lattice theory

This study investigates the application of algebraically structured lattices—such as ideal lattices—in lattice-based cryptography and secure wireless communication. By leveraging tools from algebraic number theory to construct structured lattices, the work systematically analyzes their mathematical properties through the interplay of lattice geometry, theta function theory, and classical problems in number theory and geometry, including Minkowski’s conjecture and sphere packing. The research demonstrates that such structured lattices play a pivotal role in enhancing both the efficiency and security of cryptographic schemes and in improving secrecy performance in wireless communication systems. In doing so, it fosters deep interdisciplinary connections among lattice theory, number theory, cryptography, and information theory, thereby opening new avenues for synergistic innovation across these fields.

This work proposes a parameterized information-flow framework that unifies confidentiality and integrity through their joint interaction, leveraging the duality between open and closed modalities in modal type theory. Traditional approaches model these security properties separately, leading to redundant reasoning, complex specifications, and degradation mechanisms that often undermine modularity and abstraction. In contrast, the proposed framework naturally supports downgrading operations without requiring additional extensions, while remaining compatible with strong noninterference guarantees and practical declassification needs. It preserves full noninterference and not only reproduces but also strengthens mechanisms such as robust declassification, demonstrating their complete compatibility with modular design and abstraction.

This paper studies the minimum cut problem in directed graphs under additional constraints. While the set of all minimum cuts forms a distributive lattice, imposing constraints typically renders the problem NP-hard. To address this, we model constraints as lattice-linear predicates and—novelty—integrate them with max-flow preprocessing, introducing *k-transfer predicates* and a *strong push mechanism*. We design a parallel polynomial-time algorithm to efficiently compute sublattice-irreducible elements satisfying regular constraints; provide a succinct representation and enumeration scheme for the feasible sublattice; and, for non-lattice-linear constraints, propose an exact algorithm based on *poset slicing* and *predicate propagation*, outperforming brute-force enumeration. Our main contributions are: (i) establishing a unified lattice-theoretic framework for constrained minimum cuts; (ii) enabling efficient parallel computation; (iii) achieving succinct representation of solution sublattices; and (iv) supporting scalable, structured enumeration of feasible cuts.

This paper addresses the redundancy problem in event lattices within information structures. We propose a minimal sufficient representation method based on partially ordered sets (posets): under mild conditions, we construct a simplified poset that is order-isomorphic to the original event lattice. This representation preserves all event relations and knowledge content, enabling a rational agent to reconstruct the full lattice losslessly from the reduced structure alone. Theoretically, we establish the first Minimal Sufficient Representation Theorem for event lattices, rigorously proving both the completeness and uniqueness of the derived poset. Practically, the method substantially reduces structural complexity in knowledge reasoning, offering a novel theoretical foundation and computational framework for distributed knowledge modeling and collaborative inference in multi-agent systems.

This paper investigates the preservation of decidability under disjoint combinations of first-order theories. Addressing classical combination conditions—stability-infinity, shininess, strong politeness, and tameness—we introduce, for the first time, Galois connections to characterize their intrinsic algebraic structure, thereby constructing induced complete lattice models that systematically unify existing and novel combination frameworks. Our approach precisely determines the maximal sets of theories extendable by each condition, thereby refuting several long-standing open conjectures. Moreover, within this algebraic framework, we derive a family of new combination theorems, providing a unified foundation for constructing broader classes of decidable theory combinations. The work integrates model-theoretic, order-theoretic, and categorical perspectives, enabling a formal reconstruction and boundary characterization of combination properties.

This work addresses the property testing of $k$-submodular functions, a high-dimensional generalization of submodularity defined over partial partitions of a ground set that simultaneously satisfies diminishing marginal returns and pairwise monotonicity constraints. Accounting for structural differences under $\ell_p$ and Hamming distances, the paper introduces two types of local refutation patterns—violating squares and triangles—and analyzes the combinatorial barriers arising from their conflicting repair requirements. Leveraging techniques including implicit learning over hypergrids, partial-partition filters, ideal repairs, pseudo-DNF representations, and product-domain learning, the authors construct a constant-query, non-adaptive, one-sided tester under $\ell_p$ distance. For the Hamming distance, they design subexponential-query testers for two constituent subproperties and present the first adaptive tester for monotone $k$-submodularity with bounded value ranges.

This work addresses the challenge of ensuring information-flow security when dynamically extending security lattices in concurrent systems. By extending an existing type system, it introduces—for the first time within the π-calculus—a scalable security lattice mechanism that supports runtime insertion of new security levels. The authors rigorously establish non-interference through reduction semantics and bisimulation equivalence. This approach overcomes the limitations of traditional static, binary security lattices by providing a formal verification framework that guarantees strict information isolation between high- and low-security levels, even as security policies are dynamically adjusted at runtime.

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 proposes the first three-tier hierarchical explanatory framework for lattice-based post-quantum cryptography (PQC), such as ML-KEM and ML-DSA, to enhance the technical intelligibility and transparency of PQC security assumptions. The theoretical tier characterizes security boundaries through computational complexity classifications; the mathematical tier deepens structural understanding of lattice problems by integrating combinatorial Hodge theory and polyhedral geometry; and the experimental tier implements an empirical Julia-based platform to quantitatively analyze the behavior of lattice basis reduction algorithms like LLL and BKZ in low dimensions. While introducing no new attacks or hardness results, this framework systematically bridges formal proofs, mathematical foundations, and implementation characteristics, substantially strengthening the structural interpretability of PQC security assumptions.

This work addresses the lack of a systematic algebraic framework for composition and decomposition in existing logic programming, which hinders modular analysis and construction. It introduces set-like operations for propositional Horn logic programs and establishes, for the first time, that any minimal logic program can be precisely decomposed into Krom programs—where each rule contains at most one premise atom—and that the global least model can be faithfully reconstructed from the semantics of its components. For general logic programs, an effective approximate decomposition method is also provided. By integrating the algebraic structure of propositional logic programs, least model semantics, and Krom decomposition techniques, this study lays a novel algebraic foundation for modular decomposition and semantic reconstruction, thereby advancing compositional reasoning and program construction in logic programming.