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 planez = 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 hyperplane Ωn-

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 setFxi|a1,…,an-1 = { ωΩ | 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) = { ωΩ | 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:

Index Value of x1

Value of x2

Value of x3

Value of x4

0,0,0 Ω {0} {0} Ω

0,0,1 Ω {0} {0} Ω

0,0, … Ω {0} {0} Ω

0,1,0 Ω {1} {1} Ω

0,1,1 Ω {1} {1} Ω

… Ω … … Ω

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 s

κλ F = 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, s3

2 =

x1 x2 x3 x4

x1 Ω Ø Ø Ø

x2 Ø Ø Ω Ø

x3 Ø Ω Ø Ø

x4 Ø Ø Ø Ω

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)

aΩ

• Component-wise operation

Conclusion

• When cylindrification is defined as union of instantiations -

• Matrix representations of relations form a cylindrical algebra.