Enhancing the Expressiveness of Spider Diagram Systems

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Enhancing the Expressiveness of Spider Diagram
Systems

• Gem Stapleton and John Howse
• Visual Modelling Group
• University of Brighton, UK
• Supported by the Leverhulme Trust

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Constraint Diagrams Example
Introduced by Kent (OOPSLA 1997)
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Reasoning

• To reason about specifications.
• Sound and complete fragments.
• Automated TP to provide practical support.
• Spider diagrams fragment.

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Spider Diagrams
A
B
There is an element in A and everything that is
in B is also in A.
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Spider Diagrams
B
A
There are exactly two elements in B and
everything that is in A is also in B.
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Spider Diagrams with Constants
There exists x in B that is not in A.
A
B
c
d1
The individual c is not in A or B.
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Compound Diagrams
Connectives. and or
U
C
B
A
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Building Blocks of Spider Diagrams

• Euler Diagrams with shading
• Feet (represent objects)
• round
• square
• Lines (represent disjunction)
• connect feet BUT only feet of the same type
• Spider labels
• label square feet BUT only one label per
  spider...
• Logical Connectives

s,t,
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Expressiveness
Theorem All unitary spider diagrams are
satisfiable.

• Theorem
• The spider diagram language is equivalent to
  MFOLe.

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The Expressiveness of Unitary Diagrams
Can express there are at least n elements in
A, there are at most m elements in A, the
individual c is in A, the individuals c and d
are distinct etc. Exact expressiveness later

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The Expressiveness of Unitary Diagrams
No semantically equivalent unitary diagram
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The Expressiveness of Unitary Diagrams
No semantically equivalent unitary diagram
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The Expressiveness of Unitary Diagrams
No semantically equivalent unitary diagram
A
A
s
t
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The Expressiveness of Unitary Diagrams
How do we express there are two distinct
individuals, one is s in A or t outside A and the
other is t in A or u outside A.
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Observations

• More concise than compound diagrams.
• Many simple statements cannot be made.
• Unnatural ways of making statements.
• Automated unitary reasoning is easier to control
  (D2K4, VLC 04).
• Easier to automatically draw unitary than
  compound (D2K2, VL/HCC 04).
• Increase expressiveness Generalizations.

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Generalizing Spiders
Constant spider labelling
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Generalizing Spiders
Constant spider labelling
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Generalizing Spiders

Multiple typed spiders
There is an element in U-A or s is in A and A is
a subset of s
A
s
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Generalizing Spiders

Placing of feet
A
s is in A or t is in A and A1
s
t
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Generalizing Spiders
How do we express there are two distinct
individuals, one is s in A or t outside A and the
other is t in A or u outside A.

A
A
u
A
s
s
t
u
t
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Efficiency
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Flexibility
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Previously...

• Euler Diagrams with shading
• Feet (represent objects)
• round
• square
• Lines (represent disjunction)
• connect feet BUT only feet of the same type
• Spider labels
• label square feet BUT only one label per
  spider...
• Logical Connectives

s,t,
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Now

• Euler Diagrams with shading
• Feet (represent objects)
• round
• square
• Lines (represent disjunction)
• connect ANY feet (not just same type/different
  zones)
• Spider labels
• label square feet (not just used once/one per
  c. spider)
• Logical Connectives

s,t,
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Generalizing Spiders

Theorem The generalizations increase the
expressiveness of the unitary system. How much?
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Non-generalized Expressiveness
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Non-generalized Expressiveness

• one for each spider

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Non-generalized Expressiveness

• one for each spider
• describes the set xi
• belongs to and, if
• necessary, identifies xi with a constant.

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Non-generalized Expressiveness

• one for each spider
• describes the set xi
• belongs to and, if
• necessary, identifies xi with a constant.

each spider denotes a distinct element
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Non-generalized Expressiveness

• one for each spider
• describes the set xi
• belongs to and, if
• necessary, identifies xi with a constant.

each spider denotes a distinct element
places an upper bound on set cardinality
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Example
A
s
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Example
A
s
P1
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Example
A
s
P2
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Example
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Generalized Expressiveness

• one for each spider

each spider denotes a distinct element
places an upper bound on set cardinality
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Generalized Expressiveness

• one for each spider
• a disjunction of
• Pis that assert which
• set xi belongs to and,
• if necessary, identifies xi with a constant.

each spider denotes a distinct element
places an upper bound on set cardinality
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Example
A
s
t
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Example
A
s
t
R1
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Example
A
s
t
R2
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Further Limitations
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Further Limitations

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Further Limitations
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Conclusion and further work

• By generalizing spiders we have increased the
  expressiveness of the unitary system
• -- efficiency
• -- flexibility.
• There are still simple statements that cannot be
  made.
• Can we make the unitary system as expressive as
  the whole system?
• When do the generalizations help ATP?
• What is a good choice of strand and tie
  semantics?
• When do the generalizations help people?
• Constraint diagram expressiveness.

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Thank you

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