Antichain Algorithms for Finite Automata

1. Antichain Algorithms for Finite Automata Laurent DoyenLSV, ENS Cachan & CNRS Joint work with Jean-François Raskin

2. Acknowledgments J-F. Raskin M. De Wulf N. Maquet … S. Raje, E. Filiot, N. Jin, … T. Henzinger D. Berwanger K. Chatterjee

3. Antichains everywhere!

4. Antichains everywhere!

5. Antichains for… 200 states vs. 5000 states

6. Antichains for… 10 states vs. 120 states

7. Antichains for… 50 times faster…

8. Antichains for… http://www.antichains.be

9. Reachability

10. Reachability Given a (directed) graph G=(V,E),

11. Reachability Given a graph G=(V,E), two sets Init V and Final V, Final Init

12. Reachability Given a graph G=(V,E), two sets Init V and Final V, is there a (finite) path from Init to Final ?

13. Reachability Given a graph G=(V,E), two sets Init V and Final V, is there a (finite) path from Init to Final ?

14. Reachability Given a graph G=(V,E), two sets Init V and Final V, NLogSpace-complete is there a (finite) path from Init to Final ?

15. Reachability Given a graph G=(V,E), two sets Init V and Final V, NLogSpace-complete …when the graph is given explicitly ! is there a (finite) path from Init to Final ?

16. Reachability Reachability is NLogSpace-complete. • In practice, the graph is implicit: • symbolic graphs • subset constructions • pushdown graphs • transition system of Petri nets • region graph of timed automata • …

17. Running Example

18. Running example Given finite automata A1 and A2, deciding if L(A1)  L(A2) is PSpace-complete. (even deciding L(A2) = Σ*) Subset construction

19. Subset construction Init: sets containing initial states of A Final: sets containing no accepting states of A Final … {1,2,3} {1,2,3} {1,2,4} {1,3} {1,4} {1,2} {2,3} {1,2} {1,3} {3} {1} {2} {1} Init

20. Subset construction Init: sets containing initial states of A Final: sets containing no accepting states of A Final … {1,2,3} {1,2,3} {1,2,4} {1,3} {1,4} {1,2} {2,3} {1,2} {1,3} {3} {1} {2} {1} Init

21. Subset construction B: Bc: Successor in Ac of set s = set of successors in A of elements in s

22. Subset construction B: Bc: Successor in Ac of set s = set of successors in A of elements in s

23. Subset construction … Bc: Successor in Ac of set s  set of successors in A of elements in s

24. Backward Algorithm

25. Reachability Backward algorithm One-step predecessors of states in s  V: Final Init

26. Reachability Backward algorithm One-step predecessors of states in s  V: Final Init

27. Reachability Backward algorithm One-step predecessors of states in s  V: Final Init

28. Reachability Backward algorithm One-step predecessors of states in s  V: Final Init

29. Reachability Backward algorithm One-step predecessors of states in s  V: Final Init

30. Reachability Backward algorithm One-step predecessors of states in s  V: Final Init

31. Reachability Backward algorithm If then Final is reachable from Init Final Init

32. Reachability Backward algorithm Final Init

33. Reachability Backward algorithm Fixpoint such that Final Init

34. Antichain Algorithms

35. Structure in graphs Final Init

36. Structure in graphs Final Init Graph is partially ordered…

37. Structure in graphs Final Init Final Graph is monotone…

38. Structure in graphs A pre-order ≤f over V is a forward simulation if v2 ≤fv1≡ v2 forward-simulates v1 ≤f implies ≤f ≤f for some

39. Structure in graphs A pre-order ≤f over V is a forward simulation if v2 ≤fv1≡ v2 forward-simulates v1 Running example:  is a forward simulation ≤f implies  ≤f ≤f for some

40. Structure in graphs A pre-order ≤f over V is a forward simulation if v2 ≤fv1≡ v2 forward-simulates v1 Running example:  is a forward simulation ≤f implies   ≤f ≤f for some

41. Structure in graphs A forward simulation ≤f is compatible with Final if ≤f ≤f implies

42. Structure in graphs A forward simulation ≤f is compatible with Final if Running example Final {1,2,3} ≤f ≤f implies {1,3} {1,2} {2,3} {3} {1} {2} Equivalently, Final is ≤f-downward-closed.

43. Structure in graphs Theorem A pre-order ≤f over V is a forward simulation if and only if preserves ≤f-downward-closedness. i.e., if is ≤f-downward-closed, then is ≤f-downward-closed.

44. Structure in graphs Theorem A pre-order ≤f over V is a forward simulation if and only if preserves ≤f-downward-closedness. i.e., if is ≤f-downward-closed, then is ≤f-downward-closed. ≤f

45. Structure in graphs Theorem A pre-order ≤f over V is a forward simulation if and only if preserves ≤f-downward-closedness. i.e., if is ≤f-downward-closed, then is ≤f-downward-closed. ≤f ≤f

46. Structure in graphs Theorem A pre-order ≤f over V is a forward simulation if and only if preserves ≤f-downward-closedness. i.e., if is ≤f-downward-closed, then is ≤f-downward-closed. ≤f ≤f

47. Antichain algorithm ≤f forward simulation compatible with Final. Backward algorithm B(i) is ≤f-downward-closed for all i ≥ 0 !

48. Antichain algorithm ≤f forward simulation compatible with Final. Backward algorithm B(i) is ≤f-downward-closed for all i ≥ 0 ! Use ≤f-maximal elements as a symbolic representation.

49. Antichain algorithm ≤f forward simulation compatible with Final. Backward algorithm B(i) is ≤f-downward-closed for all i ≥ 0 ! Antichain algorithm

50. Antichain algorithm ≤f forward simulation compatible with Final. Backward algorithm B(i) is ≤f-downward-closed for all i ≥ 0 ! Antichain algorithm