1. Directed Graph Reachability in the Presence of Variability Michał Antkiewicz Oct 26, 2015 University of Waterloo http://gsd.uwaterloo.cahttp://necsis.ca http://clafer.org

2. We’d like to… 1.Model directed graphs with variability 2.Enforce or verify reachability Hands on mini-tutorial: http://t3-necsis.cs.uwaterloo.ca:8091/graphReachability

3. Requirements Directed graph or multi-graph Optional edges or multiple targets Edges have weight Optional nodes Enforce or verify that in all possible graphs a given node is reachable from another node N1 N2 N3? N4 E4?(2) E1?(1) E2?(2) E3?(5) E5?(1)

4. Modeling using Clafer Directed graph Outgoing edges disappear with the node Edges have weight Directed multi-graph abstract Node * abstract Edge -> Node * weight -> integer [ this.dref > 0 ] abstract Node * abstract Edge ->> Node *

5. Modeling a Concrete Graph with Variability abstract Node * abstract Edge ->> Node * weight -> integer [ this > 0 ] N1 N2 N3? N4 E1?(1) E3?(5) E4?(2) E5?(1) E2?(2) N1 : Node 1 E1 : Edge -> N2 ? [ weight = 1 ] E2 : Edge -> N2 ? [ weight = 2 ] E3 : Edge -> N3 ? [ weight = 5 ] N2 : Node E4 : Edge -> N3 ? [ weight = 2 ] N3 : Node ? E5 : Edge -> N4 ? [ weight = 1 ] N4 : Node Reference refinement of: abstract Edge ->> Node *

6. A Few Possible Graphs N1 N2 N3 N4N1 N2 N3 N4 E1(1) E5(1) … N1 N2 N3 N4 E1(1) E3(5) E4(2) E5(1) E2?(2) … N1 N2 N4N1 N2 N4 E1(1) … N1 N2 N4 E1(1) E2(2) …

7. Enforcing reachability of N4 from N1 N4 is in the set of nodes reachable from N1 “N1.Edge” N2, N3 “N1.Edge.Edge” = “(N2, N3).Edge” N3, N4 “N1.^Edge” Transitive closure (^) is “N1.Edge ++ N1.Edge.Edge ++ …” Finally “[ N4 in N1.^Edge ]” N1 N2 N3 N4 E1?(1) E3?(5) E4?(2) E5?(1) E2?(2)

8. Unfortunately, transitive closure is not yet* implemented in Clafer We must escape to Alloy to write that constraint * It is at the top of the list….

9. Translation to Alloy abstract Node * abstract Edge ->> Node * weight -> integer abstract sig Node { r_Edge : set Edge } abstract sig Edge { Edge_ref : one Node, r_weight : one weight } { one @r_Edge.this } sig weight { weight_ref : one Int } { one @r_weight.this } [ N4 in N1.^Edge ] [alloy| fact { N4 in N1.^(r_Edge.Edge_ref) } |]

10. A Few Possible Graphs N1 N2 N3 N4 E1?(1) E4?(2) E5?(1) N1 N2 N3 N4 E3?(5) E5?(1) N1 N2 N3 N4 E4?(2) E5?(1) E2?(2) N1 N2 N3 N4 E1?(1) E3?(5) E4?(2) E5?(1) E2?(2) N1 N2 N3 N4 E1?(1) E3?(5) E5?(1) E2?(2) N1 N2 N3 N4 E1?(1) E4?(2) E5?(1) E2?(2)

11. Exercises http://t3-necsis.cs.uwaterloo.ca:8091/graphReachability

12. Q1: Can you make N1 prohibited (that is give it cardinality 0) and get an instance? Why? A: No. “N1.^Edge” will be an empty set because N1 is empty and N4 cannot be a subset of an empty set.

13. Q2: Make the edges E1 and E2 mutually exclusive. Can they be put into a feature group as follows. Why? A1: “[E1 xor E2]”. A2: No, edges must be nested directly under nodes. N1 N2 N3 N4 E1?(1) E2?(2) E3?(5) E4?(2) E5?(1) xor N1 : Node 1 xor group E1 : Edge -> N2 ? E2 : Edge -> N2 ?

14. Q3: Make the edge E2 point to either N2 or N3. A: “E2 : Edge -> N2 ++ N3 ?”. N1 N2 N3 N4 E1?(1) E2?(2) E3?(5) E4?(2) E5?(1) xor N1 N2 N3 N4 E1?(1) E2?(2) E3?(5) E4?(2) E5?(1) N2++N3 E2->N3 E2->N2

15. Q4: Change the graph such that N4 is always reachable from N1 with minimal # of edges. A: “N3 : Node E5 : Edge -> N4 [ lone (E1 ++ E2) ] [ E3 xor E4 ] [ E4 E1 || E2 ]” [alloy| assert N4isReachable { N4 in N1.^(r_Edge.Edge_ref) } check N4isReachable for 1 but 5 Edge, 4 Node, 5 weight |] N1 N2 N3 N4 E1?(1) E2?(2) E3?(5) E4?(2) E5 (1) mux xor

16. Directed Graph Reachability in the Presence of Variability Thank You! http://gsd.uwaterloo.cahttp://necsis.ca http://clafer.org