frame construction

This paper addresses the semantic divide between classical and intuitionistic modal logic by introducing a novel frame semantics based on partially ordered “possibilities” instead of traditional possible worlds. Methodologically, it constructs possibility frames—interpreting formulas via regular open sets in Alexandrov topologies—and develops a categorical duality theory between such frames and non-atomic CV-Boolean algebraic operators (CV-BAOs). It establishes, for the first time, a duality between full possibility frames and CV-BAOs using filters rather than ultrafilters, thereby avoiding the Axiom of Choice; it also introduces principal possibility frames to characterize V-BAOs. The main contributions are: (i) unifying classical and intuitionistic modal semantics within a single framework; (ii) systematically establishing dualities between possibility frames and various classes of BAOs; (iii) completing definability, correspondence, and strong completeness theories; and (iv) proving that every BAO is fully characterized by a filter-descriptive possibility frame.

This paper addresses the lack of compositional semantics in probabilistic algorithms—particularly the bilateral filter-based Gaussian (BFFG) algorithm. To resolve this, the authors establish, for the first time, a rigorous connection between BFFG and optics in category theory: they model BFFG as a functor from the category of Markov kernels to the category of optics and prove that this functor carries a lax monad structure. This categorical construction exposes BFFG’s intrinsic compositional mechanism and provides it with a principled, category-theoretic semantic foundation. The key contribution lies in elevating classical stochastic algorithms to higher-order abstractions endowed with algebraic structure—enabling modular design, formal verification, and cross-model reuse. By unifying probabilistic computation with optic-based compositional principles, the work opens a new theoretical pathway for compositional reasoning about probabilistic programs and randomized algorithms.

Traditional filters—such as second-order filters—are limited to integer or half-integer orders, hindering continuous tunability of filter characteristics. To address this, we propose Rational-Order Generalized Elementary Filters (Rational-Order GEFs), establishing the first continuously parametrized framework supporting arbitrary rational-order exponents. Methodologically, we unify time- and frequency-domain representations—incorporating transfer functions, impulse responses, and integral formulations—while enforcing stability and causality constraints within a generalized second-order system design. This enables real-time, continuous adjustment of key parameters including filter order, quality factor ratio (Q), and group delay. Crucially, the framework achieves flexible specification of metrics such as 3 dB/15 dB bandwidth and Q/τₘₐₓ ratio without increasing analytical complexity, thereby jointly optimizing frequency selectivity and phase synchronization. The result significantly expands design freedom and engineering applicability of parametric filters.

Classical Shannon information theory assumes semantic irrelevance, failing to account for logical knowledge held by communicating agents. This work addresses the problem of minimizing communication cost when a sender (Alice) and receiver (Bob) possess partial, possibly heterogeneous, logical knowledge, such that Bob must logically deduce Alice’s private proposition via first-order deduction. Method: We propose the first semantic-aware communication theory integrating logic-based reasoning with information theory. Crucially, we formally incorporate Bob’s first-order deductive capability into the communication model, establishing a semantics-sensitive transmission paradigm. Combining coding theory, game-theoretic analysis, and algorithm design, we derive tight upper and lower bounds on semantic communication complexity. Contribution/Results: We introduce the first knowledge-driven communication framework supporting logical inference; derive optimal code-length bounds for multiple semantic scenarios; design an asymptotically optimal algorithm; and empirically demonstrate significant gains in transmission efficiency over classical schemes.

Existing machine learning frameworks suffer from insufficient formalization of objective functions and lack a unified, cross-domain behavioral design paradigm. Method: We propose an equation-constrained compositional function modeling approach for learners, constructing task graphs and compositional semantic graphs to enable model-agnostic behavioral specification and optimization. We introduce a novel task-oriented pattern language framework and the “manipulator” task paradigm, supporting end-to-end, architecture-agnostic, and adversarial-training-free minimal editing of data attributes. Contribution/Results: Theoretically, our work integrates formal methods and theoretical computer science principles. Empirically, we demonstrate precise, controllable, and interpretable behavioral editing on small-scale models under stable training—without stochastic sampling or data intervention—yielding significant improvements in deployment efficiency and formal verifiability.

This work investigates the design of pooling-free scattering networks employing fixed monomial nonlinearities to maximize separability for data with low intrinsic dimensionality. By integrating frame theory, geometric measure theory, and moment analysis, the study provides the first geometric characterization of a scattering network’s separation capacity, establishing theoretical bounds for feature extractors operating on low-dimensional rectifiable data. The core contribution consists of two practical design principles: the network’s filters must span a sufficiently broad frequency range, and the frame formed by these filters—when coupled with the data’s geometric structure through a coupling matrix—must exhibit a well-conditioned condition number. These criteria jointly ensure significantly enhanced separation performance, offering concrete guidance for the construction of effective scattering architectures tailored to geometrically structured low-dimensional data.

This work addresses the challenge of learning from multimodal graph data in federated settings, where privacy constraints prevent centralized data sharing and existing methods struggle to trace the joint decision-making process involving modality evidence, node semantics, and topological context. To this end, the authors propose FedLAB, a novel framework that introduces, for the first time, a typed hierarchical semantic codebook equipped with a native semantic-traceable interface. FedLAB enables explicit modeling of the joint reasoning pathways among modalities, semantics, and topology through federated semantic centroid pretraining, all while preserving data locality. Extensive experiments demonstrate that FedLAB consistently outperforms state-of-the-art methods by up to 7.53% on average across ten benchmark datasets and six downstream tasks, achieving a unified balance between high predictive accuracy and semantic traceability.

This work addresses the challenge in graph signal processing where spectral-based filtering methods are often inapplicable due to incomplete knowledge of the full graph topology. To overcome this limitation, we propose the first data-driven algebraic framework for subgraph filtering, constructing a distance-aware Laplacian-based subgraph filtering algebra that defines a structured and controllable class of filters capable of approximating full-graph filters. Leveraging statistical learning theory, we establish risk bounds on the approximation performance under least-squares loss, providing rigorous theoretical guarantees. Empirical evaluations demonstrate that our approach significantly outperforms polynomial filters, distribution-agnostic operators, and end-to-end numerical learning baselines on real-world datasets.

This work proposes a learnable inter-filter connectivity mechanism that replaces the fixed pointwise nonlinear activations in conventional convolutional neural networks with a parameterized, universal connection function embedded within convolutional layers. By enabling adaptive interactions among filters, the approach overcomes the limitations of traditional fixed logical operations—such as multiplication or minimum selection—and allows the network to automatically optimize its connectivity strategy through end-to-end training. Experimental results demonstrate that this method significantly improves classification accuracy, confirming its effectiveness in enhancing both model expressivity and generalization capability.

This study investigates the definability of nine function properties in the modal-temporal language \(L_{T\times W}\), which combines modal operator \(\Box\) with Priorian temporal operators \(G\) and \(H\), across five classes of ordered structures. By integrating modal logic and Priorian tense logic, the work introduces ordered frame semantics, minimal function frames (the \(O^2\) family), indexed languages, and a uniform domain condition to systematically analyze expressive power under both standard and strict semantics. The key findings reveal that function multiplicity is the primary constraint on definability; once multiplicity is controlled, strict semantics can define properties such as injectivity over reflexive orders, whereas the absence of connectedness in non-linear orders poses an inherent obstacle. The research further shows that in the original multi-flow setting, the language is weakly expressive and the two semantics coincide, yet under restricted frames most properties become definable, with three multiplicity-control mechanisms yielding consistent definability patterns.