# Outline for January 12, 2007

1. Greetings and Felicitations!
2. Take-Grant
1. Counterpoint to HRU result
2. Symmetry of take and grant rights
3. Islands (maximal subject-only tg-connected subgraphs)
4. Bridges (as a combination of terminal and initial spans)
3. Sharing
1. Definition: can·share(r, x, y, G0) true iff there exists a sequence of protection graphs G0, ..., Gn such that G0 |-* Gn using only take, grant, create, remove rules and in Gn, there is an edge from x to y labeled r
2. Theorem: can·share(r, x, y, G0) iff there is an edge from x to y labeled r in G0, or all of the following hold:
1. there is a vertex y′ with an edge from y′ to y labeled 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.
4. Model Interpretation
1. ACM very general, broadly applicable; Take-Grant more specific, can model fewer situations
2. Theorem: G0 protection graph with exactly one subject, no edges; R set of rights. Then G0 |-* G iff G is a finite directed graph containing subjects and objects only, with edges labeled from nonempty subsets of R, and with at least one subject with no incoming edges
3. Example: shared buffer managed by trusted third part
5. Stealing
1. Definition: can·steal(r, x, y, G0) true iff there is no edge from x to y labeled r in G0, and there exists a sequence of protection graphs G0, ..., Gn such that G0 |-* Gn in which:
1. Gn has an edge from x to y labeled r
2. There is a sequence of rule applications ρ1, ..., ρn such that Gi-1 |- Gi; and
3. For all vertices v, w in Gi-1, if there is an edge from v to y in G0 labeled r, then ρi is not of the form "v grants (r to y) to w"
2. Example