1 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)...
TRANSCRIPT
1
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
4
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
6
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
12
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
14
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
15
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
16
The lift function for edge set R is:(R) = A(R) K(R) D(R)
Combining the Preceding Definitions …
17
We can use function (R) to formally define t e as follows:
t e =def t ({e})
Formal Definition of t e
t
e