The Expressiveness of Spider Diagrams Augmented with Constants

1
The Expressiveness of Spider Diagrams Augmented
with Constants

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

2
Overview

• Related work - Shin
• Syntax of spider diagrams
• Semantics of spider diagrams
• The expressiveness of spider diagrams
• Augment spider diagrams with constants
• syntax
• semantics
• Does augmenting spider diagrams with constants
  increase expressiveness?

3
Overview

• Related work - Shin
• Syntax of spider diagrams
• Semantics of spider diagrams
• The expressiveness of spider diagrams
• Augment spider diagrams with constants
• syntax
• semantics
• Does augmenting spider diagrams with constants
  increase expressiveness?

4
Shins Venn-II system

• Augments Venn diagrams with x-sequences

A
B
x-sequence
5
Shins Venn-II system
Semantics
and
A
B
6
Shins Venn-II system
Semantics
and
A
B
More than one x-sequence does not imply more
than one element
7
Shins Venn-II system

• Connecting diagrams

A
B
A
B
d2
d1
or
8
Expressiveness of Venn-II

• 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.
• Cannot express arbitrary finite lower and upper
  bounds

9
Overview

• Related work - Shin
• Syntax of spider diagrams
• Semantics of spider diagrams
• The expressiveness of spider diagrams
• Augment spider diagrams with constants
• syntax
• semantics
• Does augmenting spider diagrams with constants
  increase expressiveness?

10
Syntax
Contours
A
B
Shaded zone
Zone
Existential spider
Region set of zones
11
Syntax
Contours
A
B
Shaded zone
Zone
Existential spider
Region set of zones
12
Syntax
Contours
A
B
Shaded zone
Zone
Existential spider
Region set of zones
13
Syntax
Contours
A
B
Shaded zone
Zone
Existential spider
Region set of zones
14
Syntax
Contours
A
B
Shaded zone
Zone
Existential spider
Region set of zones
15
Syntax

• Euler diagram based

16
Syntax

• Euler diagram based
• Well-formedness conditions
• no concurrency

A
B
17
Syntax

• Euler diagram based
• Well-formedness conditions
• no concurrency
• no self intersection (simple)

A
B
C
18
Syntax

• Euler diagram based
• Well-formedness conditions
• no concurrency
• no self intersection (simple)
• no disconnected zones

A
B
19
Overview

• Related work - Shin
• Syntax of spider diagrams
• Semantics of spider diagrams
• The expressiveness of spider diagrams
• Augment spider diagrams with constants
• syntax
• semantics
• Does augmenting spider diagrams with constants
  increase expressiveness?

20
Informal Semantics
A
B
There is an element in A and everything that is
in B is also in A
21
Informal Semantics
B
A
There are exactly two elements in B and
everything that is in A is also in B
22
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.

23
Compound Diagrams

• Negation not

Connectives. and or
U
C
B
A
24
Overview

• Related work - Shin
• Syntax of spider diagrams
• Semantics of spider diagrams
• The expressiveness of spider diagrams
• Augment spider diagrams with constants
• syntax
• semantics
• Does augmenting spider diagrams with constants
  increase expressiveness?

25
Expressiveness of Spider Diagrams

• Spider diagrams
• Equivalent to monadic first order logic with
  equality.
• The strategy to prove this
• (1) Convert each diagram into a logic sentence.
• (2) Find minimal models for a sentence. Draw a
  diagram for each minimal model.

26
Expressiveness of Spider Diagrams

• More expressive than Venn-II
• Can express arbitrary finite lower and upper
  cardinality constraints
• Not clear that we can talk about named
  individuals
• contrast there exists x such that x is a cat
• spot is a cat

27
Overview

• Related work - Shin
• Syntax of spider diagrams
• Semantics of spider diagrams
• The expressiveness of spider diagrams
• Augment spider diagrams with constants
• syntax
• semantics
• Does augmenting spider diagrams with constants
  increase expressiveness?

28
Syntax
constant spider
Contours
S
constant spider label
A
B
Shaded zone
Zone
Existential spider
Region set of zones
29
Syntax
constant spider
Contours
S
constant spider label
A
B
Shaded zone
Zone
Existential spider
Region set of zones
30
Syntax
constant spider
Contour label
S
constant spider label
A
B
Shaded zone
Zone
Existential spider
Fixed label set used for both contours and
spiders
Region set of zones
31
Informal Semantics
A
B
S
The individual S is in A and there exists another
element in A distinct from S. B is a subset of A
32
Informal Semantics
B
A
S
T
The individuals S and T are distinct. The
individuals S and T are in B. A is a subset of B.
33
Semantics

• Constant spiders represent individuals
• Distinct constant spiders in a single unitary
  diagram represent distinct individuals
• Formally, zones map to sets and constant spiders
  map to single element sets

34
Overview

• Related work - Shin
• Syntax of spider diagrams
• Semantics of spider diagrams
• The expressiveness of spider diagrams
• Augment spider diagrams with constants
• syntax
• semantics
• Does augmenting spider diagrams with constants
  increase expressiveness?

35
Expressiveness

• Constant spiders are syntactic sugar
• no increase in expressiveness
• Diagram with constants -gt diagram without
  constants

36
Semantic Equivalence
37
Example

• Semantic equivalence

A
B
A
B
C
C
d2
d1
38
Example

• Semantic equivalence

A
B
A
B
C
C
d2
d1
A
B
C
d3
39
Expressiveness

• Aim diagram with constants -gt diagram without
  constants

A
B
C
d1
Formally, C represents a single element set in
any model
40
Expressiveness

• Expanding constant spiders

A
B
A
B
C
C
d2
d1
Formally, C represents a single element set in
any model
41
Expressiveness

• Expanding constant spiders

A
B
A
B
C
C
d2
d1
42
Expressiveness

• Expanding constant spiders

A
B
A
B
C
C
d2
d1
A
B
C
d3
43
Expressiveness

• Expanding constant spiders

A
B
A
B
A
B
C
C
C
d2
d4
d1
A
B
A
B
C
C
d5
d3
44
Expressiveness

• Expanding constant spiders

A
B
A
B
A
B
C
C
C
d2
d4
d1
A
B
A
B
C
C
d5
d3
Not well-formed
45
Expressiveness

• We require a systematic way of expanding
  constants to give well-formed diagrams.
• One node -gt can ensure result is well-formed

46
Expressiveness

• Splitting Spiders

A
B
A
B
A
B
C
C
C
d1
d3
d2
47
Expressiveness

• Theorem
• Every spider diagram is semantically equivalent
  to a spider diagram where all spiders have just
  one node.

48
Expressiveness

• Theorem
• Augmenting the spider diagram language with
  constant spiders does not increase expressiveness
• Proof Strategy
• Split all the spiders.
• Replace each constant spider by a contour with
  shading and an existential.

49
Expressiveness

• Theorem
• Augmenting the spider diagram language with
  constant spiders does not increase expressiveness
• Corollary
• The language of spider diagrams augmented with
  constants is equivalent to MFOPLe

50
Conclusion and Further Work

• Augmented the spider diagram language with
  constants.
• Augmenting with constant spiders no increase in
  expressiveness.
• The augmented language is equivalent in
  expressive power to monadic first order logic
  with equality.
• Establish whether constant spiders increase
  usability
• Develop sound and complete rules for augmented
  system
• Is the language of spider diagrams with constants
  but without existentials equally expressive?

51
The end

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