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Linear Representation of Relational Operations. Kenneth A. Presting University of North Carolina at Chapel Hill. Relations on a Domain. Domain is an arbitrary set, Ω Relations are subsets of Ω n All examples used today take Ω n as ordered tuples of natural numbers,

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Linear representation of relational operations

Linear Representation of Relational Operations

Kenneth A. Presting

University of North Carolina at Chapel Hill


Relations on a domain
Relations on a Domain

  • Domain is an arbitrary set, Ω

  • Relations are subsets of Ωn

  • All examples used today take Ωn as ordered tuples of natural numbers,

    Ωn = {(ai)1≤i≤n | ai  N }

  • All definitions and proofs today can extend to arbitrary domains, indexed by ordinals


Graph of a relation
Graph of a Relation

  • We want to study relations extensionally, so we begin from the relation’s graph

  • The graph is the set of tuples, in the context of the n-dimensional space

  • n-ary relation → set of n-tuples

  • Examples:

    x2 + y2 = p → points on a circle, in a plane

    z = nx + my + b → points in a plane, in 3-space


Hyperplanes and lines
Hyperplanes and Lines

  • Take an n-dimensional Cartesian product, Ωn, as an abstract coordinate space.

  • Then an n-1 dimensional subspace, Ωn-1, is an abstract hyperplane in Ωn.

  • For each point (a1,…,an-1) in the hyperplaneΩn-1, there is an abstract “perpendicular line,” Ω x {(a1,…,an-1)}


Illustration
Illustration

Graph, Hyperplane, Perpendicular Line, and Slice


Slices of the graph
Slices of the Graph

  • Let F(x1,…,xn) be an n-ary relation

  • Let the plain symbol F denote its graph:

    F = {(x1,…,xn)| F(x1,…,xn)}

  • Let a1,…,an-1 be n-1 elements of Ω

  • Then for each variable xi there is a set

    Fxi|a1,…,an-1 = { ωΩ | F(a1,…,ai-1,ω,ai,…, an-1}

  • This set is the xi’s which satisfy F(…xi…) when all the other variables are fixed


The matrix of slices
The Matrix of Slices

  • Every n-ary relation defines n set-valued functions on n-1 variables:

    Fxi(v1,…,vn-1) = { ωΩ | F(v1,…,vi-1,ω,vi,…,vn-1) }

  • The n-tuple of these functions is called the “matrix of slices” of the relation F


Properties of the matrix
Properties of the Matrix

  • Each slice is a subset of the domain

  • Each function Fxi(v1,…,vn-1) : Ωn-1 → 2Ω maps vectors over the domain to subsets of the domain

  • Application to measure theory


Inverse map matrices to relations
Inverse Map: Matrices to Relations

  • Two-stage process, one step at a time

  • Union across columns in each row:

    RowF(v1,…,vn-1) =

    n | i<j → ai = vj

    U { <ai>  Ωn | i=j → ai Fxj(v1,…,vn-1) }

    j=1 | i>j → ai = vj-1

  • Union of n-tuples from every row:

    F = U<vi>Ωn-1 RowF(v1,…,vn-1)


Properties of the slicing maps
Properties of the Slicing Maps

  • Map from relations to matrices is injective but not surjective

  • Inverse map from matrices to relations is surjective but not injective

  • Not all matrices in pre-image of a relation follow it homomorphically in operations


Boolean operations on matrices
Boolean Operations on Matrices

  • Matrices treated as vectors

  • i.e., Direct Product of Boolean algebras

    • Component-wise conjunction

    • Component-wise disjunction

    • Component-wise complementation


Cylindrical algebra operations
Cylindrical Algebra Operations

  • Diagonal Elements

    • Images of diagonal relations, operate by logical conjunction with operand relation

  • Cylindrifications

    • Binding a variable with existential quantifier

  • Substitutions

    • Exchange of variables in relational expression


The diagonal relations
The Diagonal Relations

  • Matrix images of an identity relation, xi = xj

  • Example. In four dimensions, x2 = x3 maps to:


Axioms for diagonals
Axioms for Diagonals

  • Universal Diagonal

    • dκκ = 1

  • Independence

    • κ {λ,μ} → cκ dλμ = dλμ

  • Complementation

    • κλ→ cκ (dκλ • F) • cκ (dκλ• ~F) = 0


Cylindrical identity elements
Cylindrical Identity Elements

  • 1 is the matrix with all components Ω, i.e. the image of a universal relation such as xi=xi

  • 0 is the matrix with all components Ø, i.e. the image of the empty relation


Diagonal operations are boolean
Diagonal Operations are Boolean

  • Boolean conjunction of relation matrix with diagonal relation matrix

  • Example


Substitution is not boolean
Substitution is not Boolean

  • Substitution of variables permutes the slices – not a component-wise operation

  • Composition of Diagonal with Substitution

    sκλF = cκ ( dκλ • F )

  • If we assume Boolean arithmetic, then standard matrix multiplication suffices


Boolean matrix multiplication
Boolean Matrix Multiplication

  • Take union down rows, of intersections across columns


Substitution operators
Substitution Operators

  • Square matrices, indexed by all variables in all relations

  • Substitution operator is the elementary matrix operator for exchange of columns

  • Example: in a four-dimensional CA, s32 =


Axioms for cylindrification
Axioms for Cylindrification

  • Identity

    • cκ 0 = 0

  • Order

    • F + cκ F = cκ F

  • Semi-Distributive

    • cκ (F + cκ G) = cκ F + cκ G

  • Commutative

    • cκcλ F = cλcκ F


Instantiation
Instantiation

  • Take an n-ary relation, F = F(x1,…,xn)

  • Fix xi = a, that is, consider the n-1-ary relation F|xi=a = F(x1,…,xi-1,a,xi+1,…,xn)

  • Each column in the matrix of F|xi=a is:

    Fxj|xi=a(v1,…,vn-2) =

    F(v1,…,vj-1,xj,vj,…,vi-1,a,vi+1,…,vn)


Cylindrification as union
Cylindrification as Union

  • Cylindrification affects all slices in every non-maximal column

  • Each slice in F|xi is a union of slices from instantiations:

    Fxj|xi(v1,…,vn-2) = U Fxj|xi=a(v1,…,vn-2)

    aΩ

  • Component-wise operation


Conclusion
Conclusion

  • When cylindrification is defined as union of instantiations -

  • Matrix representations of relations form a cylindrical algebra.