Euler diagrams have a wide variety of uses, from information visualization to logical reasoning. In all of their application areas, the ability to automatically layout Euler diagrams brings considerable benefits. In this paper, we present a novel approach to Euler diagram generation. We develop certain graphs associated with Euler diagrams in order to allow curves to be added by finding cycles in these graphs. This permits us to build Euler diagrams inductively, adding one curve at a time. Our technique is adaptable, allowing the easy specification, and enforcement, of sets of well-formedness conditions; we present a series of results that identify properties of cycles that correspond to the well-formedness conditions. This improves upon other contributions toward the automated generation of Euler diagrams which implicitly assume some fixed set of well-formedness conditions must hold. In addition, unlike most of these other generation methods, our technique allows any abstract description to be drawn as an Euler diagram. To establish the utility of the approach, a prototype implementation has been developed.

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Abstract. Concept diagrams were introduced for precisely specifying ontologies in a manner more readily accessible to developers and other stakeholders than symbolic notations. In this paper, we present a case study on the use of concept diagrams in visually specifying the Semantic Sensor Networks (SSN) ontology. The SSN ontology was originally developed by an Incubator Group of the W3C. In the ontology, a sensor is a physical object that implements sensing and an observation is observed by a single sensor. These, and other, roles and concepts are captured visually, but precisely, by concept diagrams. We consider the lessons learnt from developing this visual model and show how to convert description logic axioms into concept diagrams. We also demonstrate how to merge simple concept diagram axioms into more complex axioms, whilst ensuring that diagrams remain relatively uncluttered.

Diagrammatic reasoning has the potential to be important in numerous application areas. This paper focuses on the simple, but widely used, Euler diagrams that form the basis of many more expressive logics. We have implemented a diagrammatic theorem prover, called Edith, which has access to four sound and complete sets of reasoning rules for Euler diagrams. Furthermore, for each rule set we develop a sophisticated heuristic to guide the search for a proof. This paper is about understanding how the choice of reasoning rule set affects the time taken to find proofs. Such an understanding will influence reasoning rule design in other logics. Moreover, this work specific to Euler diagrams directly benefits the many logics based on Euler diagrams. We investigate how the time taken to find a proof depends not only on the proof task but also on the reasoning system used. Our evaluation allows us to predict the best choice of reasoning system, given a proof task, in terms of time taken, and we extract a guide for defining reasoning rules for other logics in order to minimize time requirements.

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