positive semantics of projections in venn-euler diagrams

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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 DiagramsJoseph Gil TechnionElena Tulchinsky Technion

  • Venn-Euler diagrams Case for projections Positive semantics of projections Different approach : negative semantics of projectionsSeminar 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 setsTerminology

  • ABC n contours 2n zones shading to denote empty setVenn 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

    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

  • Projections

    Denoting the set of all women employeesusing 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 Venn diagram with six contours constructed using Mores 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 )

  • Case for Projections in Constraint Diagrams (cont.) Executed contour must also intersect the King contour State that Henry VIII was not executedDivides 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

    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.) 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} > 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 itMain 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 }

  • 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.)

  • CoveringDefinition 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 zonesA 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

    H~ = H I E~ = E H~ = E H I 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 equationx = 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 ProjectionsMain problem: the context of one projection includes other projections and vice versa.

    System of equations:Unknowns and constants: setsOperations: 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 formP ( 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

    Df = D B Dg = D C Df = D B Dg = D C D B C =

  • Noncontiguous Contours Problem Main idea: unify the multi-projectionsInstead of having multiple projections of the same set, we will allow the projection to be a noncontiguous contourThe mathematical representation does not know that contours are noncontiguousOnly the layout is noncontiguous.

    Df = D B Dg = D ( B C ) = Df Dg = D B C = Dg

  • Noncontiguous LayoutMay have noncontiguous contours and noncontiguous zones

  • D~ = D B The interpretation of this diagram does not include: = Df DgNoncontiguous Projection

  • SummaryContext: the collection of minimal reduced coversSemantics: computed by the intersection with the contextInteraction: solve a system of set equationsMulti-projections: basically a matter of layout

  • Related WorkNegative 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

  • BDE Negative Semantics : D~ = D ( B - E ) Positive Semantics : D~ = D BD~ E = Difference between Positive and Negative Semantics

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