Symbolic Logic and Category Theory for Reasoning-Enabled Semantic Communications

Semantic communications (SC) offers a promising paradigm to amplify the resource efficiency of forthcoming wire-less systems by transmitting only messages that hold relevance to end users. Assessing the potential of SC systems necessitates a reevaluation of semantic information. Traditionally, semantic information has been confined to set-theoretic formulations, neglecting end users' deductive and abductive reasoning capacities. Contrary to the prior art, in this paper, a fresh framework for semantic information leveraging category theory and symbolic logic is introduced. In particular, we have proposed a neurosym-bolic artificial intelligence-based semantic communication (SC) system that encodes source data, represented as a set of logical statements, into a semantic representation. The symbolic component, structured as a first-order logical language, incorporates complex logical formulas designed to facilitate logical reasoning at the receiver nodes. Furthermore, a rigorous formulation of the semantics conveyed through the logical statements are developed through the concept of Fibrations. Finally, the mapping of the logical language onto a neural network space, while preserving the compositional structure of the language, is achieved using Grothendieck toposes in category theory. A novel formulation of semantic information is proposed, deriving it as the summation of represented information-which incorporates compositional knowledge enabled by the Grothendieck topos framework-and reasoned information.