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Equivalences in Euler-based diagram systems through normal forms

2014
*
LMS Journal of Computation and Mathematics
*

In particular, we consider

doi:10.1112/s1461157014000114
fatcat:4clnrsowrjhkzf3kkj65whw2xe
*Euler**diagrams**with**shading*, and then we conservatively extend the system to include projections, which allow*for**a*much greater degree of flexibility of representation. ... AbstractThe*form*of information presented can influence its utility*for*the conveying of knowledge by affecting an interpreter's ability to reason*with*the information. ... The authors would like to acknowledge UK EPSRC grants EP/E011160: Visualisation*with**Euler**Diagrams*, and EP/J010898/1: Automatic*Diagram*Generation. Acknowledgements. ...##
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Completeness Proofs for Diagrammatic Logics
[chapter]

2012
*
Lecture Notes in Computer Science
*

Completeness has been established

doi:10.1007/978-3-642-31223-6_38
fatcat:lceeuz3bifc4jl4k47lsk2bndi
*for*such logics, including*Euler**diagrams*, spider*diagrams*and*a*fragment of the constraint*diagram*logic. ... Visual logics based on*Euler**diagrams*have recently been developed, including generalized constraint*diagrams*and concept*diagrams*. ... NF*for**Normal**Form*). ...##
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Visualizing Sets: An Empirical Comparison of Diagram Types
[chapter]

2014
*
Lecture Notes in Computer Science
*

To address this knowledge gap, this paper empirically evaluates four

doi:10.1007/978-3-662-44043-8_18
fatcat:337omwmgvfdu5derbd6uxnn4mi
*diagram*types: Venn*diagrams*,*Euler**diagrams**with**shading*,*Euler**diagrams*without*shading*, and the less well-known linear*diagrams*. ... There are*a*range of*diagram*types that can be used to visualize sets. However, there is*a*significant lack of insight into which is the most effective visualization. ... An interesting point is that the*Euler**diagrams**with**shading*and the Venn*diagrams*used in the study were all well-*formed*, whereas those that did not use*shading*were all non-well-*formed*(see e.g. ...##
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A Survey of Reasoning Systems Based on Euler Diagrams

2005
*
Electronical Notes in Theoretical Computer Science
*

*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*. ... of these

*diagrams*to represent disjunctive information, like

*a*type of disjunctive

*normal*

*form*. ...

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A normal form for spider diagrams of order

2014
*
Journal of Visual Languages and Computing
*

We define

doi:10.1016/j.jvlc.2014.10.029
fatcat:jli7flztpzbdlcd5f4c3dkm2ia
*a**normal**form**for*spider*diagrams*of order and provide an algorithm, based on the reasoning system,*for*producing*diagrams*in our*normal**form*. ... We develop*a*reasoning system*for*an*Euler**diagram*based visual logic, called spider*diagrams*of order. ... Our algorithm also contributes to (*a*) (b) Figure 1 : An*Euler*and*a*spider*diagram*. the recent interest in*normal**forms**for**Euler**diagram*based logics [8] . ...##
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Two Types of Diagrammatic Inference Systems: Natural Deduction Style and Resolution Style
[chapter]

2010
*
Lecture Notes in Computer Science
*

Since the 1990s, reasoning

doi:10.1007/978-3-642-14600-8_12
fatcat:zumqerembveatdvradsxx6wyjq
*with*Venn and*Euler**diagrams*has been studied from mathematical and logical viewpoints. ... On the one hand,*a**diagram*construed as*a*set of regions corresponds to*a*disjunctive*normal**form*formula and the inference system based on such*diagrams*corresponds to*a*resolution calculus. ...*Euler**diagrams*in the region-based framework, called*Euler**diagrams**with**shading*, are obtained by considering some*shaded*regions of Venn*diagrams*as "missing" regions. ...##
###
Using DAG Transformations to Verify Euler/Venn Homogeneous and Euler/Venn FOL Heterogeneous Rules of Inference

2003
*
Electronical Notes in Theoretical Computer Science
*

It has been shown that

doi:10.1016/s1571-0661(04)80613-3
fatcat:kemfxhsyfvgppbbesvfpwchdwe
*a*special collection of directed acyclic graphs (DAGs) can be used interchangeably*with**Euler*/Venn*diagrams*in reasoning processes [4] . ... Thus, proofs which include*Euler*/Venn*diagrams*can be thought of as proofs*with*DAGs where steps involving only*Euler*/Venn*diagrams*can be treated as particular DAG transformations. ... To define observation*for*all formulas in monadic first order logic (MFOL), we introduce*a*special*normal**form*called EVCNF. ...##
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Using DAG transformations to verify Euler/Venn homogeneous and Euler/Venn FOL heterogeneous rules of inference

2004
*
Journal of Software and Systems Modeling
*

In previous work, it has been shown that

doi:10.1007/s10270-003-0044-8
fatcat:x6bx5hlll5fshjqv4dmdsboeee
*a*special collection of directed acyclic graphs (DAGs) can be used interchangeably*with**Euler*/Venn*diagrams*in reasoning processes. ... Thus, proofs which include*Euler*/Venn*diagrams*can be thought of as proofs*with*DAGs where steps involving only*Euler*/Venn*diagrams*can be treated as particular DAG transformations. ... Acknowledgements The research reported here was supported in part by*a*contract*with*the Telecommunications Advancement Organization of Japan entitled, "*A*study of innovational interaction media*for*the ...##
###
Intuitionistic Euler-Venn Diagrams (extended)
[article]

2020
*
arXiv
*
pre-print

We present an intuitionistic interpretation of

arXiv:2002.02929v1
fatcat:oiin6blqajhxhlaq6vfhrmpimu
*Euler*-Venn*diagrams**with*respect to Heyting algebras. ... In contrast to classical*Euler*-Venn*diagrams*, we treat*shaded*and missing zones differently, to have diagrammatic representations of conjunction, disjunction and intuitionistic implication. ...*For*an*Euler*-Venn*diagram*d = (L, Z, Z * ) we will write Venn(d) = (L, Venn(L), Z * )*for*the Venn*diagram**with*the same set of*shaded*zones as d, and*Euler*(d) = (L, Z)*for*the pure*Euler**diagram**with*...##
###
Tactical Diagrammatic Reasoning

2017
*
Electronic Proceedings in Theoretical Computer Science
*

In this paper we take an interactive proof assistant

doi:10.4204/eptcs.239.3
fatcat:nvd7gb3ti5ag3p32zy5t22iv3i
*for**Euler**diagrams*, Speedith, and add tactics to its reasoning engine, providing*a*level of automation in the construction of proofs. ... Although automated reasoning*with**diagrams*has been possible*for*some years, tools*for*diagrammatic reasoning are generally much less sophisticated than their sentential cousins. ... Although our work is concerned only*with**Euler**diagrams*, we chose Speedith as the basis*for*extension due to its code base*with*effective visualisation of all*Euler**diagrams*. ...##
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On the Completeness and Expressiveness of Spider Diagram Systems
[chapter]

2000
*
Lecture Notes in Computer Science
*

Spider

doi:10.1007/3-540-44590-0_8
fatcat:4wji7m6flfggzdl75nu7aebm5i
*diagram*systems provide*a*visual language that extends the popular and intuitive Venn*diagrams*and*Euler*circles. ...*A*set of reasoning rules*for**a*spider*diagram*system is shown to be sound and complete. ... Our longer term aim is to prove similar results*for*constraint*diagrams*, and to provide the necessary mathematical underpinning*for*the development of software tools to aid the diagrammatic reasoning process ...##
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Proof-Theoretical Investigation of Venn Diagrams: A Logic Translation and Free Rides
[chapter]

2012
*
Lecture Notes in Computer Science
*

Then we study

doi:10.1007/978-3-642-31223-6_17
fatcat:jflfezsj4zgyrhxz3enxai7fx4
*a*logic translation of the Venn diagrammatic system*with*the aim of investigating how our inference rules are rendered to resolution calculus. ... Furthermore, our translation provides*a*more in-depth analysis of the free ride. In particular, we calculate how many pieces of information are obtained in the manipulation of Venn*diagrams*. ...*For*any non-empty clause set Γ = {r 1 , . . . , r m }, we denote by d(Γ), the disjunctive*normal**form*(DNF) formula (∧r 1 ) ∨ · · · ∨ (∧r m ). ...##
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Fragments of Spider Diagrams of Order and Their Relative Expressiveness
[chapter]

2010
*
Lecture Notes in Computer Science
*

Investigating the expressiveness of

doi:10.1007/978-3-642-14600-8_10
fatcat:6h3jelafjncvbcrftt3e44xgc4
*a*diagrammatic logic provides insight into how its syntactic elements interact at the semantic level. Moreover, it allows*for*comparisons*with*other notations. ... The techniques employed by Shin*for*Venn-II were adapted to allow the expressiveness of*Euler**diagrams*to be investigated. ... Acknowledgements This research*forms*part of the the Defining Regular Languages*with**Diagrams*(EPSRC grant EP/H012311/1) and Sketching*Euler**Diagrams*(EPSRC grant EP/H048480/1) projects. ...##
###
Heterogeneous Reasoning with Euler/Venn Diagrams Containing Named Constants and FOL

2005
*
Electronical Notes in Theoretical Computer Science
*

The main goal of this paper is to present the basis

doi:10.1016/j.entcs.2005.02.023
fatcat:2hhc6yricbfzdnvqca4ed3zie4
*for**a*heterogeneous*Euler*/Venn*diagram*and First Order Logic (FOL) reasoning system. ... We will begin by defining*a*homogeneous reasoning system*for**Euler*/Venn*diagrams*including named constants and show this system to be sound and complete. ... This is done to ensure that any*Euler*/Venn*diagram**with**shading*can be redrawn as*a*logically equivalent*Euler*/Venn*diagram**with*any*shaded*area removed. ...##
###
Drawing Graphs in Euler Diagrams
[chapter]

2004
*
Lecture Notes in Computer Science
*

However, we show our system being used

doi:10.1007/978-3-540-25931-2_9
fatcat:35rucpfugnhl7fdkicjekn6b4q
*with**a**form*of constraint*diagram*, the spider*diagram*[9] . ... The*shaded*zone in the drawn*diagram*corresponds to the abstract zone {*a*}. ... We would also like to thank John Taylor and Richard Bosworth at the University of Brighton*for*helpful comments on early versions of this paper. ...
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