Conceptual Orthospaces—An Embedding Framework Accounting for Negation Operators and Convexity Constraints

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Abstract One aim of neurosymbolic AI is to enhance subsymbolic learning approaches with symbolic information, e.g., in the form of background knowledge, to increase result quality, interpretability, and trustworthiness. To model subsymbolic information on facts and their similarities, embeddings into some vector space have been proven useful. Symbolic information can then be modeled directly in the space by representing concepts as some geometric objects and logical operators as geometrical operations. These embeddings provide the first step towards filling the gap between qualitative, Tarskian style semantics, which is used for deductive reasoning over the facts, and quantitative structures, which are used for representing objects, relations, and concepts for learning purposes. To enable meaningful reasoning, embeddings of concepts are not allowed to be arbitrarily shaped. Particularly convex sets turned out to be useful due to their computational advantages and due to their foundation in cognition. However, in this context, the geometric modeling of logical operators turns out to be challenging, particularly the modeling of negation. Therefore, this paper discusses a general framework for embeddings able to model an expressive negation. It is general in the sense that it allows for relaxations of the convexity constraint and for different strengths of negation. Thus, the proposed framework can act as a basis for interpretable embedding approaches of adaptable strength and thus fills a gap in the current embedding landscape by allowing for embeddings of expressive logics.