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