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Inductively Generating Euler Diagrams
"... Abstract—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 dev ..."
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Abstract—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 wellformedness conditions; we present a series of results that identify properties of cycles that correspond to the wellformedness conditions. This improves upon other contributions toward the automated generation of Euler diagrams which implicitly assume some fixed set of wellformedness 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.
Drawing Euler diagrams with circles: The theory of piercings
 IEEE TRANS. ON VISUALISATION AND COMPUTER GRAPHICS
, 2011
"... Euler diagrams are effective tools for visualizing set intersections. They have a large number of application areas ranging from statistical data analysis to software engineering. However, the automated generation of Euler diagrams has never been easy: given an abstract description of a required Eu ..."
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Cited by 11 (7 self)
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Euler diagrams are effective tools for visualizing set intersections. They have a large number of application areas ranging from statistical data analysis to software engineering. However, the automated generation of Euler diagrams has never been easy: given an abstract description of a required Euler diagram, it is computationally expensive to generate the diagram. Moreover, the generated diagrams represent sets by polygons, sometimes with quite irregular shapes that make the diagrams less comprehensible. In this paper, we address these two issues by developing the theory of piercings, where we define single piercing curves and double piercing curves. We prove that if a diagram can be built inductively by successively adding piercing curves under certain constraints, then it can be drawn with circles, which are more esthetically pleasing than arbitrary polygons. The theory of piercings is developed at the abstract level. In addition, we present a Java implementation that, given an inductively pierced abstract description, generates an Euler diagram consisting only of circles within polynomial time.
P.: Visualizing sets and settyped data: Stateoftheart and future challenges. In: Eurographics conference on Visualization (EuroVis) State of The Art Reports, Eurographics
, 2014
"... In this document, we provide supplementary material to our report. This includes: a list of the reviewed conference proceedings and journals to conduct the survey; links to available software implementations, demonstrations, presentations and any other additional material about set visualization; a ..."
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Cited by 7 (2 self)
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In this document, we provide supplementary material to our report. This includes: a list of the reviewed conference proceedings and journals to conduct the survey; links to available software implementations, demonstrations, presentations and any other additional material about set visualization; a list of theses on set visualizations. Further resources are available on the survey website
Euler Diagram Transformations
 Graph Transformations & Visual Modelling Techniques, ECEASST
"... Abstract: Euler diagrams are a visual language which are used for purposes such as the presentation of setbased data or as the basis of visual logical languages which can be utilised for software specification and reasoning. Such Euler diagram reasoning systems tend to be defined at an abstract le ..."
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Cited by 4 (1 self)
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Abstract: Euler diagrams are a visual language which are used for purposes such as the presentation of setbased data or as the basis of visual logical languages which can be utilised for software specification and reasoning. Such Euler diagram reasoning systems tend to be defined at an abstract level, and the concrete level is simply a visualisation of an abstract model, thereby capturing some subset of the usual boolean logic. The visualisation process tends to be divorced from the data transformation process thereby affecting the user’s mental map and reducing the effectiveness of the diagrammatic notation. Furthermore, geometric and topological constraints, called wellformedness conditions, are often placed on the concrete diagrams to try to reduce human comprehension errors, and the effects of these conditions are not modelled in these systems. We view Euler diagrams as a type of graph, where the faces that are present are the key features that convey information and we provide transformations at the dual graph level that correspond to transformations of Euler diagrams, both in terms of editing moves and logical reasoning moves. This original approach gives a corre
A Diagrammatic Inference System with Euler Circles,
, 2009
"... Abstract Prooftheory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the 18th century by ..."
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Abstract Prooftheory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the 18th century by Euler [1768]. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint. We propose a novel approach to the formalization of Euler diagrammatic reasoning, in which diagrams are defined not in terms of regions as in the standard approach, but in terms of topological relations between diagrammatic objects. We formalize the unification rule, which plays a central role in Euler diagrammatic reasoning, in a style of natural deduction. We prove the soundness and completeness theorems with respect to a formal settheoretical semantics. We also investigate structure of diagrammatic proofs and prove a normalization theorem.
Transforming Constraint Diagrams
, 2009
"... Constraint diagrams were proposed by Kent for the purposes of formal software specification in a visual manner. They have recently been formalized and generalized, making them more expressive. This paper presents a collection of transformations that can be applied to the socalled unitary α fragment ..."
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Constraint diagrams were proposed by Kent for the purposes of formal software specification in a visual manner. They have recently been formalized and generalized, making them more expressive. This paper presents a collection of transformations that can be applied to the socalled unitary α fragment of constraint diagrams. The transformations can be used to define inference rules in a more succinct manner than in earlier systems. We establish that the transformations are sufficient to transform any given unitary αdiagram into any other unitary αdiagram. Therefore, they are sufficient for formalizing any inference rules between such diagrams. 1
Properties of Some Euler Graphs Constructed from Euler Diagram
"... Abstract: This article reflects some theoretical results related with Euler Graph obtained from a special pattern of Euler Diagrams. ..."
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Abstract: This article reflects some theoretical results related with Euler Graph obtained from a special pattern of Euler Diagrams.
Proof theory for reasoning with Euler diagrams: a Logic Translation and Normalization
"... Abstract Prooftheoretical notions and techniques, which are developed based on sentential/symbolic representations of formal proofs, are applied to Euler diagrams. A translation of an Euler diagrammatic system into a natural deduction system is given, and the soundness and faithfulness of the tran ..."
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Abstract Prooftheoretical notions and techniques, which are developed based on sentential/symbolic representations of formal proofs, are applied to Euler diagrams. A translation of an Euler diagrammatic system into a natural deduction system is given, and the soundness and faithfulness of the translation are proved. Some consequences of the translation are discussed in view of the notion of free ride, which is one of the most basic properties of diagrams that is mainly discussed in the literature of cognitive science as an account of inferential efficacy of diagrams. The translation enables us to formalize and analyze the free ride in terms of proof theory. The notion of normal form of Euler diagrammatic proofs is investigated, and a normalization theorem is proved. Some consequences of the theorem are further discussed: in particular, an analysis of the structure of normal diagrammatic proofs; a diagrammatic counterpart of the usual subformula property; and a characterization of diagrammatic proofs compared with natural deduction proofs.
Wellmatchedness in Euler Diagrams
"... Abstract. Euler diagrams are used for visualizing setbased information. Closed curves represent sets and the relationship between the curves correspond to relationships between sets. A notation is wellmatched to meaning when its syntactic relationships are reflected in the semantic relationships ..."
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Abstract. Euler diagrams are used for visualizing setbased information. Closed curves represent sets and the relationship between the curves correspond to relationships between sets. A notation is wellmatched to meaning when its syntactic relationships are reflected in the semantic relationships being represented. Euler diagrams are said to be wellmatched to meaning because, for example, curve containment corresponds to the subset relationship. In this paper we explore the concept of wellmatchedness in Euler diagrams, considering different levels of wellmatchedness. We also discuss how the properties, sometimes called wellformedness conditions, of an Euler diagram relate to the levels of wellmatchedness. 1