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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. - PowerPoint PPT Presentation
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Positive Semantics of Projections in Venn-Euler Diagrams
Joseph Gil – Technion
Elena Tulchinsky – Technion
• Venn-Euler diagrams
• Case for projections
• Positive semantics of projections
• Different approach : negative semantics of projections
Seminar Structure
• 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
Terminology
AB
C
• n contours
• 2n zones
• shading to denote empty set
Venn Diagrams
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
AB
CD
• 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 =
• 9 zones instead of 24=16 in Venn diagram of 4 contours
ProjectionsCompany Employees
Women
Company EmployeesWomen
Denoting the set of all women employees
using projectionswithout projections
• 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
C
B
ED
F
A B
C
A
C
B
A
C
B
A
C
B AC
B
A
C
B
Q
A B
C
• A Venn diagram with six contours constructed using More’s algorithm
• A Venn diagram with six contours using projections shows the same 64 zones
Case for Projections in Constraint Diagrams
• 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 )
Kings
Queens
Executed
Henry VIII
married
Case for Projections in Constraint Diagrams (cont.)
KingsQueens
Executed
Henry VIII
married
Kings Queens
Executed
Henry VIII
married
• 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
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
B CD
BD
A
B
D
C
• Projection into an area defined by multiple contours
• D~ = D ( B + C )
• To make the strongest possible constraint we choose the minimal possible context
• D~ = D B with B A
• Multiple minimal contexts
• D~ = D ( B C )
Intuitive Context of Projection (cont.)
B2 C2B1
C1
D
B
DE
A D
• Generalization of previous examples
• D~ = D ( ( B1 + C1 ) ( B2 + C2 ) )
• Contours disjoint to projection can not take part in the context
• D~ = D B
• The context of a contour can not comprise of the contour itself
• An illegal projection
B C
z1 z2 z3
• < { B, C }, {z1, z2, z3} >
z1 = B - C
z2 = B C
z3 = C - B
z1 = { B }
z2 = { B, C }
z3 = { C }
• Each zone is represented by the set of contours that contain it
Main idea: To define a formal mathematical representation for a diagram
Mathematical Representation
Example
< { A, B, C, D, E }, {z1, z2, z3, z4, z5, z6, z7, z8, z9 } >
z1 = { A }
z4 = { A, B, D }
z7 = { A, B, C }
z2 = { A, B }
z5 = { A, C, D }
z8 = { A, E }
z3 = { A, C }
z6 = { A, B, C, D }
z9 = { E }
z1
z2 z3
z7
z4 z5z6
z8z9 A
B CD
E
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 ).
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 non-empty subsets of C, which we will call zones, such that c C, z Z, c z.
Mathematical Representation (cont.)
Covering
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)
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.
Covering is basically containment of the set of zones
A cover by a set of contours is reduced, if all “redundant” contours are remove from it
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.
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.
Projections Diagram
Interacting Projections
HE
I
• H~ = H I
• E~ = E H~ = E H I
I U
H E
• H~ = H ( I + E~ )
• E~ = E ( U + H~ )
H~ = H ( I + E ( U + H~ ) ) = H I + H E U + H E H~ = H~ +
= H E
= H I + H E U = H ( I + E U )
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 ) =
= E U + E H I + E H U = E ( U + H I )
Solving a Linear Set Equation
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
f g
B CDD
f g
B CDD
• Df = D B
• Dg = D C
• Df = D B
• Dg = D C
• D B C =
Noncontiguous Contours • Problem
• 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.
f g
B CDD • Df = D B
• Dg = D ( B C )
• = Df Dg = D B C = Dg
Noncontiguous Layout
• May have noncontiguous contours and noncontiguous zones
z9
z1
z2 z3z7
z8z9
z8
z4 z5z6
A
B C
E
E
E
D DD
B CDD
• D~ = D B
• The interpretation of this diagram does not include: = Df Dg
Noncontiguous Projection
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
BD E
• Negative Semantics : D~ = D ( B - E )
• Positive Semantics : D~ = D B
D~ E =
Difference between Positive and Negative Semantics