相关论文: Topos Theory for Generative AI and LLMs

In this survey, we provide an overview of category theory-derived machine learning from four mainstream perspectives: gradient-based learning, probability-based learning, invariance and equivalence-based learning, and topos-based learning.…

Modal logics have proved useful for many reasoning tasks in symbolic artificial intelligence (AI), such as belief revision, spatial reasoning, among others. On the other hand, mathematical morphology (MM) is a theory for non-linear analysis…

This paper introduces the Token Space framework, a novel mathematical construct designed to enhance the interpretability and effectiveness of deep learning models through the application of category theory. By establishing a categorical…

This paper explores the interplay between category theory, topology, and the algebraic theory of finite groups. Our analysis unfolds in three stages. First, we establish the foundational universe of our objects: the complete and cocomplete…

This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. Our basic…

We introduce the theory of generalised ultracategories, these are relational extensions to ultracategories as defined by Lurie. An essential example of generalised ultracategories are topological spaces, and these play a fundamental role in…

In this paper, we propose GAIA, a generative AI architecture based on category theory. GAIA is based on a hierarchical model where modules are organized as a simplicial complex. Each simplicial complex updates its internal parameters biased…

This article is an introduction to the basic generalized category theory used in recent work on an extension of the theory of categories and categorical logic, including parts of topos theory. We discuss functors, equivalences, natural…

We present a method for automatically constructing a concept hierarchy for a given domain by querying a large language model. We apply this method to various domains using OpenAI's GPT 3.5. Our experiments indicate that LLMs can be of…

Large Language Models (LLMs) have revolutionized AI systems by enabling communication with machines using natural language. Recent developments in Generative AI (GenAI) like Vision-Language Models (GPT-4V) and Gemini have shown great…

This paper presents preliminary work on a general system for integrating dependent types into substructural type systems such as linear logic and linear type theory. Prior work on this front has generally managed to deliver type systems…

Compound AI Systems (CAIS) are an emerging paradigm that integrates large language models (LLMs) with external components, including retrievers, agents, tools, and orchestrators, to overcome the limitations of standalone models in tasks…

A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We…

Complex systems of systems (SoS) are characterized by multiple interconnected subsystems. Typically, each subsystem is designed and analyzed using methodologies and formalisms that are specific to the particular subsystem model of…

Topoi are categories which have enough structure to interpret higher order logic. They admit two notions of morphism: logical morphisms which preserve all of the structure and therefore the interpretation of higher order logic, and…

We introduce an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts; such theories are obtained from representations of certain atomic two-valued toposes as toposes of…

The rise of generative large language models (LLMs) has opened new opportunities for automating knowledge representation through concept maps, a long-standing pedagogical tool valued for fostering meaningful learning and higher-order…

We consider the category Grpd(Asm$(A)$) of groupoids defined internally to the category of assemblies on a partial combinatory algebra $A$. In this thesis we exhibit the structure of a $\pi$-tribe on Grpd(Asm$(A)$) showing the category to…

Designing functional transition metal complexes (TMCs) faces challenges due to the vast search space of metals and ligands, requiring efficient optimization strategies. Traditional genetic algorithms (GAs) are commonly used, employing…

This paper explores a top-down approach to automating incremental advances in machine learning research through component-level innovation, facilitated by Large Language Models (LLMs). Our framework systematically generates novel…