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Equivalences in Euler-based diagram systems through normal forms

Andrew Fish, John Taylor
2014 LMS Journal of Computation and Mathematics  
In particular, we consider 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.  ... 
doi:10.1112/s1461157014000114 fatcat:4clnrsowrjhkzf3kkj65whw2xe

Completeness Proofs for Diagrammatic Logics [chapter]

Jim Burton, Gem Stapleton, John Howse
2012 Lecture Notes in Computer Science  
Completeness has been established 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).  ... 
doi:10.1007/978-3-642-31223-6_38 fatcat:lceeuz3bifc4jl4k47lsk2bndi

Visualizing Sets: An Empirical Comparison of Diagram Types [chapter]

Peter Chapman, Gem Stapleton, Peter Rodgers, Luana Micallef, Andrew Blake
2014 Lecture Notes in Computer Science  
To address this knowledge gap, this paper empirically evaluates four 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.  ... 
doi:10.1007/978-3-662-44043-8_18 fatcat:337omwmgvfdu5derbd6uxnn4mi

A Survey of Reasoning Systems Based on Euler Diagrams

Gem Stapleton
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.  ... 
doi:10.1016/j.entcs.2005.02.022 fatcat:mnf4zv54hvchpkranod5stucdq

A normal form for spider diagrams of order

Aidan Delaney, Gem Stapleton, John Taylor, Simon Thompson
2014 Journal of Visual Languages and Computing  
We define 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] .  ... 
doi:10.1016/j.jvlc.2014.10.029 fatcat:jli7flztpzbdlcd5f4c3dkm2ia

Two Types of Diagrammatic Inference Systems: Natural Deduction Style and Resolution Style [chapter]

Koji Mineshima, Mitsuhiro Okada, Ryo Takemura
2010 Lecture Notes in Computer Science  
Since the 1990s, reasoning 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.  ... 
doi:10.1007/978-3-642-14600-8_12 fatcat:zumqerembveatdvradsxx6wyjq

Using DAG Transformations to Verify Euler/Venn Homogeneous and Euler/Venn FOL Heterogeneous Rules of Inference

Nik Swoboda, Gerard Allwein
2003 Electronical Notes in Theoretical Computer Science  
It has been shown that 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.  ... 
doi:10.1016/s1571-0661(04)80613-3 fatcat:kemfxhsyfvgppbbesvfpwchdwe

Using DAG transformations to verify Euler/Venn homogeneous and Euler/Venn FOL heterogeneous rules of inference

Nik Swoboda, Gerard Allwein
2004 Journal of Software and Systems Modeling  
In previous work, it has been shown that 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  ... 
doi:10.1007/s10270-003-0044-8 fatcat:x6bx5hlll5fshjqv4dmdsboeee

Intuitionistic Euler-Venn Diagrams (extended) [article]

Sven Linker
2020 arXiv   pre-print
We present an intuitionistic interpretation of 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  ... 
arXiv:2002.02929v1 fatcat:oiin6blqajhxhlaq6vfhrmpimu

Tactical Diagrammatic Reasoning

Sven Linker, Jim Burton, Mateja Jamnik
2017 Electronic Proceedings in Theoretical Computer Science  
In this paper we take an interactive proof assistant 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.  ... 
doi:10.4204/eptcs.239.3 fatcat:nvd7gb3ti5ag3p32zy5t22iv3i

On the Completeness and Expressiveness of Spider Diagram Systems [chapter]

John Howse, Fernando Molina, John Taylor
2000 Lecture Notes in Computer Science  
Spider 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  ... 
doi:10.1007/3-540-44590-0_8 fatcat:4wji7m6flfggzdl75nu7aebm5i

Proof-Theoretical Investigation of Venn Diagrams: A Logic Translation and Free Rides [chapter]

Ryo Takemura
2012 Lecture Notes in Computer Science  
Then we study 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 ).  ... 
doi:10.1007/978-3-642-31223-6_17 fatcat:jflfezsj4zgyrhxz3enxai7fx4

Fragments of Spider Diagrams of Order and Their Relative Expressiveness [chapter]

Aidan Delaney, Gem Stapleton, John Taylor, Simon Thompson
2010 Lecture Notes in Computer Science  
Investigating the expressiveness of 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.  ... 
doi:10.1007/978-3-642-14600-8_10 fatcat:6h3jelafjncvbcrftt3e44xgc4

Heterogeneous Reasoning with Euler/Venn Diagrams Containing Named Constants and FOL

Nik Swoboda, Gerard Allwein
2005 Electronical Notes in Theoretical Computer Science  
The main goal of this paper is to present the basis 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.  ... 
doi:10.1016/j.entcs.2005.02.023 fatcat:2hhc6yricbfzdnvqca4ed3zie4

Drawing Graphs in Euler Diagrams [chapter]

Paul Mutton, Peter Rodgers, Jean Flower
2004 Lecture Notes in Computer Science  
However, we show our system being used 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.  ... 
doi:10.1007/978-3-540-25931-2_9 fatcat:35rucpfugnhl7fdkicjekn6b4q
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