lattice-theoretic analysis

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 work presents the first complete formalization in Lean 4, based on mathlib, of Dana Scott’s 1972 theory of continuous lattices and its application to modeling the untyped λ-calculus. The project rigorously reproduces the 43 core results from the first four sections of Scott’s original paper, covering essential constructions such as T₀-space embeddings, the Scott topology, the way-below relation, function spaces, and inverse limits, while also incorporating Milner’s corrections to the original proofs. By introducing foundational infrastructure—including bases of Scott-open sets, step functions, towers of function spaces, and the i_∞/j_∞ mapping pair—the formalization establishes the self-embedding theorem D_∞ ≅ [D_∞ → D_∞] under classical logic (with the axiom of choice), propositional extensionality, and quotient type soundness assumptions. All results are verified machine-checked without any use of “sorry”.

This paper investigates the expressive power of stable matching lattices under standard choice functions and the computational complexity of computing minimum-cost stable matchings. Addressing two open questions—(1) whether every finite lattice (including non-distributive ones) can be realized as a stable matching lattice, and (2) whether minimum-cost stable matching is NP-hard—we introduce novel techniques based on partial lattice representation, distributive closure, and join constraints. Our approach extends stable matching lattice theory to arbitrary finite lattices for the first time. We construct a computationally tractable representation framework for non-distributive lattices, circumventing the distributivity requirement of Birkhoff’s representation theorem. Moreover, we establish, under standard assumptions, that computing a minimum-cost stable matching is NP-hard. These results forge a deep connection between lattice structure and matching optimization, yielding a new structural and algorithmic paradigm for stable matching theory.

This paper investigates the algebraic structure and diversity of clones generated by projection operators, focusing on a newly introduced class termed “partition clones.” Using universal algebra and equational logic, we establish, for the first time, a complete equational axiomatization for partition clones. We prove that partition clones are precisely the subdirect products of projection clones and admit a natural bijective correspondence with Boolean algebras; furthermore, their action algebras are characterized as Bergman-type Boolean action sets. The main contributions are threefold: (i) systematic introduction and equational axiomatization of partition clones; (ii) establishment of a full categorical equivalence between partition clones and Boolean algebras; and (iii) unification of clone theory with Boolean algebra actions within a universal algebraic framework. These results provide novel theoretical tools and foundations for clone classification, algebraic semantics modeling, and the algebraization of logic.

This work addresses the problem of syntactic modeling of circuits—represented as string diagrams—in terms of their structural connectivity. We propose a novel circuit grammar framework grounded in order theory: circuits are modeled as partially ordered sets (posets) equipped with designated input and output interfaces, where connectivity is fully encoded by the partial order; circuit rewriting is formalized via structure-preserving morphisms, and a factorization theorem for such morphisms is established. Crucially, we introduce formal concept analysis into circuit theory for the first time: we prove that all circuits sharing the same connectivity correspond bijectively to a unique concept lattice, which precisely characterizes their minimal (canonical) representative. Furthermore, we construct the dual lattice structure. This establishes a rigorous, computable correspondence between circuit connectivity and lattice-theoretic structures, providing a principled, algebraic foundation for modeling quantum causality and enabling causal decomposition of unitary transformations.

This work proposes the Graphical Algebraic Geometry (GAG) framework, which for the first time rigorously formalizes polynomials, ideals, and affine varieties from commutative algebra using a diagrammatic language. By integrating tools from category theory, (co)span semantics, and algebraic geometry, GAG establishes a universal and complete compositional reasoning system for polynomial constraint satisfaction problems (#CSP). The core contributions include establishing a formal correspondence between #CSP and graph rewriting, uncovering a deep connection between GAG and the qudit ZH quantum graphical calculus, and proving that constraint rewriting in GAG is #P-hard. Furthermore, it is shown that computing amplitudes in the qudit ZH calculus requires only a constant number of oracle queries to GAG, thereby opening a novel pathway for efficient modeling of quantum computations.

The formalization of finite element methods in Rocq lacks foundational libraries for monomial orders and graded orders. Method: We develop the first comprehensive formal library, rigorously defining binary relations, total orders, monomial orders, and graded structures; we introduce a higher-order operator framework to uniformly construct and verify four canonical graded orders—lex, grevlex, and others—and design a proof reuse mechanism to systematically derive order properties. Contribution/Results: The library comprises over 700 mechanically verified lemmas, covering definitions, core operators, and essential properties—including well-foundedness and compatibility—thereby significantly enhancing both the efficiency and trustworthiness of finite element method formalization within Rocq.

This study addresses the challenge of systematically extending zero-dimensional dualities, such as Stone spaces, to continuous dualities, including compact Hausdorff spaces. To this end, it proposes a relation-based bicategorical framework that integrates distributive lattices, domain theory, and topological algebra, unifying the treatment of functional and relational morphisms by combining algebraic proximity relations with topological preorders. The transformation is achieved through a three-step procedure comprising the extension of relational morphisms, the splitting of idempotents, and the restriction of mappings. This work successfully establishes a systematic duality translation mechanism from Priestley and Stone spaces to their corresponding continuous counterparts, thereby providing a unified paradigm for duality theory bridging discrete and continuous structures.

This study addresses several long-standing conjectures by J. H. Davenport, A. Locatelli, G. K. Sankaran, and others concerning the topological properties of cells in cylindrical algebraic decomposition (CAD). For the first time, the paper systematically constructs explicit counterexamples that refute these conjectures. By integrating CAD theory from real algebraic geometry, topological construction techniques, and symbolic computation, the work reveals fundamental limitations in the existing assumptions. These findings not only clarify the theoretical boundaries of CAD but also provide a refined foundation for applications involving topological analysis of semi-algebraic sets, such as motion planning in robotics.

This study addresses the long-standing challenge of proving the finite model property (FMP) for a broad class of modal logics and rule-based systems. We introduce a novel method based on subpartition construction, integrating the framework of stable canonical rules, finite-height modal algebras, and modal space techniques—marking the first application of subpartitions to generate finite countermodels and establishing a synergistic analytical pathway linking algebraic and Kripke semantics. Our main contributions are: (1) a proof that all systems axiomatized by stable canonical formulas and rules over finite-height modal algebras possess the FMP; and (2) a characterization of a class of systems whose corresponding lattices admit splitting joins, revealing that their Kripke incompleteness degree is exactly 1. These results uniformly extend the scope of known FMP results and provide a new paradigm for investigating metalogical properties of modal systems.