Part 9. Lifting and Tubes

1. Part 9.Lifting and Tubes

2. Tubes • Edge (m,n) is lifted to form tube (M,N) • Tube (M,N) permits edge (m,n) N M n m Tube [Gail Murphy ‘95: Reflexions] [Holt ‘95 term “induce” instead of “lift”] [Krikhaar et al term “lift”.]

3. Tor LA Calif Ont Canada USA Nested Tubes &“Flow of Goods” USA ships to Canada California ships to Ontario Los Angeles ships to Toronto Tube from USA to Canada Tube from California to Ontario

4. Nested Ancestor and Descendent Tubes Exporting Importing LA Tor Calif Ont USA Canada 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 Canada LA Tor Calif Ont Ancestors Exports Imports Descendents Cousins ShipsTo Self loops ID Also: Self loop (ID) edges to pass through perimeters

6. xj xi f  xn x0 e Definition of Lifting of Edges • 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 <= j • We define • f £l e, that is, • f is lifted from e (or is sub-edge of e ) We explicitly allow f to be ID (zero length). Note that e can be a K, A, D or ID edge.

7. f £ e means f is sub-edge of e e is lifted to f e has tube f f is lowered to e We also define f = e, f < e, f ³ 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) xj xi f  xn x0 e Meaning of t £ e Note that £ is a partial ordering of edges in tree T

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) xj xi xn x0 e

9. Length of Edge (x,y) root Len(x,y) =def shortest distance from node x to node y following P, S and C edges (or 0 for an ID edge) P P P y P P Ancestor edge x Len(x,y) = 3 root C C S P C x C x C C Descendent edge y y Len(x,y) = 2 Cousin edge Len(x,y) = 4 Identity edge ID x Len(x, x) = 0

10. Lifting Shortens Edges (or keeps same length) t £ e  Len(t) £Len(e) t e

11. Part 10.Formal Definition of LiftingDefined Using Tarski Algebra

12. xj xi F  xn x0 E Approach to Formalizing Definition of Lifting • Given • a tree T and • any set of edges R • We define • The set of edges lifted from R, l(R), as follows: • For each edge E in R, l(R) contains each edge F that can be lifted from E • Definition given in terms of Tarski algebra

13.  Lifting K Edges to K Edges Eliminates non-K edges K,K(R) = Do o RK o Ao  K RK = R  K K,K(RK) K Do Ao K,K RK [See also Feijs, Krikhaar et al]

14. Lifting K to A and D Edges (and to ID edges) Allows ID edges K,A(R) = Do o RK o K  Ao K,D(R) = K o RK o Ao  DoRK = R  K K(R) = K,A(R) K,K(R) K,D(R) K,A(RK) Ao K K,A Do RK

15.  Lifting A and D Edges Allows ID edges A(R) = Do o RA o Do  Ao RA = R  Ao D(R) = Ao o RD o Ao  Do RD = R  Do Kinds of lifting: A, K, D where K consists of (K,A), (K,K), (K,D) Do Ao D(R) A(R) Ao Do RA RD A & D produce identity edges as well as A & D 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