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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 ndimensional 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 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 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
U { <ai>  Ωn
j=1
| i<j → ai = vj
| i=j → ai  Fxj(v1,…,vn-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 Value of Value of Value of
x1
x2
x3
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
sκλ 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, s32 =
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)
aΩ
• Component-wise operation
Conclusion
• When cylindrification is defined as union of
instantiations • Matrix representations of relations form a
cylindrical algebra.