Spider Diagrams

Howse, John; Taylor, John

doi:10.1112/s1461157000000942

Cited by 94 publications

(127 citation statements)

References 25 publications

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“…The implications of diagrammatic reasoning with an open universe is an interesting and open topic. Lastly, the use of absence could be incorporated into other Euler-diagram-based logics, such as spider diagrams [9], Euler/Venn diagrams [19], constraint diagrams [7] and concept diagrams [10].…”

Section: Discussionmentioning

confidence: 99%

Minimizing Clutter Using Absence in Venn- $$\text {i}^\text {e}$$

Burton

Chakraborty

Choudhury

2016

Lecture Notes in Computer Science

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Abstract. Over the last two decades substantial advances have been made in our understanding of diagrammatic logics. Many of these logics have the expressiveness of monadic first-order logic, sometimes with equality, and are equipped with sound and complete inference rules. A particular challenge is the representation of negated statements. This paper addresses the problem of how to represent negated statements involving constants, thus asserting the absence of specific individuals, in the context of Euler-diagram-based logics. Our first contribution is to explore the potential benefits of explicitly representing absence using constants, in terms of clutter reduction, and to highlight ontological issues that arise. We go on to define a measure of clutter arising from constants. By defining a set of semantics-preserving inference rules, we are able to algorithmically minimize diagram clutter, in part made possible by the inclusion of absence. Consequently, information about individuals can be represented in a minimally cluttered way.

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Section: Discussionmentioning

confidence: 99%

Minimizing Clutter Using Absence in Venn- $$\text {i}^\text {e}$$

Burton

Chakraborty

Choudhury

2016

Lecture Notes in Computer Science

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“…This section provides an overview of the syntax and semantics of spider diagrams of order, originally presented in [5] which in turn extends [11].…”

Section: Syntax and Semantics Of Spider Diagrams Of Ordermentioning

confidence: 99%

Spider Diagrams of Order and a Hierarchy of Star-Free Regular Languages

Delaney

Taylor

Thompson

Diagrammatic Representation and Inference

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Abstract. The spider diagram logic forms a fragment of constraint diagram logic and is designed to be primarily used as a diagrammatic software specification tool. Our interest is in using the logical basis of spider diagrams and the existing known equivalences between certain logics, formal language theory classes and some automata to inform the development of diagrammatic logic. Such developments could have many advantages, one of which would be aiding software engineers who are familiar with formal languages and automata to more intuitively understand diagrammatic logics. In this paper we consider relationships between spider diagrams of order (an extension of spider diagrams) and the star-free subset of regular languages. We extend the concept of the language of a spider diagram to encompass languages over arbitrary alphabets. Furthermore, the product of spider diagrams is introduced. This operator is the diagrammatic analogue of language concatenation. We establish that star-free languages are definable by spider diagrams of order equipped with the product operator and, based on this relationship, spider diagrams of order are as expressive as first order monadic logic of order.

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“…See, for example, [Stapleton 2005, Molina 2001] for surveys of various syntactic devices. Two Venn diagrams may be combined into another Venn diagram by accommodating the labels of circles and then by superposing the shaded regions, as illustrated in Fig.7.…”

Section: Linking Between Diagrams (For Disjunctive Information On Diamentioning

confidence: 99%

“…Venn-I and Venn-II systems of [Shin 1994]; Heterogeneous inference system of [Hammer 1994], Spider diagrams SD1 and SD2 of [Howse-Molina-Taylor 2000, Molina 2001], etc. For recent surveys, see [Stapleton 2005, Howse 2008.…”

Section: Linking Between Diagrams (For Disjunctive Information On Diamentioning

confidence: 99%

Diagrammatic Reasoning System with Euler Circles: Theory and Experiment Design

Mineshima

Okada

Sato

et al.

Diagrammatic Representation and Inference

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This paper is concerned with Euler diagrammatic reasoning. Proof-theory 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 many logicians. Euler diagrams were introduced in the 18th century by Leonhard Euler [1768]. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint, and there are only few proof-theoretical investigations. Accordingly, in order to fill this gap, we formalize an Euler diagrammatic inference system and prove the soundness and completeness theorems with respect to a formal set-theoretical semantics. We further consider, from a proof-theoretical viewpoint, the structure of diagrammatic proofs and manners of their construction.

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Spider Diagrams

Cited by 94 publications

References 25 publications

Minimizing Clutter Using Absence in Venn- $$\text {i}^\text {e}$$

Minimizing Clutter Using Absence in Venn- $$\text {i}^\text {e}$$

Spider Diagrams of Order and a Hierarchy of Star-Free Regular Languages

Diagrammatic Reasoning System with Euler Circles: Theory and Experiment Design

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