Formal Issues in Languages Based on Closed Curves

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Formal Issues in Languages Based on Closed Curves

• Andrew Fish and Gem Stapleton
• University of Brighton
• Supported by the Leverhulme Trust

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Important questions

• Given some information, can we represent it?
• Can we reliably interpret statements?
• Can we reason about information?

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Closed curves

• An Euler diagrams is a finite collection of
  closed curves

A
B
C
D
lots of people use this definition see the paper
for some references
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Closed curves

• An Euler diagrams is a finite collection of
  closed curves

A
B
Book
C
D
lots of people use this definition see the paper
for some references
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Closed curves

• Euler diagram application areas
• Visualizing genetic set relations (Kestler et al)

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Closed curves

• Euler diagram application areas
• Visualizing statistical data (Artes et al)

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Closed curves

• Euler diagram application areas
• Robot Trajectory data (Quick et al)

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Closed curves

• Euler diagram application areas
• Others include
• displaying the results of data base queries
• representing non-hierarchical file systems
• various UML diagrams
• constraint diagrams lots of others

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Information representation

• Given some information, can we represent it?
• Information lt-gt which minimal regions are be
  present

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Information representation

• Diagram descriptions

A
B
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Information representation

• Diagram descriptions

A
B
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Information representation

• Diagram descriptions

A
B
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Information representation

• Diagram descriptions

A
B
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Information representation

• Diagram descriptions

A
B
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Information representation

• Diagram descriptions

A
B
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Information representation

• Diagram descriptions

A
B
A
B
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Information representation

• Diagram descriptions

A
B
A
B
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Information representation

• Diagram descriptions

A
B
A
B
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Information representation

• A diagram description is a pair
• where W is a subset of
• Given a description can we represent/draw it?

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Information representation

• Theorem All descriptions are drawable.

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Information representation

• Theorem All descriptions are drawable.

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Semantic consequences

• What does this mean?

softwareEngineers
jobSeekers
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Semantic consequences

• What does this mean?

softwareEngineers
jobSeekers
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Semantic consequences

• What does this mean?

softwareEngineers
jobSeekers
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Semantic consequences

• What does this mean?

softwareEngineers
jobSeekers
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Semantic consequences

• What does this mean?

softwareEngineers
jobSeekers
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Semantic consequences

• What does this mean?

softwareEngineers
jobSeekers
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Semantic consequences

• What does this mean?

softwareEngineers
jobSeekers
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Semantic consequences

• What does this mean?

softwareEngineers
jobSeekers
PowerPoints version!
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Semantic consequences

• What does this mean?

softwareEngineers
jobSeekers
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Semantic consequences interiors

• How do we determine inside?

path
softwareEngineers
p
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Semantic consequences interiors

• How do we determine inside?

path
softwareEngineers
p
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Semantic consequences interiors

• How do we determine inside?

Each bounded component
softwareEngineers
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Semantic consequences interiors

• How do we determine inside?

Each bounded component
softwareEngineers
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Semantic consequences interiors

• How do we determine inside?
• Two methods, two different interiors
• Must specify method
• Usability

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Reasoning consequences

• Rules -- pre/post specification.
• Eg Remove a curve, A, from a diagram
• pre The curve A is in the diagram
• post The resulting diagram is identical to the
  original with the exception that A is deleted

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Reasoning consequences

• Everything drawable -gt can always reason

A
d1
Swoboda and Allwein
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Reasoning consequences

• Everything drawable -gt can always reason

A
d1
d2
Swoboda and Allwein
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Reasoning consequences

• Everything drawable -gt can always reason

A
A
d1
d2
d3
Swoboda and Allwein
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Reasoning consequences

• Everything drawable -gt can always reason

A
A
d1
d2
d3
A
Swoboda and Allwein
d4
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Reasoning consequences

• Everything drawable -gt can always reason

A
A
d1
d2
d3
A
Proof
Swoboda and Allwein
d4
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Reasoning consequences

• Can reason nicely mental map preservation

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Reasoning consequences

• Can reason nicely mental map preservation

shrink to a point
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Reasoning consequences

• Can reason nicely mental map preservation

shrink to a point
Theorem Any minimal region can be removed by
shrinking.
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Reasoning consequences

• Can shrink, but not to a point

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Allowing any closed curves
The Hilbert Space Filling Curve
See eg http//www.cut-the-knot.org/do_you_know/hi
lbert.shtml
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Answers

• Given some information, can we represent it? Yes
• Can we reliably interpret statements? No
• Can we reason about information? Yes
• Overcoming interpretation problem enforce
  well-formedness conditions

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Simple closed curves

• Lots of people enforce simplicity (see paper)
• What is a simple closed curve?

Simple, but not closed.
Closed and simple
Closed, but not simple.
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Information representation

• Cannot draw some descriptions.
• eg. one with dual graph
• This example is due to Lemon and Pratt, 1997.
• Unknown Which descriptions can be drawn.

def
ghi
abc
beh
cfi
adg
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Information representation

• Theorem Every Venn diagram description is
  drawable with simple closed curves.
• (A Venn diagram description has )

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Semantic consequences

• What does this mean?

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Semantic consequences

• What does this mean?
• Interior Obvious (Jordan Curve Theorem).
• Can reliably interpret.

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Reasoning consequences

• Shrinking minimal regions sometimes produces
  non-simple curves.

shrink to a point
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Reasoning consequences

• Not everything drawable -gt when can we reason?
• Does the proof pass through a diagram with
  non-simple curves?
• Pre and Post condition problem.

premise diagram
conclusion diagram
apply rules to write a proof
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Answers Simple case

• Given some information, can we represent it? Not
  always
• Can we reliably interpret statements? Yes
• Can we reason about information? Sometimes
  general result unknown

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Inductive Definitions

• A rectangle is a Euler diagram so is anything
  which is obtained by adding any closed curve (to
  an Euler diagram) which splits minimal regions
  into at most two components.

• This is another type of well-formedness condition
  (also frequently enforced).

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Inductive definitions

• The inductive definition does not allow minimal
  regions to be disconnected these are banned

A
B
B
A
C
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Information representation

• Theorem Every Venn diagram description is
  drawable as an inductive diagram (More, 1959).
• Unknown Which Euler diagram descriptions are
  drawable.
• What can be represented?

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Inductive definitions Venn(5)

• This symmetric drawing of Venn (5) is not
  inductive

No minimal region is disconnected Removing
any contour disconnects
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Reasoning consequences
Can we delete contours and retain connectedness
of minimal regions?
Venn(4)
Try deleting A or D.
REDRAW?
Try deleting C or D.
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Answers Inductive case

• Given some information, can we represent it?
  Unknown
• Can we reliably interpret statements? No (we used
  closed curves here)
• Can we reason about information? Sometimes
  general result unknown

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Conclusion and further work

• Using closed curves brings semantic consequences
• Enforcing wfcs also has consequences
• Identify subdivisions of the well-formedness
  conditions to answer Yes
• Identify which descriptions are drawable under
  various wfcs.

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Thank you

• www.cmis.brighton.ac.uk/research/vmg