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

A B C n contours 2 n 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 A B 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 =  9 zones instead of 2 4 =16 in Venn diagram of 4 contours

Projections Company Employees Women Company EmployeesWomen Denoting the set of all women employees using projections without 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 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.) Kings Queens 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 C D B D 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 C2 B1 C1 D B D E AD 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 z1z2z3 z 1 = B - C z 2 = B C z 3 = C - B z 1 = { B } z 2 = { B, C } z 3 = { 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 z 1 = { A } z 4 = { A, B, D } z 7 = { A, B, C } z 2 = { A, B } z 5 = { A, C, D } z 8 = { A, E } z 3 = { A, C } z 6 = { A, B, C, D } z 9 = { E } z1 z2z3 z7 z4z5z6 z8 z9 A B C D 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 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, 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 H E 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 x 1 = P 1 (  1,...,  m, x 2,..., x n )... x n = P n (  1,...,  m, x 1,..., x n-1 ) where x 1,..., x n are the values of p  P ( unknowns ),  1,...,  m are the values of c  C ( constants ), P 1,..., P n are multivariate positive set polynomial over  1,...,  m and x 1,..., x n. Lemma Every multivariate set polynomial P over variables  1,...,  k, x can be rewritten in a “linear” form P (  1,...,  k, x ) = P 1 (  1,...,  k ) x + P 2 (  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 C D D f g B C D D D f = D B D g = D C D f = D B D g = 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. fg B C D D D f = D B D g = D ( B C )  = D f D g = D B C = D g

Noncontiguous Layout May have noncontiguous contours and noncontiguous zones z9 z1 z2z3 z7 z8 z9 z8 z4 z5 z6 A B C E E E DD D

B C D D D~ = D B The interpretation of this diagram does not include:  = D f D g 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

B DE Negative Semantics : D~ = D ( B - E ) Positive Semantics : D~ = D B D~ E =  Difference between Positive and Negative Semantics