Outline for January 21, 1999

1.	Greetings and felicitations!

a.	Please get your project proposals in as soon as possible, so you can get started

b.	Hand out new homework; point out #6 and review rules of attacking lassen over the network

2.	Take-Grant

a.	Show bridges (as a combination of terminal and initial spans)

b.	Show islands (maximal subject-only tg-connected subgraphs)

c.	canÖshare(r, x, y, G0) iff there is an edge from x to y labelled r in G0, or all of the following hold: (1) there 
is a vertex  y'' with an edge from y' to y labelled r; (2) there is a subject y' which terminally spans to y'', 
or y' = y''; (3)  there is a subject x' which initially spans to x, or x' = x; and (4) there is a sequence of 
islands I1, ..., In connected by bridges for which x' is in I1 and y' is in In .

d.	Describe canÖsteal; don't state theorem

3.	Lattice models

a.	poset,  the relation

b.	highest and lowest

c.	Set of classes SC is a partially ordered set under relation  with GLB (greatest lower bound), LUB (least 
upper bound) operators 

d.	Note: is reflexive, transitive, antisymmetric 

e.	Examples: (A, C)  (A', C') iff A  A' and C is a subset of C';
LUB((A, C), (A', C')) = (max(A, A'), �(C, C')), GLB((A, C), (A', C')) = (min(A, A'), ¥(C, C'))

4.	Decidability vs. Undecidability

a.	Notion of type; subject, object types

b.	SPM: types fixed at creation

c.	tickets: X/r (apply right r to entity X), X/r:c (c copy flag, so can give right away)

d.	link predicate: X possesses tickets in set dom(X). Then link(X, Y) = rights over X and Y ? dom(X) and 
dom(Y), and true; basically, everything connecting X and Y

e.	copying X/r:c from dom(X) to dom(Z): needs X/rc ? dom(X), link(X, Z), and t(X)/r:c ? f(t(X), t(Z)), f 
filter function

f.	Do TG example pp. 82-83

g.	create: can-create (cc) relation says that subject of type a can create entities of type b iff cc(a, b) holds

h.	create rule: crp(a, b) = set of tickets added to dom(A), crc(a, b) = set of tickets added to dom(B)

i.	attenuation: cr(a, b) = crp(a,b)|crc(a, b) is attenuating if: (1) crc(a, b) ¼ crp(a, b) and a/r:c ? crp(a, b) ? 
self/r:c ? crp(a, b)

j.	cyclic: cc(a, b) ? cc(b, a)

k.	if attenuating acyclic, it's decidable; so that is sufficient. Open question: is it necessary?