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Spider diagrams
"... The use of diagrams in mathematics has traditionally been restricted to guiding intuition and communication. With rare exceptions such as Peirce’s α and β systems, purely diagrammatic formal reasoning has not been in the mathematicians or logicians toolkit. This paper develops a purely diagrammatic ..."
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Cited by 79 (32 self)
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The use of diagrams in mathematics has traditionally been restricted to guiding intuition and communication. With rare exceptions such as Peirce’s α and β systems, purely diagrammatic formal reasoning has not been in the mathematicians or logicians toolkit. This paper develops a purely diagrammatic reasoning system of ‘spider diagrams ’ that builds on Euler, Venn and Peirce diagrams. The system is known to be expressively equivalent to first order monadic logic with equality. We develop two levels of diagrammatic syntax: an ‘abstract ’ syntax that captures the structure of diagrams and a ‘concrete’ syntax that captures topological properties of drawn diagrams. A number of simple diagrammatic transformation rules are given and the resulting reasoning system is shown to be sound and complete. 1
Drawing areaproportional Venn and Euler diagrams
 In Proceedings of Graph Drawing 2003
, 2003
"... Abstract. We consider the problem of drawing Venn diagrams for which each region’s area is proportional to some weight (e.g., population or percentage) assigned to that region. These areaproportional Venn diagrams have an enhanced ability over traditional Venn diagrams to visually convey informatio ..."
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Cited by 46 (1 self)
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Abstract. We consider the problem of drawing Venn diagrams for which each region’s area is proportional to some weight (e.g., population or percentage) assigned to that region. These areaproportional Venn diagrams have an enhanced ability over traditional Venn diagrams to visually convey information about data sets with interacting characteristics. We develop algorithms for drawing areaproportional Venn diagrams for any population distribution over two characteristics using circles and over three characteristics using rectangles and nearrectangular polygons; modifications of these algorithms are then presented for drawing the more general Euler diagrams. We present results concerning which population distributions can be drawn using specific shapes. A program to aid further investigation of areaproportional Venn diagrams is also described. 1
Layout metrics for Euler Diagrams
 7th International Conference on Information Visualisation IEEE Computer
, 2003
"... We present an aesthetics based method for drawing Euler diagrams. Aesthetic layout metrics have been found to be useful in graph drawing algorithms, which use metrics motivated by aesthetic principles that aid user understanding of diagrams. We have taken a similar approach to Euler diagram drawing, ..."
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Cited by 26 (13 self)
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We present an aesthetics based method for drawing Euler diagrams. Aesthetic layout metrics have been found to be useful in graph drawing algorithms, which use metrics motivated by aesthetic principles that aid user understanding of diagrams. We have taken a similar approach to Euler diagram drawing, and have defined a set of suitable metrics to be used within a hill climbing multicriteria optimiser to produce “good ” drawings. There are added difficulties when drawing Euler diagrams as they are made up of contours whose structural properties of intersection and containment must be preserved under any layout improvements. In this paper we describe our Java implementation of a pair of hill climbing variants to find good drawings, a set of metrics that measure aesthetics for good diagram layout, and issues concerning the choice of weightings for a useful combination of the metrics.
Decidability of String Graphs
 Proceedings of the 33rd Annual Symposium on the Theory of Computing
, 2003
"... We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a minimal drawing realizing a string graph in the plane. This upper bound confirms a conjecture by Kratochvl and Matousek [KM91] and settles th ..."
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Cited by 25 (5 self)
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We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a minimal drawing realizing a string graph in the plane. This upper bound confirms a conjecture by Kratochvl and Matousek [KM91] and settles the longstanding open problem of the decidability of string graph recognition (Sinden [Sin66], Graham [Gra76]). Finally we show how to apply the result to solve another old open problem: deciding the existence of Euler diagrams, a fundamental problem of topological inference (Grigni, Papadias, Papadimitriou [GPP95]). The general theory of Euler diagrams turns out to be as hard as secondorder arithmetic.
Generating Readable Proofs: A Heuristic Approach to Theorem Proving with Spider Diagrams
 Proc. Diagrams 2004 LNAI 2980
, 2004
"... Abstract. An important aim of diagrammatic reasoning is to make it easier for people to create and understand logical arguments. We have worked on spider diagrams, which visually express logical statements. Ideally, automatically generated proofs should be short and easy to understand. An existing p ..."
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Cited by 22 (12 self)
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Abstract. An important aim of diagrammatic reasoning is to make it easier for people to create and understand logical arguments. We have worked on spider diagrams, which visually express logical statements. Ideally, automatically generated proofs should be short and easy to understand. An existing proof generator for spider diagrams successfully writes proofs, but they can be long and unwieldy. In this paper, we present a new approach to proof writing in diagrammatic systems, which is guaranteed to find shortest proofs and can be extended to incorporate other readability criteria. We apply the A ∗ algorithm and develop an admissible heuristic function to guide automatic proof construction. We demonstrate the effectiveness of the heuristic used. The work has been implemented as part of a spider diagram reasoning tool. 1
Using Euler Diagrams in Traditional Library Environment
 in &quot;Electronic Notes in Computer Science
, 2005
"... In this paper, we present a new graphical interface for traditional library environments, which allows the user to elaborate easily and efficiently new strategies in search processes. This tool is based on two linked interactive Euler diagram representations. The first one is an interactive represen ..."
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Cited by 18 (1 self)
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In this paper, we present a new graphical interface for traditional library environments, which allows the user to elaborate easily and efficiently new strategies in search processes. This tool is based on two linked interactive Euler diagram representations. The first one is an interactive representation of the structures composing the documentary kernel of the library. The user may navigate and select items, making their own understanding of the database content, structure and access. The second one is a set based visualization of the results of a composed query. This allows the user to validate his search context and to elaborate strategies to go through the results. The association of both interfaces generates a tool that allows the user to elaborate the main search strategies through graphical manipulations.
On the Completeness and Expressiveness of Spider Diagram Systems
 PROC. DIAGRAMS 2000, EDINBURGH, SEPT 2000. LNAI 1889
, 2000
"... Spider diagram systems provide a visual language that extends the popular and intuitive Venn diagrams and Euler circles. Designed to complement objectoriented modelling notations in the specification of large software systems they can be used to reason diagrammatically about sets, their cardinalit ..."
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Cited by 17 (7 self)
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Spider diagram systems provide a visual language that extends the popular and intuitive Venn diagrams and Euler circles. Designed to complement objectoriented modelling notations in the specification of large software systems they can be used to reason diagrammatically about sets, their cardinalities and their relationships with other sets. A set of reasoning rules for a spider diagram system is shown to be sound and complete. We discuss the extension of this result to diagrammatically richer notations and also consider their expressiveness. Finally, we show that for a rich enough system we can diagrammatically express the negation of any diagram.
General Euler Diagram Generation
 In Diagrams
, 2008
"... Abstract. Euler diagrams are a natural method of representing settheoretic data and have been employed in diverse areas such as visualizing statistical data, as a basis for diagrammatic logics and for displaying the results of database search queries. For effective use of Euler diagrams in practica ..."
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Cited by 16 (11 self)
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Abstract. Euler diagrams are a natural method of representing settheoretic data and have been employed in diverse areas such as visualizing statistical data, as a basis for diagrammatic logics and for displaying the results of database search queries. For effective use of Euler diagrams in practical computer based applications, the generation of a diagram as a set of curves from an abstract description is necessary. Various practical methods for Euler diagram generation have been proposed, but in all of these methods the diagrams that can be produced are only for a restricted subset of all possible abstract descriptions. We describe a method for Euler diagram generation, demonstrated by implemented software, and illustrate the advances in methodology via the production of diagrams which were difficult or impossible to draw using previous approaches. To allow the generation of all abstract descriptions we may be required to have some properties of the final diagram that are not considered nice. In particular we permit more than two curves to pass though a single point, permit some curve segments to be drawn concurrently, and permit duplication of curve labels. However, our method attempts to minimize these bad properties according to a chosen prioritization.
Evaluating the comprehension of Euler diagrams
 Proc. Euler Diagrams
, 2005
"... We describe an empirical investigation into layout criteria that can help with the comprehension of Euler diagrams. Our work is intended to inform automatic Euler diagram layout research by confirming the importance of various Euler diagram aesthetic criteria. The three criteria under investigation ..."
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Cited by 16 (5 self)
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We describe an empirical investigation into layout criteria that can help with the comprehension of Euler diagrams. Our work is intended to inform automatic Euler diagram layout research by confirming the importance of various Euler diagram aesthetic criteria. The three criteria under investigation were: smoothness, zone area equality and edge closeness. Subjects were asked to interpret diagrams with different combinations of levels for each of the criteria. Results for this investigation indicate that, within the parameters of the study, all three criteria are important for understanding Euler diagrams and we have a preliminary indication of the ordering of importance for the criteria. 1.
Ensuring the Drawability of Extended Euler Diagrams for up to 8 Sets
 PROC. DIAGRAMS 2004. LNAI 2980
, 2003
"... This paper shows by a constructive method the existence of a diagrammatic representation called extended Euler diagrams for any collection of sets X1 , ..., Xn , n 9. These diagrams are adapted for representing sets inclusions and intersections: each set X i and each non empty intersection of a sub ..."
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Cited by 16 (2 self)
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This paper shows by a constructive method the existence of a diagrammatic representation called extended Euler diagrams for any collection of sets X1 , ..., Xn , n 9. These diagrams are adapted for representing sets inclusions and intersections: each set X i and each non empty intersection of a subcollection of X1 , ..., Xn is represented by a unique connected region of the plane. Starting with an abstract description of the diagram, we define the dual graph G and reason with the properties of this graph to build a planar representation of the X1 , ..., Xn . These diagrams will be used to visualize the results of a complex request on any indexed video databases. In fact, such a representation allows the user to perceive simultaneously the results of his query and the relevance of the database according to the query.