Generalizing spiders

Abstract

Recent times have seen various formal diagrammatic logics and reasoning systems emerging [1, 4, 5, 7]. Many of these logics are based on the popular and intuitive Euler diagrams; see [6] for an overview. The diagrams in figure 1 are all based on Euler diagrams and are examples of unitary diagrams. Compound diagrams are formed by joining unitary diagrams using connectives such as ‘or’. We generalize the syntax of spider diagrams (of which d 3 in figure 1 is an example), increasing the expressiveness of the unitary system. These generalizations give rise to a more natural way of expressing some statements because there is an explicit mapping from the statement to a generalized diagram. Our theoretical motivation is to provide the necessary underpinning required to develop efficient automated theorem proving techniques: developing such techniques for compound systems is challenging and enhancing the expressiveness of unitary diagrams will enable more theorems to be proved efficiently.