Positive Semantics of Projections
in Venn-Euler Diagrams
Joseph Gil – Technion
Elena Tulchinsky – Technion
Seminar Structure
• Venn-Euler diagrams
• Case for projections
• Positive semantics of projections
• Different approach : negative semantics of
projections
Terminology
• contour - simple closed plane curve
• district - set of points in the plane enclosed by a contour
• region - union, intersection or difference of districts
• zone - region having no other region contained within it
• shading - denote the empty set
• projection, context - another way of showing the
intersection of sets
Venn Diagrams
B
A
C
• n contours
• 2n zones
• shading to denote empty set
Venn Diagrams (cont.)
The simple and symmetrical Venn diagrams of four and five
contours
Venn diagram disadvantages:
– Difficult to draw
– Most regions take some pondering before it is clear
which combination of contours they represent
Venn-Euler Diagrams
B
A
C
D
• The notation of Venn-Euler diagram is obtained by a
relaxation of a demand that all contours in Venn diagrams
must intersect
• The interpretation of this diagram includes:
D  (C - B) - A
and
ABC = 
Projections
Women
Company Employees
Company Employees
Women
without projections
using projections
Denoting the set of all women employees
• A projection is a contour, which is used to denote an
intersection of a set with a context
• Dashed iconic representation is used to distinguish
projections from other contours
• Use of projections potentially reduces the number of zones
Case for Projections
• A Venn diagram with six
contours constructed using
More’s algorithm
• A Venn diagram
with six contours
using projections
shows the same 64
zones
Q
A
D
B
A
C
B
A
B
C
C
B
A
C
A
B
E
C
A
A
C
B
C
B
A
C
F
B
Case for Projections in Constraint
Diagrams
Queens
Kings
married
Executed
Henry VIII
• The sets Kings and Queens are disjoint
• The set Kings has an element named Henry VIII
• All women that Henry VIII married were queens
• There was at least one queen Henry VIII married who was
executed
• Divides the plane into 5 disjoint areas ( zones )
Case for Projections in Constraint
Diagrams (cont.)
• Executed contour must also
intersect the King contour
• State that Henry VIII was not
executed
•Divides the plane into 8 disjoint
areas
• Using of spider to refrain from
stating whether or not Henry VIII
was executed
• Draws the attention of the reader
to irrelevant point
Queens
Kings
married
Henry VIII
Executed
Queens
Kings
married
Henry VIII
Executed
Questions
• Context What is the context with which a projection
intersects?
• Interacting Projections What if two or more projections
intersect?
• Multi-Projections Can the same set be projected more
than once into a diagram? Can these two projections
intersect?
Intuitive Context of Projection
C
B
D
• Projection into an area defined
by multiple contours
• D~ = D ( B + C )
• To make the strongest possible
constraint we choose the minimal
possible context
A
B
D
• D~ = D B with B  A
B
C
D
• Multiple minimal contexts
• D~ = D ( B C )
Intuitive Context of Projection (cont.)
B1
B2
C2
• Generalization of previous
examples
D
• D~ = D ( ( B1 + C1 ) ( B2 + C2 ) )
C1
B
E
D
• Contours disjoint to projection can
not take part in the context
• D~ = D B
A
D
• The context of a contour can not
comprise of the contour itself
• An illegal projection
Mathematical Representation
Main idea: To define a formal mathematical representation for a
diagram
C
B
z1
z2
z3
• < { B, C }, {z1, z2, z3} >
z1 = B - C
z1 = { B }
z2 = B C
z2 = { B, C }
z3 = C - B
z3 = { C }
• Each zone is represented by the set of contours that contain it
Example
z9
E
z2
B
A
z1
z8
z7
z4 z6
z5
D
z3
C
< { A, B, C, D, E }, {z1, z2, z3, z4, z5, z6, z7, z8, z9 } >
z1 = { A }
z2 = { A, B }
z3 = { A, C }
z4 = { A, B, D }
z5 = { A, C, D }
z6 = { A, B, C, D }
z7 = { A, B, C }
z8 = { A, E }
z9 = { E }
Mathematical Representation (cont.)
Definition A diagram is a pair < C, Z > of a finite set C of
objects, which we will call contours, and a set Z of nonempty subsets of C, which we will call zones, such that c 
C, z  Z, c  z.
Dually: The district of a contour c is d ( c ) = { z  Z | c  z }.
The district of a set of contours S is the union of the districts of its
contours d ( S ) =  c  S d ( c ).
Covering
Covering is basically
containment of the set
of zones
Definition We say that X is covered by Y if d ( X )  d ( Y
). We say that X is strictly covered by Y if the set
containment in the above is strict.
(X and Y can be sets)
A cover by a set of contours
is reduced, if all
“redundant” contours are
remove from it
Definition A set of contours S is a reduced cover of X if S
strictly covers X, X S =  , and there is no S’  S such that S’
covers X.
Territory and Context
Definition The territory of X is the set of all of its reduced covers
 ( X ) = { S  C | S is a reduced cover of X }.
Definition The context of X,  ( X )   is the maximal
information that can be inferred from what covers it, i.e., its
territory
 ( X ) =  S   ( X ) d ( S ) =  S   ( X )  c  S d ( S ).
If on the other hand  ( X ) = , we say that X is context free.
Projections Diagram
Definition A projections diagram is a diagram < C, Z >,
with some set P  C of contours which are marked as
projections. A projections diagram is legal only if all of its
projections have a context.
Interacting Projections
I
• H~ = H I
H
E
• E~ = E H~ = E H I
I
U
• H~ = H ( I + E~ )
H
E
• E~ = E ( U + H~ )
H~ = H ( I + E ( U + H~ ) ) = H I + H E U + H E H~ =  H~ +

=HE
=HI+HEU=H(I+EU)
Solving a Linear Set Equation
Lemma Let  and  be two given sets. Then, the equation
x =  x +  holds if and only if   x   +; .
• The minimal solution must be taken
• In the example: H~ =  = H ( I + E U )
E~ = E ( U + H~) = E ( U + H ( I + E U
)=
=EU+EHI+EHU=E(U+HI
)
Dealing with Interacting Projections
• Main problem: the context of one projection includes other
projections and vice versa.
• System of equations:
– Unknowns and constants: sets
– Operations: union and intersect, “polynomial equations”
• Technique: use Gaussian like elimination
System of Equations
x1 = P1 (1, . . . , m, x2, . . . , xn )
...
xn = Pn (1, . . . , m, x1, . . . , xn-1 )
where x1, . . . , xn are the values of p  P ( unknowns ),
1, . . . , m are the values of c  C ( constants ),
P1, . . . , Pn are multivariate positive set polynomial
over
1, . . . , m and x1, . . . , xn.
Lemma Every multivariate set polynomial P over variables
1, . . . , k, x can be rewritten in a “linear” form
P ( 1, . . . , k, x ) = P1 ( 1, . . . , k ) x + P2 (1, . . . , k ).
Procedure for Interacting Projections
• Solve the first equation for the first variable
• Solution is in term of the other variables
• Substitute the solution into the remaining
equations
• Repeat until the solution is free of projections
• Substitute into all other solutions
• Repeat until all the solutions are free of
projections
Multi-Projections
B
D
• Df = D B
• Dg = D C
f
g
B
D
f
C
D
C
D
g
• Df = D B
• Dg = D C
•DBC=
Noncontiguous Contours
• Problem
B
C
D
f
D
g
• Df = D B
• Dg = D ( B C )
•  = Df Dg = D B C = Dg
• Main idea: unify the multi-projections
– Instead of having multiple projections of the same set, we
will allow the projection to be a noncontiguous contour
– The mathematical representation does not know that
contours are noncontiguous
– Only the layout is noncontiguous.
Noncontiguous Layout
• May have noncontiguous contours and noncontiguous
zones
E
z9
z8
A
z8
D
E
z4
B
z2
D
z6
z7
D
z5
z9
z3
C
z1
E
Noncontiguous Projection
B
D
C
D
• D~ = D B
• The interpretation of this diagram does not include:  = Df
Dg
Summary
• Context: the collection of minimal reduced
covers
• Semantics: computed by the intersection
with the context
• Interaction: solve a system of set equations
• Multi-projections: basically a matter of
layout
Related Work
• Negative semantics: compute the semantics
of a projection based also on the contours it
does not intersect with. (Gil, Howse, Kent,
Taylor)
• Different approach. Not clear which is more
intuitive
Difference between Positive and
Negative Semantics
B
D
E
• Positive Semantics : D~ = D B
D~ E = 
• Negative Semantics : D~ = D ( B - E )