What Can Spider Diagrams Say

1
What Can Spider Diagrams Say?

• Gem Stapleton, John Howse, John Taylor,
• Visual Modelling Group
• University of Brighton, UK
• Simon Thompson
• University of Kent, UK

2
Overview

• Related work - Shin
• What are spider diagrams?
• What do spider diagrams mean?
• How expressive are spider diagrams?
• Equivalent to monadic first order logic with
  equality.

3
Overview

• Related work - Shin
• What are spider diagrams?
• What do spider diagrams mean?
• How expressive are spider diagrams?
• Equivalent to monadic first order logic with
  equality.

4
Related work

• Shins Venn-II system

and
A
B
5
Related work

• Shins Venn-II system

A
B
A
B
or
6
Related work

• Shins Venn-II system
• Equivalent to monadic first order logic without
  equality.
• The strategy to prove this
• (1) Convert each diagram into a logic sentence.
• (2) Apply syntactic operations to a sentence
  until it has no nested quantifiers. Draw a
  diagram for each simple part.

7
Overview

• Related work
• What are spider diagrams?
• What do spider diagrams mean?
• How expressive are spider diagrams?
• Equivalent to monadic first order logic with
  equality.

8
Syntax
Spider foot
Contour label
A
B
Shaded zone
Zone (A,B)
Spider
Region set of zones
9
Syntax
Spider foot
Contour label
A
B
Shaded zone
Zone (A,B)
Spider
Region set of zones
10
Syntax
Spider foot
Contour label
A
B
Shaded zone
Zone (A,B)
Spider
Region set of zones
11
Syntax
Spider foot
Contour label
A
B
Shaded zone
Zone (A,B)
Spider
Region set of zones
12
Syntax
Spider foot
Contour label
A
B
Shaded zone
Zone (A,B)
Spider
Region set of zones
13
Syntax
Spider foot
Contour label
A
B
Shaded zone
Zone (A,B)
Spider
Region set of zones
14
Overview

• Related work - Shin
• What are spider diagrams?
• What do spider diagrams mean?
• How expressive are spider diagrams?
• Equivalent to monadic first order logic with
  equality.

15
Informal Semantics
A
B
There is an element in A and everything that is
in B is also in A
16
Informal Semantics
B
A
There are exactly two elements in B and
everything that is in A is also in B
17
Informal Semantics

• Regions represent sets.
• Spiders place lower bounds on the cardinalities
  of sets.
• Shading places upper bounds on the cardinalities
  of sets.
• Missing zones represent the empty set.

18
Informal Semantics
Zones (A,B) (AB,) (B,A) (,AB)
Sets 1 2
3 is a model
B
A
Zones (A,B) (AB,) (B,A) (,AB)
Sets 2
1,3 is not a model
19
Compound diagrams

• Increase expressiveness

Connectives. and or
U
C
B
A
20
Overview

• Related work - Shin
• What are spider diagrams?
• What do spider diagrams mean?
• How expressive are spider diagrams?
• Equivalent to monadic first order logic with
  equality.

21
Measuring expressiveness
Compare diagram expressiveness with the
expressiveness of a standard language First
Order Predicate Logic (FOPL). Which fragment of
FOPL is equivalent to the spider diagram
language?
22
Expressiveness

• A fragment of FOPL equivalent in expressive power
  requires
• Monadic Predicate Symbols P1,P2,
• Equality
• Quantifiers
• Negation, And, Or
• This is monadic first order logic with equality
  (MFOL)

23
Expressiveness
24
Expressiveness
25
Expressiveness

• Tasks
• Map each spider diagram to an expressively
  equivalent sentence
• Map each sentence to an expressively equivalent
  diagram

26
Examples a-diagrams
27
Examples a-diagrams
A
B
28
Mapping diagrams to sentences
We specify a sentence for each zone (including
missing zones). For the diagram, take the
conjunction of these, giving the sentence for d.
29
Mapping diagrams to sentences
Every spider diagram, D1, is semantically
equivalent to an a-diagram, D2. Theorem Every
spider diagram is expressively equivalent to a
sentence in MFOL.
30
Expressiveness

• Tasks
• Map each spider diagram to an expressively
  equivalent sentence
• Map each sentence to an expressively equivalent
  diagram

31
Mapping sentences to diagrams
Examples

32
Mapping sentences to diagrams
Examples

33
Mapping sentences to diagrams
Examples

34
Mapping sentences to diagrams
Examples

A
35
Mapping sentences to diagrams
Examples

A
A
36
Mapping sentences to diagrams
Examples

A
B
A
37
Mapping sentences to diagrams
Examples
Question Can we extend Shins approach for
Venn-II? No.
38
Mapping sentences to diagrams
Examples
A
39
Mapping sentences to diagrams
Examples
A
Not clear!
40
Studying models of diagrams
Consider the models for a diagram There is
usually an infinite class but some pattern
amongst the infinity of models (A,) (,A)
1 2,3 1 2,3,4 1 2,3,4,5
...
41
Studying models of diagrams
(A,B) (AB,) (B,A) (,AB) 1
2 3 1 2,4
3 1 2,4,5 3 1
2 3,6 1
2,4 3,6 1 2,4,5
3,6 ...
m
42
Studying models of diagrams
(A,B) (AB,) (B,A) (,AB) 1
2 3 1 2,4
3 1 2,4,5 3 1
2 3,6 1
2,4 3,6 1 2,4,5
3,6 ...
m
43
Studying models of diagrams
Can we draw a diagram with this model pattern?
(A,B) (AB,) (B,A) (,AB) 1,2
3 1 2
3 1 2,4 3 1
2,4,5 3 1 2 3,6
1 2,4 3,6 1
2,4,5 3,6 ...
m
m cannot be enlarged
n
m
44
Studying models of diagrams
Can we draw a diagram with this model pattern?
(A,B) (AB,) (B,A) (,AB) 1,2
3 1 2
3 1 2,4 3 1
2,4,5 3 1 2 3,6
1 2,4 3,6 1
2,4,5 3,6 ...
m
A
B
n
A
B
45
Mapping sentences to diagrams
If a sentence has a class of models which satisfy
this pattern i.e. a finite set of models plus
some extensions of models then we can draw a
diagram.
46
Mapping sentences to diagrams

• Given a sentence and a model, we have sets, some
  of which can be enlarged.
• If the sets are already large then they can be
  enlarged.
• A large set is one whose cardinality is at least
  the maximum number of nested quantifiers.
• There are only a finite number of models that
  have small sets.

47
Mapping sentences to diagrams

• Example
• Let S be the sentence with 1 nested
  quantifier.
• A model for S
• (A,) (,A)
• 1

We can deduce that (A,) can be enlarged. We
cannot deduce (,A) can be enlarged.
48
Mapping sentences to diagrams

• Theorem
• There is a finite set of models for sentence S,
  whose cones contain precisely the models for S.
• We call such a set of models a classifying set of
  models for S.

49
Mapping sentences to diagrams

• Strategy to find a diagram for S
• Given a finite classifying set of models for S,
• draw a diagram for each of these models
• no of spiders in each zone size of the
  represented set
• shade a zone if the represented set has
  cardinality less than the maximum number of
  nested quantifiers

50
Mapping sentences to diagrams

• Example
• The sentence
• maximum 1 nested quantifier
• (A,) (,A)
• 1
• 1 2
• 1

51
Mapping sentences to diagrams

• Example
• The sentence
• maximum 1 nested quantifier
• classifying set of models
• (A,) (,A)
• 1
• 1 2

52
Mapping sentences to diagrams

• Example
• The sentence
• maximum 1 nested quantifier
• classifying set of models
• (A,) (,A)
• 1
• 1 2

53
Mapping sentences to diagrams

• Example
• The sentence
• maximum 1 nested quantifier
• classifying set of models
• (A,) (,A)
• 1
• 1 2

A
A
A
54
Mapping sentences to diagrams

• Theorem
• Every sentence in MFOL has an expressively
  equivalent spider diagram.

55
Conclusion and further work

• The spider diagram language is equivalent in
  expressive power to monadic first order logic
  with equality.
• Spider diagrams properly increase expressiveness
  over Venn-II.
• Establish the expressive power of the constraint
  diagram language.

56
The end

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