Conceptual orthospaces—Convexity meets negation

Abstract

Neural networks and in general subsymbolic learning approaches perform well on usual learning tasks, but they are black boxes lacking desired properties such as explainability or trustworthiness. Fully integrated symbolic/subsymbolic systems account for these properties, e.g., through the injection of symbolic information into a subsymbolic learning framework. Systems relying on embeddings can be considered as specific examples of such systems where facts, expressed in some logic, are embedded into a continuous space. Instances mentioned in the facts are mapped to points and the relations/concepts mentioned in the facts are mapped to a region in that space. This allows to exploit geometric regularities in order to tackle typical learning tasks such as link prediction. 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. However, to enable meaningful reasoning, embeddings of relations and concepts are not allowed to be shaped arbitrarily. Especially, convex sets turned out to be appropriate due to their computational advantages and due to their foundation in cognition. Convexity can be defined with a ternary betweenness relation and concepts can be defined as betweenness-closed (= convex) sets. Though many interesting phenomena of cognitive reasoning can be explained in such a framework, it is at least not obvious how to use betweenness for other, more logico-formal aspects of reasoning that, e.g., require defining logical operators. In particular, for the logical operator of negation other mathematical structures such as the orthoframes of Goldblatt (1974) [13] have proven more useful. In this paper, we provide results on the connection between convexity and orthoframes. In particular, we investigate the construction of convex concepts equipped with an orthogonality relation. We give a universal construction of a betweenness relation over an orthoframe and show that based on Euclidean betweenness (thus using the classical notion of convexity) and under some natural restrictions, convex cones are the only structures capable of modeling such a space.