Related papers: Morpho-logic from a Topos Perspective: Application…
A general definition of mathematical morphology has been defined within the algebraic framework of complete lattice theory. In this framework, dealing with deterministic and increasing operators, a dilation (respectively an erosion) is an…
Mathematical morphology contributes many profitable tools to image processing area. Some of these methods considered to be basic but the most important fundamental of data processing in many various applications. In this paper, we modify…
A new model of symbol grounding is presented, in which the structures of natural language, logical semantics, perception and action are represented categorically, and symbol grounding is modeled via the composition of morphisms between the…
We define the notion of a model of higher-order modal logic in an arbitrary elementary topos $\mathcal{E}$. In contrast to the well-known interpretation of (non-modal) higher-order logic, the type of propositions is not interpreted by the…
A new set of mathematical morphology (MM) operators adaptive to illumination changes caused by variation of exposure time or light intensity is defined thanks to the Logarithmic Image Processing (LIP) model. This model based on the physics…
Several tasks in artificial intelligence require to be able to find models about knowledge dynamics. They include belief revision, fusion and belief merging, and abduction. In this paper we exploit the algebraic framework of mathematical…
Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal…
We establish a novel connection between two research areas in non-classical logics which have been developed independently of each other so far: on the one hand, input/output logic, introduced within a research program developing logical…
Morphisms, structure preserving maps, are everywhere in Mathematics as useful tools for thinking and problem solving, or as objects to study. Here, we argue that the idea of operations being compatible across two domains goes beyond its…
Several logical operators are defined as dual pairs, in different types of logics. Such dual pairs of operators also occur in other algebraic theories, such as mathematical morphology. Based on this observation, this paper proposes to…
In this paper we provide a unifying description of different types of semantics of modal logic found in the literature via the framework of topological categories. In the style of categorical logic, we establish an exact correspondence…
Symbolic learning represents the most straightforward approach to interpretable modeling, but its applications have been hampered by a single structural design choice: the adoption of propositional logic as the underlying language.…
Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements &…
We propose the design of novel categorical generative AI architectures (GAIAs) using topos theory, a type of category that is ``set-like": a topos has all (co)limits, is Cartesian closed, and has a subobject classifier. Previous theoretical…
Large Language Models (LLMs) have demonstrated impressive capabilities in structured reasoning and symbolic tasks, with coding emerging as a particularly successful application. This progress has naturally motivated efforts to extend these…
Applying logic in the area of conceptual modeling has been investigated widely, yet there has been limited uptake of logic-based conceptual modeling in industry. According to some researchers, another formalization of such tools as EER or…
In this work, we introduce Contextual Analog Logic with Multimodality (CALM). CALM unites symbolic reasoning with neural generation, enabling systems to make context-sensitive decisions grounded in real-world multi-modal data. Background:…
Mathematical reasoning and optimization are fundamental to artificial intelligence and computational problem-solving. Recent advancements in Large Language Models (LLMs) have significantly improved AI-driven mathematical reasoning, theorem…
Spatial aspects of computation are becoming increasingly relevant in Computer Science, especially in the field of collective adaptive systems and when dealing with systems distributed in physical space. Traditional formal verification…
Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal…