categorical semantics

This work unifies the modeling of inference rules and proof structures of formal systems within a categorical framework. It introduces a met-variable context representation based on Cartesian PROPs, encodes assumptions and conclusions using spans, and constructs a symmetric monoidal category of proofs with met-variable substitution as the sole primitive operation. This approach is the first to uniformly embed both inference rules and proof structures into the semantics of symmetric monoidal categories, thereby supporting compositional and reusable handling of hypotheses. The authors implement an open-source verification algorithm and surface syntax, successfully encoding formulas, axioms, and representative derivations of first-order logic, and release a functional proof checker.

This work addresses the formal modeling of interactive concurrent processes by proposing a multi-categorical parameterized term rewriting system. For the first time, process interaction is characterized as a confluent and terminating rewrite relation, and it is shown that the resulting term structures naturally form a virtual double category. The central contribution lies in the construction of a denotational semantic functor from this computational syntax to the free corner construction in free monoidal categories, thereby providing a rigorous semantic foundation and mathematical guarantee for interactive behavior.

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.

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.

This paper addresses untyped lambda calculus with global read-write state, developing a unified semantic framework for effectful functional computation. Methodologically, it pioneers the integration of intersection type systems with monadic algebraic effects semantics, concurrently defining operational semantics, denotational semantics, and a type system—while proving type preservation under reduction and expansion of state-term configurations. The contributions are threefold: (1) establishing completeness of type safety and convergence characterization; (2) employing intersection types to precisely capture termination behavior of stateful computations; and (3) providing a theoretically rigorous foundation—combining semantic precision and type-based guarantees—for functional languages with global state.

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 addresses the challenge of ensuring compositionality for process languages under trace equivalence—rather than merely strong bisimilarity—by extending the abstract GSOS framework of Turi and Plotkin to Kleisli categories. It introduces a generalized notion of De Simone laws suitable for this categorical setting and proves that the induced operational semantics is compositional with respect to coalgebraic trace equivalence. Building on this foundation, the paper derives a novel De Simone format that enables compositional reasoning about trace equivalence in probabilistic systems. The approach not only recovers the classical compositionality results for De Simone rules in non-probabilistic settings but also establishes, for the first time, a rule format yielding compositional trace semantics for probabilistic concurrency.

This work addresses the open problem of uniformly constructing distributive laws between monads in substructural settings lacking structural rules such as exchange, contraction, and weakening. Building on Tronin’s syntactic category \( \mathcal{W} \), the paper introduces two novel structures—\( \mathcal{W} \)-operadic and \( \mathcal{W} \)-commutative monads—and shows that, under specific conditions, they admit a canonical, representation-independent distributive law from \( S T \) to \( T S \). The approach employs a refinement technique that broadens its applicability, subsuming Varacca and Winskel’s indexed valuation model as a special case. This constitutes the first general framework capable of accommodating both established and newly discovered distributive laws, enabling extensive monad compositions over the category of sets and significantly extending prior results.

This work establishes a duality between relations among computational systems—such as bisimulation—and relations among logical predicates, thereby enabling cross-system logical reasoning. By extending Tarski duality and Thomason duality to the relational level for the first time, and integrating tools from category theory, Kripke semantics, and infinitary modal logic, the authors construct a dual framework that systematically links system relations with predicate relations. Building on this foundation, they develop a novel proof system capable of formally relating formulas across distinct systems. The resulting framework provides a robust theoretical basis for program logics and verification of concurrent systems, while significantly broadening the scope of classical duality theory within relational semantics.

This work addresses the challenge of formally characterizing global, structural, and emergent behaviors in large-scale autonomous component systems—such as robotic swarms—where traditional formal methods fall short. It proposes a multilayer semantic model for the Software Component Ensemble Language (SCEL) grounded in category theory and sheaf theory, interpreting the system as a sheaf over a topological space. For the first time, the sheaf-theoretic “gluing” operation is aligned with distributed information sharing, while sheaf cohomology is leveraged to quantify system failures. This reframes verification as an analysis of geometric structures, effecting a paradigm shift from computational reasoning to mathematical-geometric analysis and offering deep structural insights for designing robust autonomous systems.