A Constraint Diagram Reasoning System

1
A Constraint Diagram Reasoning System

• G. Stapleton, J. Howse and J. Taylor
• University of Brighton, UK

2
Overview

• Diagrammatic systems
• What is a constraint diagram syntax
• What do constraint diagrams mean semantics
• Satisfiability
• Diagrammatic reasoning

3
Diagrammatic Systems

• Venn-Peirce Diagrams
• express whether a set is empty or non-empty
• sound and complete reasoning rules

A
B
4
Diagrammatic Systems

• Spider Diagrams
• extend Venn-Peirce diagrams
• express lower and upper bounds on cardinalities
  of sets
• sound and complete reasoning rules

A
B
5
Diagrammatic Systems

• Constraint Diagrams
• extend spider diagrams
• far more expressive than spider diagrams
• they allow universal quantification and
  relational navigation
• can be used to model software systems

f
A
6
Syntax of unitary diagrams

• Contours

A
B
7
Syntax of unitary diagrams

• Contours
• Zones
• Regions are sets
• of zones

A
B
three zones
8
Syntax of unitary diagrams

• Contours
• Zones
• Shaded zones

A
B
shaded zone
9
Syntax of unitary diagrams

• Contours
• Zones
• Shaded zones
• Existential spiders

A
B
existential spiders
10
Syntax of unitary diagrams

• Contours
• Zones
• Shaded zones
• Existential spiders
• Universal spiders

A
B
universal spider
11
Syntax of unitary diagrams

• Contours
• Zones
• Shaded zones
• Existential spiders
• Universal spiders
• Habitats

A
B
12
Syntax of unitary diagrams

• Contours
• Zones
• Shaded zones
• Existential spiders
• Universal spiders
• Habitats
• Arrows

A
B
f
arrow
13
Syntax of compound diagrams

• If D1 and D2 are constraint diagrams so
  are (or) (and)
• with brackets when necessary.

14
Semantics of unitary diagrams

• Regions denote sets.
• A intersection B is empty

A
B
15
Semantics of unitary diagrams

• Regions denote sets. An existential spider
  denotes the existence of an element in
  the set represented by its habitat.
• The cardinality of A is at least 1. The
  cardinality of U-A is at least 1.

A
B
16
Semantics of unitary diagrams

• Regions denote sets.
• An existential spider
• denotes the existence
• of an element in the set
• represented by its habitat.
• In a shaded region, all of
• the elements are represented by existential
  spiders.
• The cardinality of B is at most 1.

A
B
17
Semantics of unitary diagrams

• Arrow represent relations
• The is an x in B such
• that for all y in
• U-(A union B), y.fx.

A
B
f
18
Semantics of unitary diagrams

• This diagram asserts that
• A and B are disjoint,
• B is not empty and
• there exists x in U-(A union B) such that for all
  a in A the relational image of a under the
  relation f is x.

B
A
f
19
Semantics of unitary diagrams

• This diagram could equally well be interpreted as
  
• for all a in A, there exists an x in U-(A union
  B) such that the relational image of a
• under f is x',
• but we will not allow such a reading.

B
A
f
20
Satisfiability

• We map regions to subsets of U
• and map arrows labels to relations on U
• An interpretation is said to satisfy a unitary
  diagram if it agrees with the meaning of the
  diagram.

21
Satisfiability

• In an alpha-diagram all spiders inhabit exactly
  one zone.
• We will identify whether or not a unitary
  alpha diagram is satisfiable.

A
B
f
f
22
Satisfiability

• A satisfiable diagram

A
B
f
f
23
Satisfiability

• An unsatisfiable diagram.
• We say the two arrows are incompatible.

A
f
f
d
24
Satisfiability

• Theorem
• Unitary alpha-diagram d is satisfiable if and
  only if d does not contain incompatible arrows.

25
Diagrammatic reasoning

• Reasoning rules transform one diagram into
  another.
• We will illustrate some of the reasoning rules.

26
Diagrammatic reasoning

• Inconsistency.
• If d contains incompatible arrows we may replace
  d with any diagram.

A
f
f
d1
d2
27
Diagrammatic reasoning

• Erasure of an arrow

28
Diagrammatic reasoning

• Introduction of an arrow universal deduction.

l
A
A
l
l
d2
d1
29
Diagrammatic reasoning

• Introduction of an arrow spider to contour.

l
A
A
l
l
d2
d1
30
Diagrammatic reasoning

• Splitting existential spiders.

B
A
B
A
f
f
B
A
f
31
Alternative Semantics incorrectly splitting
spiders

• If for all comes before there exists we cannot
  split spiders.

B
A
B
A
f
f
B
A
f
32
Diagrammatic reasoning

• Disjunctifying diagrams -- alpha diagrams with
  the same zones.
• Example 1

A


A
f

A
f






g
g
d
d1
d2
33
Diagrammatic reasoning

• Disjunctifying diagrams -- alpha diagrams with
  the same zones.
• Example 2

B
B
A
A
A
B

f


f
f







g
g




d2

d1
d



A
B
A
B
A
B

f
f


g
g
g
d'
d3
d4
34
Soundness and Completeness

• Theorem
• The system is sound and complete.

35
Conclusion and Further Work

• We have
• defined a constraint diagram language with
  restricted semantics,
• identified syntactic criteria for
  satisfiability,
• given sound and complete reasoning rules.

36
Conclusion and Further Work

• Short term plans
• relax semantic constraints,
• investigate the expressiveness of the system.
• Long term plans
• develop tools to support modelling and
  reasoning with constraint diagrams.