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Part 9.Lifting and Tubes

2

Tubes

M N

m n

Tube

•Edge (m,n) is lifted to form tube (M,N)•Tube (M,N) permits edge (m,n)

[Gail Murphy ‘95: Reflexions] [Holt ‘95 term “induce” instead of “lift”] [Krikhaar et al term “lift”.]

3

Nested Tubes &“Flow of Goods”

USA

LA

Calif

Canada

Tor

Ont

USA ships to CanadaCalifornia ships to Ontario

Los Angeles ships to Toronto

Tube from USA to CanadaTube from California to Ontario

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Nested Ancestor and Descendent Tubes

USA

LA

Calif

Canada

Tor

Ont

Exporting Importing

Ancestor/descendent tubes go beyond the “usual” meaning of lifting

“Flow” vs “dependency” (or “visibility”)

Term “import” is inconsistently used.

5

LA Ships to Toronto:What lifting can occur?

USA

LA

Calif

Canada

Tor

Ont

Exports Imports

ShipsTo

Ancestors Descendents

Cousins

Also: Self loop (ID) edges to pass through perimeters

Self loops ID

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Given a tree T with edge e = (x0 xn)

with shortest path between x0 and xn (x0 x1 … xn), where path follows P, S and C edges (or follows an ID edge)

where

f = (xi xj) and i <= jWe define

f e, that is,

f is lifted from e (or is sub-edge of e )

Definition of Lifting of Edges

We explicitly allow f to be ID (zero length). Note that e can be a K, A, D or ID edge.

x0

f

e

xn

xixj

7

f e means

f is sub-edge of ee is lifted to fe has tube ff is lowered to e

We also define f e, f e, f e, f e

in the obvious way. We extend this definition to triples, so we write

F E

when F =(w t x), E = (y u z), and (w x) (y z)

Meaning of t e

Note that is a partial ordering of edges in tree T

x0

f

e

xn

xixj

8

Definition of Length of Edge

Given: a tree T with edge E = (x0 xn)

with shortest path between x0 and xn (x0 x1 … xn), where path follows P, S and C edges (or follows an ID edge)

We define:Len(E) =def n = (number of non-ID edges on shortest path)

x0e

xn

xixj

9

Length of Edge (x,y)

Len(x,y) =def shortest distance from node x to node y following P, S and C edges (or 0 for an ID edge)

x Len(x, x) = 0

x

y

Len(x,y) = 3

root

xy

Cousin edge

Identity edge ID

Ancestor edge

Len(x,y) = 4

y

x

Len(x,y) = 2

root

Descendent edge

P C

C

S

PP

PP

P

CC

CC

10

Lifting Shortens Edges (or keeps same length)

t e Len(t) Len(e)

t

e

11

Part 10.Formal Definition of Lifting

Defined Using Tarski Algebra

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Approach to Formalizing Definition of Lifting

Given a tree T and any set of edges R

We defineThe set of edges lifted from R, (R), as follows:For each edge E in R, (R) contains each edge F that can be lifted from E

Definition given in terms of Tarski algebra

x0

F

E

xn

xixj

13

Lifting K Edges to K Edges

K,K(R) = Do o RK o Ao K RK = R K

K,K(RK)

RK

[See also Feijs, Krikhaar et al]

K

AoDo K,K

Eliminates non-K edges

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Lifting K to A and D Edges (and to ID edges)

K,A(R) =Do o RK o K Ao

K,D(R) = K o RK o Ao Do RK = R K

RK

K,A(RK)Ao

K(R) = K,A(R) K,K(R) K,D(R)

KDo

K,A

Allows ID edges

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Lifting A and D Edges

A(R) = Do o RA o Do Ao RA = R Ao

D(R) = Ao o RD o Ao Do RD = R Do

RA RD

A(R) D(R)

Kinds of lifting:A, K, D

where K consists of(K,A), (K,K), (K,D)Do

Do

Ao

Ao

A & D produce identity edges as well as A & D edges

Allows ID edges

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The lift function for edge set R is:(R) = A(R) K(R) D(R)

Combining the Preceding Definitions …

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We can use function (R) to formally define t e as follows:

t e =def t ({e})

Formal Definition of t e

t

e