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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
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.
Towards a Diagrammatic Reasoning System for Description Logics
, 2006
"... Diagrammatic reasoning is a tradition of visual logic that allows sentences that are equivalent to first order logic to be written in a visual or structural form: typically for improved usability. A calculus for the diagram is then defined that allows wellformed formulas to be derived. This calculu ..."
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Diagrammatic reasoning is a tradition of visual logic that allows sentences that are equivalent to first order logic to be written in a visual or structural form: typically for improved usability. A calculus for the diagram is then defined that allows wellformed formulas to be derived. This calculus is intended to simulate logical inference. Description logics (DLs) have become a popular subset of first order logic that have decidable tableau theorem provers and are sound and complete. Our paper explores whether several existing wellknown diagrammatic reasoning systems are compatible with DLs. We provide translations between the DL ALCI and a appropriate subset of Peirce’s relation graphs, termed RG ALCI. A precise formal elaboration, where relation graphs are for example defined in terms of mathematical graph theory, goes beyond the scope of this paper. We will provide a semiformal approach instead.
The expressiveness of spider diagrams augmented with constants.
 Journal of Visual Languages and Computing, available online,
, 2008
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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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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.
The Advent of Formal Diagrammatic Reasoning Systems
, 2009
"... In knowledge representation and reasoning systems, diagrams have many practical applications and are used in numerous settings. Indeed, it is widely accepted that diagrams are a valuable aid to intuition and help to convey ideas and information in a clear way. On the other side, logicians have viewe ..."
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In knowledge representation and reasoning systems, diagrams have many practical applications and are used in numerous settings. Indeed, it is widely accepted that diagrams are a valuable aid to intuition and help to convey ideas and information in a clear way. On the other side, logicians have viewed diagrams as informal tools, but which cannot be used in the manner of formal argumentation. Instead, logicians focused on symbolic representations of logics. Recently, this perception was overturned in the mid 1990s, first with seminal work by Shin on an extended version of Venn diagrams. Since then, certainly a growth in the research field of formal reasoning with diagrams can be witnessed. This paper discusses the evolution of formal diagrammatic logics, focusing on those systems which are based on Euler and VennPeirce diagrams, and Peirces existential graphs. Also discussed are some challenges faced in the area, some of which are specifically related to diagrams.
Conceptual Spider Diagrams
"... Conceptual Graphs are a common knowledge representation system which are used in conjunction with an explicit type hierarchy of the domain. However, this means the interpretation of information expressed in conceptual graphs requires the combined use of information from different sources, which is n ..."
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Conceptual Graphs are a common knowledge representation system which are used in conjunction with an explicit type hierarchy of the domain. However, this means the interpretation of information expressed in conceptual graphs requires the combined use of information from different sources, which is not always an easy cognitive task. Though it is possible to explicitly represent the type hierarchy with Conceptual Graphs with Cuts, this less natural expression of the type hierarchy information is not as easy to interpret and soon takes up a lot of space. Now, one of the main advantages of Euler diagrambased notations like Spider diagrams is the natural diagrammatic representation of hierarchies. However, Spider diagrams lack facilities such as the ability to represent general relationships between objects which is necessary for knowledge representation tasks. We bring together the most pertinent features of both of these notations, creating a new hybrid notation called Conceptual Spider Diagrams. We provide formal syntax and semantics of this new notation, together with examples demonstrating its capabilities.
A Diagrammatic Reasoning System for ALC
"... Description logics (DLs) are a wellknown family of knowledge representation (KR) languages. The notation of DLs has the style of a variablefree first order predicate logic. In this paper a diagrammatic representation of the DL ALC – based on Peirce’s existential graphs – is presented and a set of ..."
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Description logics (DLs) are a wellknown family of knowledge representation (KR) languages. The notation of DLs has the style of a variablefree first order predicate logic. In this paper a diagrammatic representation of the DL ALC – based on Peirce’s existential graphs – is presented and a set of transformation rules on these graphs provided. As the transformation rules modify the diagrammatic representation of ALC this produces a diagrammatic calculus. Some examples present in the paper illustrate the use and properties of this calculus.