Linear Representation of Relational Operations

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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

Kenneth A. Presting

University of North Carolina at Chapel Hill

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
• 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
• 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

Graph, Hyperplane, Perpendicular Line, and Slice

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
• 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
• 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
• 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
• 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
• Matrices treated as vectors
• i.e., Direct Product of Boolean algebras
• Component-wise conjunction
• Component-wise disjunction
• Component-wise complementation
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
• Matrix images of an identity relation, xi = xj
• Example. In four dimensions, x2 = x3 maps to:
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
• 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
• Boolean conjunction of relation matrix with diagonal relation matrix
• Example
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
• Take union down rows, of intersections across columns
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
• 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
• 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 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
• When cylindrification is defined as union of instantiations -
• Matrix representations of relations form a cylindrical algebra.