Corresponding Regions in Euler Diagrams

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Corresponding Regions in Euler Diagrams

• John Howse, Gem Stapleton, Jean Flower
• and John Taylor
• Visual Modelling Group
• University of Brighton, UK.
• http//www.it.brighton.ac.uk/research/vmg/

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Overview

• What is a correspondence relation.
• Why a correspondence relation needed.
• What has been done before.
• Informal syntax and semantics of Euler diagrams.
• Corresponding regions in Venn diagrams.
• Corresponding regions in Euler diagrams.

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An Euler Diagram

• This diagram asserts

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Example

• The shaded regions are corresponding.

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Example

• The shaded regions are corresponding.

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Example

• The shaded regions are corresponding.

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Example

• The shaded regions are not corresponding.

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Why do we need a correspondence relation?

• So we can reason diagrammatically
• Practical applications
• Some UML notations.
• Constraint diagrams.
• Development of software tools
• must work at the diagrammatic level.

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Previous Work

• Shin
• Counterpart relation on regions in Venn diagrams.
• Sound and complete reasoning rules in Venn-Peirce
  diagrams.
• Hammer
• Counterpart relation on minimal regions in Euler
  diagrams with same label sets.
• Sound and complete reasoning rules for a simple
  Euler system.

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Example
Shin identifies the shaded regions as
corresponding.
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Example

• Hammer identifies the shaded regions as
  corresponding.

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Informal Syntax and Semantics

• Unitary Euler diagrams
• Syntax
• Labels, denoted L(d).

A
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Informal Syntax and Semantics

• Unitary Euler diagrams
• Syntax
• Labels, denoted L(d).
• Zones.

A
Zone 2
Zone 1
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Informal Syntax and Semantics

• Unitary Euler diagrams
• Syntax
• Labels, denoted L(d).
• Zones.
• Regions.

A
Zone 2
Zone 1
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Informal Syntax and Semantics

• Unitary Euler diagrams
• Syntax
• Labels, denoted L(d).
• Zones.
• Regions.
• A Venn diagram is an Euler diagram containing all
  possible zones.

A
Zone 2
Zone 1
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Informal Syntax and Semantics

• Unitary Euler diagrams

• Semantics
• Regions denote, possibly empty, sets.
• missing zones denote the empty set.

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Example

• This diagram has one missing zone, .
• The diagram asserts that

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Informal Syntax and Semantics

• Compound diagrams
• A compound diagram is a set of unitary diagrams
  taken in disjunction.

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Informal Syntax and Semantics

• Multi-diagrams
• A multi-diagram is a set of compound diagrams
  taken in conjunction.

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Informal Syntax and Semantics

• In disjunction cannot deduce B is empty.

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Informal Syntax and Semantics

• In disjunction cannot deduce B is empty.
• In conjunction B is empty.

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Venn Diagram Case
The shaded regions correspond

• Zonal regions they have representation
  which we call a split.

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Algebra of Splits

• A split is a pair of sets, , such that

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Algebra of Splits
A split is a pair of sets, , such that
Splits for the two shaded zones in d1 are
and . The split for the shaded
zone in d2 is .
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Algebra of Splits

• We want and to be the same as .

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Algebra of Splits

• Introduction of a label

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Algebra of Splits

• Introduction of a label
• We wanted

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Algebra of Splits

• Introduction of a label
• We wanted
• Set and , introduce C to obtain

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Algebra of Splits

• Addition
• Commutative.
• Associative.
• Each element is idempotent

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Mapping Regions to Splits

• Define a function from regions to splits
• zones

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Corresponding Zonal Regions
Define a function from regions to
splits zones regions
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Corresponding Zonal Regions

• Zonal regions zr1 and zr2 are corresponding if
  and only if
• Corresponding zonal regions represent the same
  set.

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Example

• The two shaded zonal regions are corresponding

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Corresponding Regions

• Two regions, r1 and r2, are corresponding iff
• Corresponding regions represent the same set.

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Example

• These regions are corresponding

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Corresponding Regions

• Theorem The correspondence
  relation is an equivalence relation on regions
  of Venn diagrams.

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Euler Diagrams

• The shaded zone in d2 is missing from d1. in
  the context of d1.

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Context of a Unitary Diagram

• The context of d1 is

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Example

• Which regions in d2 correspond to the shaded
  region?

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Suggested Solutions

• Which is correct?
• Are both correct?

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Disjunction

• We can only deduce that the shaded zone in the
  bottom diagram represents the empty set.

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Disjunction

• The first solution is correct.

This zone may be non-empty
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The Context of a Compound Diagram

• The set of splits associated with the
  intersection of the shaded zones in the component
  diagrams is called the context of the compound
  diagram.

Any split in the context is zero.
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Conjunction

• We can deduce that all missing zones represent
  the empty set.

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Conjunction
This zone is empty

• Both solutions are correct.

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The Context of a Multi-diagram

• The set of splits associated with the union of
  the shaded zones in the component diagrams is
  called the context of the multi-diagram.
• Any split in the context of a multi-diagram is
  zero.

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Algebra of Splits

• The context is

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Algebra of Splits

• With context

introducing C
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Algebra of Splits
With context
adding zero
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Example
These sums of splits are equal in context
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Corresponding Regions

• Two regions, r1 and r2, are corresponding in
  context if and only if

modulo the context.
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Example

• This multi-diagram has context
• The shaded regions correspond

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Example

• However, removing d2 affects the context.
• This multi-diagram has an empty context.
• The shaded regions no longer correspond.

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Euler Diagrams

• Theorem Given a fixed context, the
  correspondence relation is an equivalence
  relation on regions of Euler diagrams.

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Summary

• Correspondence relation on regions in Euler
  diagrams with different label sets.
• Shown this relation is an equivalence relation.
• Generalises work by Shin and Hammer.