construct two categories—the theory category (objects: first-order theories; morphisms: theory extensions) and the model category (objects: Henkin models; morphisms: model homomorphisms)—and define two core functors: one induced by syntactic Henkin construction, the other by semantic finite-satisfiability extensions. Contribution/Results: We prove, for the first time, that these functors are naturally isomorphic, thereby unifying proof-theoretic and model-theoretic perspectives at the categorical level. This yields a structural explanation of both theorems and provides a conceptual framework for extending categorical semantics to non-classical logics.

This study addresses the problem of diachronic and cross-deployment identity in AI systems, which arises due to retraining and environmental shifts after deployment. The authors propose a formal framework grounded in category theory, defining an AI system’s type as a triple comprising its technical functionality, trustworthiness profile, and the mapping between them. They construct a reachability category via lifecycle paths that preserve trustworthiness, employing time-admissible functors and natural transformations to model system evolution and enable historical comparisons. A hierarchical criterion for identity is introduced: weak identity, based on equality of trustworthiness, and strong identity, grounded in state isomorphism or natural isomorphism of histories. This framework elucidates the structural basis for AI identity judgments, clarifies the identity conditions necessary for transferring accountability claims and governance mechanisms across versions, and demonstrates that categorical identity alone is insufficient to guarantee such transference.

Compositional systems lack a structured, rewriteable mathematical foundation. Method: This work proposes a category-theoretic rewriting framework wherein structured cospans serve as the fundamental syntactic units; it introduces, for the first time, the coupling of structured cospans with double-pushout (DPO) rewriting, yielding a unified theory supporting both traced and trace-free semantics. The framework enables inductive, structure-preserving decomposition of closed systems and establishes a sound correspondence between syntax (structured cospans) and semantics (DPO rewriting). Contribution/Results: It provides the first categorical integration of structured cospans with DPO rewriting; defines two distinct rewriting paradigms—traceable and trace-free; and delivers the first mathematically rigorous, compositional, and rewriteable foundation for systems science, enabling cross-disciplinary modeling and formal analysis of complex systems.

Traditional database models, grounded in the set-valued functor paradigm, lack native support for algebraic operations—such as numerical comparison and arithmetic—and exhibit a fundamental semantic and computational gap with programming languages. To address this, we propose an algebraic database model that systematically embeds multiple Lawvere theories into a unified categorical semantics framework, thereby coherently formalizing schemas, instances, schema transformations, and queries. Leveraging a proarrow equipment—a double-categorical structure—we integrate all model components, enabling direct expression and execution of algebraic operations (e.g., addition, order comparison) within data constraints and queries. This approach bridges the foundational disconnect between database theory and programming language semantics, yielding a verifiable algebraic semantics for databases and establishing computational completeness.

This study addresses the theoretical challenges of proving the correctness of automatic differentiation (AD) and constructing free categorical structures in higher-order typed languages. By treating programming languages as freely generated structures through categorical semantics, the proposed approach leverages Cartesian closed categories, Grothendieck constructions, and forward- and reverse-mode AD techniques to develop structure-preserving program transformations, with particular emphasis on differentiating higher-order and recursive languages. The contributions include novel mathematical results such as free distributive and extensive categories, the establishment of a dialogue mechanism between abstract theory and computational practice, and the formalization of an AD theoretical foundation guided by categorical semantics. Ultimately, this work enables reliable and practical program transformations for automatic differentiation.

Existing functorial coalgebras struggle to adequately model continuous-time transition systems. This work introduces graded coalgebras based on graded monoids and, for the first time, applies them to capture continuous-time evolutionary behavior, establishing a semantic correspondence with Feller–Dynkin processes. By leveraging graded distributive laws and terminal coalgebra constructions, the paper provides sufficient conditions for the existence of terminal coalgebras and defines two semantic notions: branching time and trace semantics. Furthermore, it proposes an accompanying coalgebraic modal logic that precisely characterizes state invariance and expressive power over system behaviors.

This work investigates the relationship between two categories of transfer structures under topological constraints. By integrating algebraic and topological methods with the theory of adjoint functors in category theory, the authors construct a pair of contravariant adjoint functors that establish a deep connection between these categories. This construction yields the finest possible family of contravariant adjoints when accounting for the topological restrictions imposed on both objects and morphisms. The result not only reveals an intrinsic symmetry of transfer structures within a topological setting but also significantly advances the understanding of structural relationships between categories subject to such topological constraints.

This work addresses the limitations of traditional institutional approaches, which rely on signature morphisms to handle variables indirectly, thereby necessitating cumbersome side conditions in logical formalizations. Within the framework of D-institutions, the paper introduces functor categories for the first time to directly model variable structures, defines a category for predicate logic, and formalizes the construction of complex sentences as functorial operations. Building on this foundation, the authors develop a corresponding proof system and establish its completeness. This approach substantially simplifies the axiomatic presentation of variable-rich logical systems, enhancing both formal rigor and notational conciseness.

This work addresses the lack of a rigorous and intuitive compositional theory for the Universal Composability (UC) security framework in static systems by introducing, for the first time, category-theoretic and string diagram techniques to develop a graphical formal model of UC. This approach not only relaxes the original UC framework’s constraints and corrects certain technical oversights but also extends its applicability to diverse computational models, including quantum computation. The authors establish graphically verifiable composition theorems that demonstrate the equivalence of the new framework to traditional UC while simultaneously offering greater expressiveness and broader applicability.