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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 94 (34 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
Generating euler diagrams
 In Proceedings of Diagrams 2002
, 2002
"... Abstract. This article describes an algorithm for the automated generation of any Euler diagram starting with an abstract description of the diagram. An automated generation mechanism for Euler diagrams forms the foundations of a generation algorithm for notations such as Harel’s higraphs, constrain ..."
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Cited by 65 (23 self)
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Abstract. This article describes an algorithm for the automated generation of any Euler diagram starting with an abstract description of the diagram. An automated generation mechanism for Euler diagrams forms the foundations of a generation algorithm for notations such as Harel’s higraphs, constraint diagrams and some of the UML notation. An algorithm to generate diagrams is an essential component of a diagram tool for users to generate, edit and reason with diagrams. The work makes use of properties of the dual graph of an abstract diagram to identify which abstract diagrams are “drawable ” within given wellformedness rules on concrete diagrams. A Java program has been written to implement the algorithm and sample output is included. 1 Introduction and
Spider diagrams: A diagrammatic reasoning system
 Journal of Visual Languages and Computing
, 2001
"... Spider diagrams combine and extend Venn diagrams and Euler circles to express constraints on sets and their relationships with other sets. These diagrams can be used in conjunction with objectoriented modelling notations such as the Unified Modeling Language. This paper summarises the main syntax a ..."
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Cited by 46 (12 self)
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Spider diagrams combine and extend Venn diagrams and Euler circles to express constraints on sets and their relationships with other sets. These diagrams can be used in conjunction with objectoriented modelling notations such as the Unified Modeling Language. This paper summarises the main syntax and semantics of spider diagrams. It also introduces inference rules for reasoning with spider diagrams and a rule for combining spider diagrams. This system is shown to be sound but not complete. Disjunctive diagrams are considered as one way of enriching the system to allow combination of diagrams so that no semantic information is lost. The relationship of this system of spider diagrams to other similar systems, which are known to be sound and complete, is explored briefly.
Drawing graphs in Euler diagrams
 In Proceedings of 3rd International Conference on the Theory and Application of Diagrams, volume 2980 of LNAI
, 2004
"... Abstract. We describe a method for drawing graphenhanced Euler diagrams using a three stage method. The first stage is to lay out the underlying Euler diagram using a multicriteria optimizing system. The second stage is to find suitable locations for nodes in the zones of the Euler diagram using a ..."
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Cited by 17 (6 self)
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Abstract. We describe a method for drawing graphenhanced Euler diagrams using a three stage method. The first stage is to lay out the underlying Euler diagram using a multicriteria optimizing system. The second stage is to find suitable locations for nodes in the zones of the Euler diagram using a force based method. The third stage is to minimize edge crossings and total edge length by swapping the location of nodes that are in the same zone with a multicriteria hill climbing method. We show a working version of the software that draws spider diagrams. Spider diagrams represent logical expressions by superimposing graphs upon an Euler diagram. This application requires an extra step in the drawing process because the embedded graphs only convey information about the connectedness of nodes and so a spanning tree must be chosen for each maximally connected component. Similar notations to Euler diagrams enhanced with graphs are common in many applications and our method is generalizable to drawing Hypergraphs represented in the subset standard, or to drawing Higraphs where edges are restricted to connecting with only atomic nodes. 1
A Survey of Reasoning Systems Based on Euler Diagrams
 EULER DIAGRAMS 2004 PRELIMINARY VERSION
, 2004
"... Euler diagrams have been used for centuries as a means for conveying ideas in an intuitive, informal way. Recently much research has been conducted to develop formal, diagrammatic reasoning systems based on Euler diagrams. Most of these systems extend Euler diagrams by adding further syntax to incre ..."
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Cited by 15 (2 self)
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Euler diagrams have been used for centuries as a means for conveying ideas in an intuitive, informal way. Recently much research has been conducted to develop formal, diagrammatic reasoning systems based on Euler diagrams. Most of these systems extend Euler diagrams by adding further syntax to increase expressiveness. In this paper we survey such systems and draw comparisons between them. Key words: Visual Logic, Diagrammatic Reasoning.
Reasoning with Constraint Diagrams
 School of Computing, Mathematical and Information Sciences
, 2004
"... Constraint diagrams are designed for the formal specification of software systems. However, their applications are broader than this since constraint diagrams are a logic that can be used in any formal setting. This document summarizes the main results presented in my PhD thesis, the focus of which ..."
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Cited by 14 (3 self)
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Constraint diagrams are designed for the formal specification of software systems. However, their applications are broader than this since constraint diagrams are a logic that can be used in any formal setting. This document summarizes the main results presented in my PhD thesis, the focus of which is on a fragment of the constraint diagram language, called spider diagrams, and constraint diagrams themselves. In the thesis, sound and complete systems of spider diagrams and constraint diagrams are presented and the expressiveness of the spider diagram language is established. 1
Towards a Formalization of Constraint Diagrams
 In: Proc. 2002 IEEE CS International Symposium on HumanCentric Computing Languages and Environments (HCC
, 2001
"... Geared to complement UML and to the specification of large software systems by nonmathematicians, constraint diagrams are a visual language that generalizes the popular and intuitive Venn diagrams and Euler circles, and adds facilities for quantifying over elements and navigating relations. The lan ..."
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Cited by 13 (5 self)
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Geared to complement UML and to the specification of large software systems by nonmathematicians, constraint diagrams are a visual language that generalizes the popular and intuitive Venn diagrams and Euler circles, and adds facilities for quantifying over elements and navigating relations. The language design emphasizes scalability and expressiveness while retaining intuitiveness. Spider diagrams form a subset of the notation, leaving out universal quantification and the ability to navigate relations. Spider diagrams have been given a formal definition. This paper extends that definition to encompass the constraint diagram notation. The formalization of constraint diagrams is nontrivial: it exposes subtleties concerned with the implicit ordering of symbols in the visual language, which were not evident before a formal definition of the language was attempted. This has led to an improved design of the language.
Typesyntax and tokensyntax in diagrammatic systems
 In Proceedings FOIS2001: 2nd International Conference on Formal Ontology in Information Systems
, 2001
"... The uptake in the software industry of notations for designing systems visually has been accelerated with the standardization of the Unified Modeling Language (UML). The formalization of diagrammatic notations is important for the development of essential tool support and to allow reasoning to take ..."
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Cited by 10 (4 self)
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The uptake in the software industry of notations for designing systems visually has been accelerated with the standardization of the Unified Modeling Language (UML). The formalization of diagrammatic notations is important for the development of essential tool support and to allow reasoning to take place at the diagrammatic level. Focusing on an extended version of Venn and Euler diagrams (which was developed to complement UML in the specification of software systems), this paper presents two levels of syntax for this system: typesyntax and tokensyntax. Tokensyntax is about particular diagrams instantiated on some physical medium, and typesyntax provides a formal definition with which a concrete representation of a diagram must comply. While these two levels of syntax are closely related to each other, the domains of typesyntax and tokensyntax are ontologically and the other concrete. We discuss the roles of typesyntax and tokensyntax in diagrammatic systems and show that it is important to consider both levels of syntax in diagrammatic reasoning systems and in developing software tools to support such systems.
Towards a default reading for constraint diagrams
 Proceedings of Diagrams 2004, LNAI 2980, pp 51– 65
, 2004
"... Abstract. Constraint diagrams are a diagrammatic notation which may be used to express logical constraints. They were designed to complement the Unified Modeling Language in the development of software systems. They generalize Venn diagrams and Euler circles, and include facilities for quantificatio ..."
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Cited by 7 (5 self)
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Abstract. Constraint diagrams are a diagrammatic notation which may be used to express logical constraints. They were designed to complement the Unified Modeling Language in the development of software systems. They generalize Venn diagrams and Euler circles, and include facilities for quantification and navigation of relations. Due to the lack of a linear ordering of symbols inherent in a diagrammatic language which expresses logical statements, some constraint diagrams have more than one intuitive meaning. We generalize, from an example based approach, to suggest a default reading for constraint diagrams. This reading is usually unique, but may require a small number of simple user choices.
A Survey of Euler Diagrams
, 2013
"... Euler diagrams visually represent containment, intersection and exclusion using closed curves. They first appeared several hundred years ago, however, there has been a resurgence in Euler diagram research in the twentyfirst century. This was initially driven by their use in visual languages, where ..."
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Cited by 6 (1 self)
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Euler diagrams visually represent containment, intersection and exclusion using closed curves. They first appeared several hundred years ago, however, there has been a resurgence in Euler diagram research in the twentyfirst century. This was initially driven by their use in visual languages, where they can be used to represent logical expressions diagrammatically. This work lead to the requirement to automatically generate Euler diagrams from an abstract description. The ability to generate diagrams has accelerated their use in information visualization, both in the standard case where multiple grouping of data items inside curves is required and in the areaproportional case where the area of curve intersections is important. As a result, examining the usability of Euler diagrams has become an important aspect of this research. Usability has been investigated by empirical studies, but much research has concentrated on wellformedness, which concerns how curves and other features of the diagram interrelate. This work has revealed the drawability of Euler diagrams under various wellformedness properties and has developed embedding methods that meet these properties. Euler diagram research surveyed in this paper includes theoretical results, generation techniques, transformation methods and the development of automated reasoning systems for Euler diagrams. It also overviews application areas and the ways in which Euler diagrams have been extended.