Categorical semiotics: Foundations for Knowledge Integration
Carlos Leandro
TL;DR
This work tackles the problem of integrating knowledge across diverse models by proposing a rigorous, graphical, and fuzzy specification framework grounded in Ehresmann sketches and sign systems. It extends algebraic specification methods to handle both deterministic and non-deterministic architectures, enabling seamless knowledge integration across domain experts and machine-generated insights. The core contributions include the development of monoidal logics, Omega-sets, multi-morphisms, multi-diagrams, libraries of components, and semiotic systems, together with semantic notions for limits, colimits, and integration of semiotics. The approach offers a principled pathway to model, query, and merge heterogeneous knowledge representations with formal semantics, potentially enhancing interoperability, reasoning, and learning in complex AI systems.
Abstract
The integration of knowledge extracted from diverse models, whether described by domain experts or generated by machine learning algorithms, has historically been challenged by the absence of a suitable framework for specifying and integrating structures, learning processes, data transformations, and data models or rules. In this work, we extend algebraic specification methods to address these challenges within such a framework. In our work, we tackle the challenging task of developing a comprehensive framework for defining and analyzing deep learning architectures. We believe that previous efforts have fallen short by failing to establish a clear connection between the constraints a model must adhere to and its actual implementation. Our methodology employs graphical structures that resemble Ehresmann's sketches, interpreted within a universe of fuzzy sets. This approach offers a unified theory that elegantly encompasses both deterministic and non-deterministic neural network designs. Furthermore, we highlight how this theory naturally incorporates fundamental concepts from computer science and automata theory. Our extended algebraic specification framework, grounded in graphical structures akin to Ehresmann's sketches, offers a promising solution for integrating knowledge across disparate models and domains. By bridging the gap between domain-specific expertise and machine-generated insights, we pave the way for more comprehensive, collaborative, and effective approaches to knowledge integration and modeling.