design filtration techniques

This paper addresses the challenge of modeling stable paths in spaces lacking an explicit metric structure. We propose a covering-based filtration method grounded in the generalized Steinhaus distance, establishing a covering-driven theory of stable paths. Methodologically, we integrate Steinhaus filtering with the Mapper algorithm, formally define path stability via strict matching filtration stability, and prove its equivalence and determinacy with respect to Čech and Vietoris–Rips filtrations. Path stability is quantified using persistent homology and the bottleneck distance. Our key contribution is the first formulation of a covering filtration framework, accompanied by rigorous stability theorems, enabling topological analysis of data endowed with natural coverings—even in the absence of a metric. Experiments demonstrate that our approach generates interpretable, smooth genre-transition sequences on a movie dataset and significantly enhances model interpretability regarding subgroup relationships in FashionMNIST, validating its efficacy in recommender systems and interpretable machine learning.

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.

This work investigates monoid structures on the category of indexed containers to provide an algebraic characterization of monads on $mathbf{Set}^I$. The problem addressed is the lack of an intrinsic, compositional definition of monoids in the indexed container setting that faithfully captures monads. Methodologically, we define the tensor product as composition of the corresponding endofunctors, yielding the first internal, compositional definition of monoids in indexed containers; we rigorously prove a bijective correspondence between such monoids and monads on $mathbf{Set}^I$. Our contribution unifies classical effectful constructions—including state, writer, and free monads—within a single categorical framework, thereby establishing a foundational basis for effect modeling in dependently typed programming. The approach integrates higher-order category theory with dependent type theory, and all definitions and equivalence proofs have been fully formalized in Cubical Agda. The core innovation lies in establishing a precise, formally verified correspondence between indexed container monoids and monads, together with the first complete mechanized construction and proof.

This paper addresses the fragmentation and lack of interoperability among structural complexity measures across graph theory, geometric group theory, and dynamical systems. Methodologically, it introduces a unified “structured decomposition” framework grounded in category theory: (i) it is the first to formalize diverse domain-specific decomposition paradigms categorically; (ii) it establishes a general duality theory linking decompositions to object completions; and (iii) it defines composable width functors enabling cross-model quantification, comparison, and translation of structural complexity. Key contributions include: a unified categorical characterization of over ten complexity parameters—including treewidth, layered treewidth, and hypergraph treewidth—revealing their intrinsic structural relationships; and a novel parameterized tractability paradigm for NP-hard problems, grounded in decomposition width. The framework achieves both theoretical unification and algorithmic realizability.

This paper addresses the computational inefficiency of the lcm-filtration method in redundancy analysis, which stems from repeated construction of equivalent ideals. We propose a novel stepwise filtration framework grounded in algebraic topology and lattice theory; it tracks only non-equivalent generator additions and avoids redundant lcm-ideal constructions, thereby substantially reducing computational complexity. To our knowledge, this work presents the first systematic comparative analysis of lcm- versus stepwise filtrations, uncovering the algebraic nature of redundancy structures. Empirical evaluation across complex network analysis, system signature modeling, and sensitivity assessment demonstrates that the proposed framework reduces computational steps by 35%–62% on average, without compromising accuracy. It thus enables faster identification of critical components and more efficient reliability analysis.

This work addresses the problem of characterizing behavioral equivalence for time-varying stateful processes based solely on finite input-output observations, without relying on inaccessible internal states. To this end, it introduces a novel structure called the “discard double category,” which provides a unified framework for modeling partial, nondeterministic, probabilistic, and quantum processes, and constructs a functorial semantics into the free feedback category. By leveraging preordered enriched monoidal categories and closed relations over compact Hausdorff spaces, the paper establishes a categorical compactness theorem. This framework not only subsumes Willems’ behavioral theory of linear time-invariant systems as a special case but also offers a unified, internal-state-free foundation for behavioral semantics across a broad spectrum of process types.

本文通过建立代数架构理论，解决软件变更中的结构保存、全局一致性及变更分类等问题，提出了一种几何与局部模型等价的重构定理。

This work investigates the construction of filter lambda-models endowed with the sensibility property—namely, that all unsolvable terms are interpreted as the least element in the model. By viewing intersection type theories as specific meet-semilattices and leveraging morphisms in their dual categories together with the Tait–Girard computability method, the paper establishes, for the first time, a connection between sensibility and morphisms of meet-semilattices. The main contribution consists of two classes of intersection type theories that induce sensible filter models: the effective class not only supports concrete model constructions but also generalizes Mendler’s criterion to intersection types and head-normalizable terms. Nevertheless, a complete characterization of sensible filter models remains an open problem.

研究针对过滤气泡问题，通过开发静态和动态逻辑系统来推理过滤气泡的形成与维持机制。

本文通过群论和编码理论构建代数框架，分类并优化数据组织与编码的代数结构，提出一种基于公理签名的分类器，并展示了具体的编码构造和优化方法。