Compiling with Decomposition

1. Compiling with Decomposition By Paul Elliott

2. Outline • Implicant Generator • Using Dependency Information • Decomposition Algorithm

3. Implicant Generator • Two components • solution generator • Iterative deepening • Enumerates all solutions from shortest to longest • Solutions come from search groups • Prunes supersets of solutions • Satisfies conflicts generated • solution tester • Tests for Valid, Satisfiable, Unsatisfiable • Valid testing is clause directed • Domain aware • Aware of search groups

4. Overall Algorithm

5. Generator

6. Generator (Continued)

7. Thruster Model • All variables have 2 element domains • Propellant Tank: F1 • Three Valves: V1, V2, V3 • A Thruster: R1 • Pressure measured at P1 and P3

8. Thruster Variable Constraint Hypergraph

9. Generator Search Tree

10. Tester

11. Tester (Continued) • Propagate(Assignment, Parity) • Reduce Domain of Variable • If Domain is empty, unsatisfiable, done • Satisfy or reduce clauses naming V • If Empty clause created, unsatisfiable, done • Propagate Unit Clauses

12. Tester Search Tree

13. Performance

14. Performance

15. Outline • Implicant Generator • Using Dependency Information • Decomposition Algorithm

16. Using Dependency Information • Goal Function • Observation Function • Transition Function

17. Goal Function Algorithm

18. Example • Compile the goal (P3 = High)

19. Example (2) • Two decomposition sets • {F1, V1, V2, V3, P3} • {P3, R1} • All sets mention P3, so reduce all sets • {F1, V1, V2, V3} • {R1} • Two solutions • (F1=Full) and (V1=Open) and (V2=Open) • (F1=Full) and (V1=Open) and (V3=Open)

20. Example Part 2 • Added Twist • Assume (V2=Open) and (V3=Open) conflict • Creates new goal table • (F1=Full) and (V1=Open) and (V2=Open) and (V3=Closed) • (F1=Full) and (V1=Open) and (V3=Open) and (V2=Closed)

21. Observation Function Algorithm

22. Observation Hypergraph

23. Observation Example • Decomposition • {F1, P1}, {P1, V1, V2, V3, P3}, {P3, R1, T1} • Dissents • First Set • (F1=Filled) and (P1=Low) • (F1=Empty) and (P1=High) • Third Set • (P3=Low) and (T1=Thrust) • (R1=Off) and (T1=Thrust) • (P3=High) and (R1=On) and (T1=NoThrust)

24. Generator Search Tree

25. Transition Algorithm

26. Transition Hypergraph

27. Transition Example • Compile the transition R1:(On->Off) • Gather nodes connected to the guard • {R1, CR1, R1’} • Remove R1 and R1’ • {CR1} • Unsatisfy and get the same compiled guard • (CR1=TurnOff)

28. Outline • Implicant Generator • Using Dependency Information • Decomposition Algorithm

29. Partitioning • Each compilation step involves unsatisfying the model over some set of variables and extra constraints • Partition the model by the settable variables • Finds bounded regions of the constraints space • Solution cannot be formed from variables that do not connect directly • Abstract away the indirect connection as the settable variable in-between the two variables

30. A Concept B C • Constraints • (A = B) • (C != B) • A and C are independent when B is set • Conflicts: • (A != B) • (C = B)

31. Generating Partitioning Algorithm • Three Functions • Gather Partition Space • Gather Reachable Variables • Remove Redundant Sets

32. Generate Partition Space

33. Gather Reachable Variables

34. Remove Redundant Sets

35. Total Running Time • Generate Partition Space • O(E2 * V+ E * V log V) • Gather Reachable • O(E * V log V + E * V) = O(E * V log V) • Remove Redundant Sets • O(E2 * V)

36. Partition Variables Vp = {So, Ss} = {F1, V1, V2, V3, R1, P1, P3, T1} Partitioning Example

37. Decomposition of Hypergraph

38. Trace of the Decomposition • Gather all groups of connected variables • 6 groups (Edges 1, 2, 15, 16,17, and the rest) • {F1, P1} • {F1, P1} • {P3, R1, T1} • {P3, R1, T1} • {R1, T1} • {P1, V1, V2, V3, P3}

39. Removing Redundant Sets • 3 Unique overlapping sets exist • {F1, P1} • {P3, R1, T1} • {P1, V1, V2, V3, P3}