apply order theory

This work addresses the automatic inference of closed-form bounds for recursively defined functions—such as operator fixed points or solutions to functional equations—arising in program cost analysis, loop acceleration, and hybrid system verification. We introduce the *B-bound abstract domain*, which approximates numerical functions via conjunctions of predefined bounding functions, enabling synthesis of highly nonlinear invariants. To systematically lift Galois connections from value domains to function spaces, we design an *abstract domain functor*. Our approach integrates constraint-driven abstract construction, higher-order abstract interpretation, operator fixed-point theory, and symbolic-numerical dimensionality reduction. Experiments demonstrate that the framework efficiently handles multivariate, piecewise, and non-discrete functions; significantly improves nonlinear invariant inference; simplifies transition function design; and achieves end-to-end automation across diverse verification and analysis tasks.

This work investigates the construction of context-sensitive, fine-grained distance metrics for higher-order programs to precisely characterize the relationship between input and output errors. To this end, it introduces a quasi-quasi-metric framework that provides a metric foundation for differential logical relations and leverages cartesian closed categories to support compositional reasoning about program distances. The main contributions include revealing the intrinsic metric nature of differential logical relations, developing differential pre-logical relations and clarifying their structural distinctions from quantitative pre-logical relations—such as the absence of a coarsest relation—and proving that the quasi-quasi-metric spaces induced by type interpretations satisfy strong transitivity, indistinguishability, and weak symmetry. Furthermore, the paper establishes a principle for compositional reasoning about program distances and elucidates their underlying partial order structure.

This work proposes a formalization of algorithms within an intensional computability framework and clarifies their relationship to implementations in computational models. Treating computational models as monoid actions on configuration spaces, programs are modeled as dynamical systems constrained by such actions. Algorithms are defined as finite directed graphs of partial maps over edge-labeled abstract data structures, explicitly separating control flow from data operations. By leveraging tools from category theory, dynamical systems theory, and graph theory, the approach constructs a rigorous semantic framework that, for the first time, treats algorithms as abstract specifications of computational behavior and precisely characterizes the structure-preserving implementation relation between programs and algorithms, thereby deepening our understanding of the nature of computation.

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 paper establishes a general theory of relational indices and cores to address the lack of structured selection principles in point-free reasoning. Methodologically, it (i) formally defines relational indices and cores for the first time; (ii) proves that the index and core of a difference function must be bijective; (iii) introduces a restricted Axiom of Choice—requiring every partial equivalence relation to admit an index; and (iv) systematically investigates its logical consequences within a point-free axiomatic framework. Key contributions include: (i) deriving the “all-or-nothing” axiom as a strict logical consequence of the restricted choice principle, thereby unifying the foundations of both point-based and point-free reasoning; (ii) reconstructing a more concise and elegant axiomatization of relation algebra; and (iii) providing novel point-free logical tools applicable to program verification and formal semantics.

This study establishes a theoretical foundation for emerging representation models and uncovers their deep connections with classical order-theoretic structures. By leveraging tools from higher category theory—specifically 2-categories and adjoint functors—it constructs, for the first time, a 2-adjunction between representation models and the category of preordered morphisms. This framework not only embeds representation theory naturally within the classical setting of ordered algebraic structures, thereby validating its theoretical coherence, but also significantly extends the applicability of classical order theory to representation-theoretic investigations. The work thus offers novel conceptual perspectives and formal machinery for advancing research at the intersection of these domains.

This work addresses the challenge of quantitative reasoning about program equivalence for code involving branching constructs such as case statements. It proposes an additive-disjunctive extension of quantalic linear lambda calculus, incorporating additive disjunction into this formal framework for the first time. By integrating categorical logical gluing techniques, Banach-space-based probabilistic models, and quantum computational semantics, the paper develops an equational system capable of supporting quantitative program equivalence reasoning. Under continuity assumptions, the system enjoys approximate completeness and has been successfully applied to the quantitative analysis of Cauchy sequences arising from random walks. This advances program semantics beyond classical notions of equivalence toward a quantitative paradigm grounded in functional analysis.

This work investigates the effective decomposition of higher-arity relations over finite domains into binary relations, with applications in constraint satisfaction problems, clone theory, and relational databases. By interpreting relations as partially defined graphs of multi-valued functions and leveraging functional completeness and function decomposition techniques from many-valued logic, the authors first reduce arbitrary high-arity relations to ternary ones. Subsequently, they achieve full binarization through a transformation that replaces disjunctions with existential quantifiers. This approach yields an elementary and computationally efficient constructive proof, unifying and rederiving Pierce’s reduction theorem for finite domains. Moreover, it demonstrates that the graph of any Sheffer function suffices to generate all relations, thereby establishing a computable theoretical foundation for the aforementioned fields.

This work addresses the challenges of formalizing set theory in dependently typed proof assistants, where boilerplate code is often excessive and integration with typed libraries is difficult. Building upon Lean 4’s Mathlib, the authors develop a set-theoretic framework over a model of ZFC that incorporates relational calculus, lightweight automation with predictable behavior, and standard set constructions. Crucially, it enables seamless interoperability between ZFC objects and Lean’s native types for the first time, supporting hybrid set/type reasoning. The framework substantially reduces boilerplate in scalable set-theoretic proofs while remaining fully compatible with Mathlib. Its expressiveness and practicality are demonstrated through complete formalizations of Boolean values, natural numbers, integers, and the Curry isomorphism theorem.

This study addresses the complexity in point-based constructions arising from varying morphism definitions in modal logic by establishing a Stone-type duality between an algebraic category equipped with paired modal operators and a category of topological spaces endowed with binary relations. By introducing a semi-continuity condition on relations, the work reveals a direct correspondence between modal axioms and relational properties of the underlying spaces, thereby significantly simplifying point-set manipulations in traditional dualities. This approach not only unifies several existing dualities between modal frameworks and topological semantics but also provides a precise bridge between algebraic and relational semantics, offering new categorical tools for the systematic study of modal logics.