Topological Semantics Research Articles

Intuitionistic S4 as a logic of topological spaces

Abstract We design and study various topological semantics for the diamond-free intuitionistic modal logic $\textsf{iS4}$, an intuitionistic analogue of $\textsf{S4}$. Ultimately we prove that ordinary topological spaces can be used as semantics, using the specialization order to interpret intuitionistic implication and the interior for the modality. Some of our soundness and completeness results are mechanised in Coq.

Indoor mobility data encoding with TSTM-in: A topological-semantic trajectory model

The growing ubiquity of location/activity sensing technologies has created unprecedented opportunities for research on human spatiotemporal interaction behavior in mobile environments. However, existing studies of human mobility need to sufficiently account for the association of indoor scenes with the semantics of human behavior. This paper introduces TSTM-in, a trajectory model that combines trajectory data and indoor scenes using topological semantic modeling, semantic trajectory reconstruction, and trajectory queries. The model effectively manages indoor semantic trajectory data and extracts topological behavioral semantics by incorporating important points across a trajectory to reflect the semantics of key points connected to indoor corridors and regions. These topological semantics facilitate the creation of a flexible intersection-based indoor semantic trajectory reconstruction. Reconstructed semantic trajectories represent human mobility by integrating semantic data sets along the time axis. A case study with real-world trajectory queries from travelers demonstrates the model's effectiveness. TSTM-in realizes the association of indoor scenes with human behavior semantics, supporting the construction of mobile object management applications for indoor scenes and providing scientific and reasonable spatiotemporal semantic information description for location service-based intelligent cities.

Living and dying on the edge in the digital age. An interview with Andrei Vieru on why closure and boundaries matter in science, art, and life

ABSTRACT This conversation elaborates on the latest work of the Paris-based writer, philosopher, translator, mathematician and pianist Andrei Vieru, where he maintains that human freedom is inseparable from the idea of closure. Vieru’s model is rooted in the topological semantics of modal logic and provides valuable epistemological insights for the broader community of scholars (death and grief scholars, social scientists, humanists) as well as for policymakers, civil social actors and, generally, for all existentially concerned individuals in times of eroding democracies and pressing global risks. In this dialogue conducted during 2023 in face-to-face sessions and via email exchanges between Paris and Bucharest, Vieru reflects on what is at stake philosophically, artistically and existentially when humans are confronted with the radical openness of possibilities brought by contemporary challenges like global crises and digital technologies.

Identification of gene biomarkers for brain diseases via multi-network topological semantics extraction and graph convolutional network

BackgroundBrain diseases pose a significant threat to human health, and various network-based methods have been proposed for identifying gene biomarkers associated with these diseases. However, the brain is a complex system, and extracting topological semantics from different brain networks is necessary yet challenging to identify pathogenic genes for brain diseases.ResultsIn this study, we present a multi-network representation learning framework called M-GBBD for the identification of gene biomarker in brain diseases. Specifically, we collected multi-omics data to construct eleven networks from different perspectives. M-GBBD extracts the spatial distributions of features from these networks and iteratively optimizes them using Kullback–Leibler divergence to fuse the networks into a common semantic space that represents the gene network for the brain. Subsequently, a graph consisting of both gene and large-scale disease proximity networks learns representations through graph convolution techniques and predicts whether a gene is associated which brain diseases while providing associated scores. Experimental results demonstrate that M-GBBD outperforms several baseline methods. Furthermore, our analysis supported by bioinformatics revealed CAMP as a significantly associated gene with Alzheimer's disease identified by M-GBBD.ConclusionCollectively, M-GBBD provides valuable insights into identifying gene biomarkers for brain diseases and serves as a promising framework for brain networks representation learning.

Wansing's bi-intuitionistic logic: semantics, extension and unilateralisation

ABSTRACT The well-known algebraic semantics and topological semantics for intuitionistic logic ( Int ) is here extended to Wansing's bi-intuitionistic logic ( 2 Int ). The logic 2 Int is also characterised by a quasi-twist structure semantics, which leads to an alternative topological characterisation of 2 Int . Later, notions of Fregean negation and of unilateralisation are proposed. The logic 2 Int is extended with a ‘Fregean negation’ connective ∼ , obtaining 2 In t ∼ , and it is showed that the logic N 4 ⋆ (an extension of Nelson's paraconsistent logic) results to be the unilateralisation of 2 In t ∼ via ∼ . The logic 2 In t ∼ is also characterised by a Kripke-style semantics, a twist structure semantics and a topological semantics.

Dynamic Cantor Derivative Logic

Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as $d$-logics. Unlike logics based on the topological closure operator, $d$-logics have not previously been studied in the framework of dynamical systems, which are pairs $(X,f)$ consisting of a topological space $X$ equipped with a continuous function $f\colon X\to X$. We introduce the logics $\bf{wK4C}$, $\bf{K4C}$ and $\bf{GLC}$ and show that they all have the finite Kripke model property and are sound and complete with respect to the $d$-semantics in this dynamical setting. In particular, we prove that $\bf{wK4C}$ is the $d$-logic of all dynamic topological systems, $\bf{K4C}$ is the $d$-logic of all $T_D$ dynamic topological systems, and $\bf{GLC}$ is the $d$-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where $f$ is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems $\bf{wK4H}$, $\bf{K4H}$ and $\bf{GLH}$. The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological $d$-logics. Furthermore, our result for $\bf{GLC}$ constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation -- something known to be impossible over the class of all spaces.

Fixed point logics and definable topological properties

AbstractModal logic enjoys topological semantics that may be traced back to McKinsey and Tarski, and the classification of topological spaces via modal axioms is a lively area of research. In the past two decades, there has been interest in extending topological modal logic to the language of the mu-calculus, but previously no class of topological spaces was known to be mu-calculus definable that was not already modally definable. In this paper, we show that the full mu-calculus is indeed more expressive than standard modal logic, in the sense that there are classes of topological spaces (and weakly transitive Kripke frames), which are mu-definable but not modally definable. The classes we exhibit satisfy a modally definable property outside of their perfect core, and thus we dub them imperfect spaces. We show that the mu-calculus is sound and complete for these classes. Our examples are minimal in the sense that they use a single instance of a greatest fixed point, and we show that least fixed points alone do not suffice to define any class of spaces that is not already modally definable.

Combining prompt learning with contextual semantics for inductive relation prediction

Inductive relation prediction for knowledge graphs aims to predict missing relations between two new entities. Most previous studies on relation prediction are limited to the transductive setting and could not be applied to it. Recently, some inductive methods have been proposed to handle it by learning the topological semantics. However, they solely rely on structural information, disregarding the role of prior knowledge. In cases of sparse structures, this limitation is magnified, thereby hindering the inductive ability. Prior knowledge can not only filter out invalid topological structures but also complement the topological semantics. To this end, We propose a novel inductive model, PLCS, which incorporates prompt learning with contextual semantics to fully exploit prior knowledge. To filter out irrelevant topological structures, we innovatively employ hard prompts to mine prior knowledge in pre-trained language models (PLMs) as the basis for subgraph extraction. Additionally, we enhance semantic representation by integrating relation text descriptions into relation embeddings during initialization, supplementing topological semantics. The experimental results on four benchmark datasets show the superiority of PLCS over existing state-of-the-art methods.

Carnap’s Problem for Intuitionistic Propositional Logic

We show that intuitionistic propositional logic is Carnap categorical: the only interpretation of the connectives consistent with the intuitionistic consequence relation is the standard interpretation. This holds with respect to the most well-known semantics relative to which intuitionistic logic is sound and complete; among them Kripke semantics, Beth semantics, Dragalin semantics, topological semantics, and algebraic semantics. These facts turn out to be consequences of an observation about interpretations in Heyting algebras.

Learning Topology-Specific Experts for Molecular Property Prediction

Recently, graph neural networks (GNNs) have been successfully applied to predicting molecular properties, which is one of the most classical cheminformatics tasks with various applications. Despite their effectiveness, we empirically observe that training a single GNN model for diverse molecules with distinct structural patterns limits its prediction performance. In this paper, motivated by this observation, we propose TopExpert to leverage topology-specific prediction models (referred to as experts), each of which is responsible for each molecular group sharing similar topological semantics. That is, each expert learns topology-specific discriminative features while being trained with its corresponding topological group. To tackle the key challenge of grouping molecules by their topological patterns, we introduce a clustering-based gating module that assigns an input molecule into one of the clusters and further optimizes the gating module with two different types of self-supervision: topological semantics induced by GNNs and molecular scaffolds, respectively. Extensive experiments demonstrate that TopExpert has boosted the performance for molecular property prediction and also achieved better generalization for new molecules with unseen scaffolds than baselines. The code is available at https://github.com/kimsu55/ToxExpert.

Topological duality for orthomodular lattices

AbstractA class of ordered relational topological spaces is described, which we call orthomodular spaces. Our construction of these spaces involves adding a topology to the class of orthomodular frames introduced by Hartonas, along the lines of Bimbó's topologization of the class of orthoframes employed by Goldblatt in his representation of ortholattices. We then prove that the category of orthomodular lattices and homomorphisms is dually equivalent to the category of orthomodular spaces and certain continuous frame morphisms, which we call continuous weak p‐morphisms. It is well‐known that orthomodular lattices provide an algebraic semantics for the quantum logic . Hence, as an application of our duality, we develop a topological semantics for using orthomodular spaces and prove soundness and completeness.

Justified belief, knowledge, and the topology of evidence

We propose a new topological semantics for evidence, evidence-based justifications, belief, and knowledge. Resting on the assumption that an agent’s rational belief is based on the available evidence, we try to unveil the concrete relationship between an agent’s evidence, belief, and knowledge via a rich formal framework afforded by topologically interpreted modal logics. We prove soundness, completeness, decidability, and the finite model property for the associated logics, and apply this setting to analyze key epistemological issues such as “no false lemma” Gettier examples, misleading defeaters, undefeated justification versus undefeated belief, as well as the defeasibility theories of knowledge.

On algebraic and topological semantics of the modal logic of common knowledge S4CI

Abstract For the modal logic $\textsf {S4}^{C}_{I}$, we identify the class of completable $\textsf {S4}^{C}_{I}$-algebras and prove for them a Stone-type representation theorem. As a consequence, we obtain strong algebraic and topological completeness of the logic $\textsf {S4}^{C}_{I}$ in the case of local semantic consequence relations. In addition, we consider an extension of the logic $\textsf {S4}^{C}_{I}$ with certain infinitary derivations and establish the corresponding strong completeness results for the enriched system in the case of global semantic consequence relations.

Visual learning graph convolution for multi-grained orange quality grading

The quality of oranges is grounded on their appearance and diameter. Appearance refers to the skin’s smoothness and surface cleanliness; diameter refers to the transverse diameter size. They are visual attributes that visual perception technologies can automatically identify. Nonetheless, the current orange quality assessment needs to address two issues: 1) There are no image datasets for orange quality grading; 2) It is challenging to effectively learn the fine-grained and distinct visual semantics of oranges from diverse angles. This study collected 12 522 images from 2 087 oranges for multi-grained grading tasks. In addition, it presented a visual learning graph convolution approach for multi-grained orange quality grading, including a backbone network and a graph convolutional network (GCN). The backbone network’s object detection, data augmentation, and feature extraction can remove extraneous visual information. GCN was utilized to learn the topological semantics of orange feature maps. Finally, evaluation results proved that the recognition accuracy of diameter size, appearance, and fine-grained orange quality were 99.50, 97.27, and 97.99%, respectively, indicating that the proposed approach is superior to others.

A topological model of trajectories with road network space

AbstractSpatiotemporal topological semantic information has attracted a lot of attention recently when exploring high‐level human movement features in geographic contexts. Although topological semantics between a trajectory and individual spatial objects have been studied extensively, few studies have focused on movements in the specific context of spatial infrastructures, such as road networks. This study proposes a topological trajectory model that integrates the topological semantics of trajectory data related to road networks and regions in a unified framework. The proposed model considers key points during a trajectory and establishes a topological intersection model to describe key points’ semantics. With the topological semantics of the trajectory–road network obtained, the road network’s semantic relation is derived directly from the topological relation between road networks and regions. This model can intuitively express the semantic relations of the three elements, trajectory, road network, and region, which helps spatial queries and analyses of network constrained movements. A case study of real traffic data shows that more detailed semantic information can be obtained based on this model.

Closure System and Its Semantics

It is well known that topological spaces are axiomatically characterized by the topological closure operator satisfying the Kuratowski Closure Axioms. Equivalently, they can be axiomatized by other set operators encoding primitive semantics of topology, such as interior operator, exterior operator, boundary operator, or derived-set operator (or dually, co-derived-set operator). It is also known that a topological closure operator (and dually, a topological interior operator) can be weakened into generalized closure (interior) systems. What about boundary operator, exterior operator, and derived-set (and co-derived-set) operator in the weakened systems? Our paper completely answers this question by showing that the above six set operators can all be weakened (from their topological counterparts) in an appropriate way such that their inter-relationships remain essentially the same as in topological systems. Moreover, we show that the semantics of an interior point, an exterior point, a boundary point, an accumulation point, a co-accumulation point, an isolated point, a repelling point, etc. with respect to a given set, can be extended to an arbitrary subset system simply by treating the subset system as a base of a generalized interior system (and hence its dual, a generalized closure system). This allows us to extend topological semantics, namely the characterization of points with respect to an arbitrary set, in terms of both its spatial relations (interior, exterior, or boundary) and its dynamic convergence of any sequence (accumulation, co-accumulation, and isolation), to much weakened systems and hence with wider applicability. Examples from the theory of matroid and of Knowledge/Learning Spaces are used as an illustration.

Topological semantics of conservativity and interpretability logics

AbstractWe introduce and develop a topological semantics of conservativity logics and interpretability logics. We prove the topological compactness theorem of consistent normal extensions of the conservativity logic $\textbf {CL}$ by extending Shehtman’s ultrabouquet construction method to our framework. As a consequence, we prove that several extensions of $\textbf {CL}$ such as $\textbf {IL}$, $\textbf {ILM}$, $\textbf {ILP}$ and $\textbf {ILW}$ are strongly complete with respect to our topological semantics.

Neighbourhood semantics and labelled calculus for intuitionistic infinitary logic

Abstract Neighbourhood semantics for intuitionistic logic extended with countable conjunctions and disjunctions is introduced and shown equivalent to topological semantics, with an indirect completeness proof as payoff. The new semantics is used to obtain a labelled sequent calculus with good structural properties. In particular, admissibility of weakening and contraction, invertibility with preservation of height for each rule and cut elimination are shown. Finally, a direct Tait–Schütte–Takeuti style form of completeness via the extraction of a countermodel from a failed proof search is proved.

Prototypes, poles, and tessellations: towards a topological theory of conceptual spaces

The aim of this paper is to present a topological method for constructing discretizations (tessellations) of topological conceptual spaces. The method works for a class of topological spaces that the Russian mathematician Pavel Alexandroff defined more than 80 years ago. The aim of this paper is to show that Alexandroff spaces, as they are called today, have many interesting properties that can be used to explicate and clarify a variety of problems in philosophy, cognitive science, and related disciplines. For instance, recently, Ian Rumfitt used a special type of Alexandroff spaces to elucidate the logic of vague concepts in a new way. Moreover, Rumfitt’s class of Alexandroff spaces can be shown to provide a natural topological semantics for Susanne Bobzien’s “logic of clearness”. Mainly due to the work of Peter Gardenfors and his collaborators, conceptual spaces have become an increasingly popular tool of dealing with a variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics and philosophy. For Gardenfors’s conceptual spaces, geometrically defined discretizations (so-called Voronoi tessellations) play an essential role. These tessellations can be shown to be extensionally equivalent to topological tessellations that can be constructed for Alexandroff spaces in general. Thereby, Rumfitt’s and Gardenfors’s constructions turn out to be special cases of an approach that works for a more general class of spaces, namely, for weakly scattered Alexandroff spaces. The main aim of this paper is to show that this class of spaces provides a convenient framework for conceptual spaces as used in epistemology and related disciplines in general. Weakly scattered Alexandroff spaces are useful for elucidating problems related to the logic of vague concepts, in particular they offer a solution of the Sorites paradox (Rumfitt). Further, they provide a semantics for the logic of clearness (Bobzien) that overcomes certain problems of the concept of higher-order vagueness. Moreover, these spaces help find a natural place for classical syllogistics in the framework of conceptual spaces. The specialization order of Alexandroff spaces can be used to refine the all-or-nothing distinction between prototypical and non-prototypical stimuli in favor of a gradual distinction between more or less prototypical elements of conceptual spaces. The greater conceptual flexibility of the topological approach helps avoid some inherent inadequacies of the geometrical approach, for instance, the so-called “thickness problem”. Finally, it is shown that the Alexandroff spaces offer an appropriate framework to deal with digital conceptual spaces that are gaining more and more importance in the areas of artificial intelligence, computer science and related disciplines.

Some Formal Semantics for Epistemic Modesty

Given the frequency of human error, it seems rational to believe that some of our own rational beliefs are false. This is the axiom of epistemic modesty. Unfortunately, using standard propositional quantification, and the usual relational semantics, this axiom is semantically inconsistent with a common logic for rational belief, namely KD45. Here we explore two alternative semantics for KD45 and the axiom of epistemic modesty. The first uses the usual relational semantics and bisimulation quantifiers. The second uses a topological semantics and standard propositional quantification. We show the two different semantics validate many of the same formulas, though we do not know whether they validate exactly the same formulas. Along the way we address various philosophical concerns.