- 62 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Linear Representation of Relational Operations' - petra-leonard

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### 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 = { 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:

Axioms for Diagonals

- Universal Diagonal
- dκκ = 1
- Independence
- κ {λ,μ} → cκ dλμ = dλμ
- Complementation
- κλ→ 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)

aΩ

- Component-wise operation

Conclusion

- When cylindrification is defined as union of instantiations -
- Matrix representations of relations form a cylindrical algebra.

Download Presentation

Connecting to Server..