1. Reduction of inductive predicates for shape analysis of circular lists Daniel Stutzman April 27, 2010

2. circular lists

3. shape analysis of circular lists

4. inductive predicates for shape analysis of circular lists

5. Reduction of inductive predicates for shape analysis of circular lists

6. Let’s make a sorted linked list struct ListNode { int data; struct ListNode* next; struct ListNode* prev; }

7. Let’s make a sorted linked list 3 0 47 0 x

8. Run-time “checker” functions boolean is_sll(ListNode* x) { return (x->prev == NULL) && is_sll2(x); } boolean is_sll2(ListNode* x, int min_data) { return (x->data >= min_data) && (x->next->prev == x) && is_sll2(x->next, x->data); }

9. Run-time “checker” functions boolean is_sll(ListNode* x) { return (x->prev == NULL) && is_sll2(x); } boolean is_sll2(ListNode* x) { return (x->next->data >= x->data) && (x->next->prev == x) && is_sll2(x->next); }

10. Run-time “checker” functions boolean is_sll(ListNode* x) { return (x->prev == NULL) && is_sll2(x); } boolean is_sll2(ListNode* x) { return (x->next == NULL) || ((x->next->data >= x->data) && (x->next->prev == x) && is_sll2(x->next)); }

11. Static inductive predicates is_sll(x) = null(x->prev) /\ is_sll2(x) is_sll2(x) = null(x->next) V ((x->next->data >= x->data) /\ (x->next->prev == x) /\ is_sll2(x->next))

12. ? 0 x is_sll2 Example in a shape domain

13. Unfolding ? 0 x is_sll2 ?

14. Unfolding ? 0 x is_sll2

15. Unfolding ? 0 0 x

16. ? 0 x is_sll2 ? ? 0 0 x V

17. Let’s make a circular list 3 0 47 0 x

18. 3 0 47 x

19. 347 x

20. Walking backwards ? ? ? x is_circular_list

21. Walking backwards ? ? ? x is_circular_list

22. Walking backwards ? ? ? x is_circular_list Error: Unable to find an appropriate edge to unfold

23. Reduction ?? x is_list

24. Reduction ?? x is_list

25. Reduction ?? is_list ? x

26. Two equivalent summaries is_bounded_list

27. Two equivalent summaries is_bounded_list

28. Two equivalent summaries is_bounded_list

29. Two equivalent summaries is_bounded_ list is_bounded_ list

30. Two equivalent summaries is_ bound ed_list is_ bound ed_list

31. Two equivalent summaries

32. is_list

33. Two equivalent summaries is_list

34. Two equivalent summaries is_list

35. Two equivalent summaries is_list

36. Two equivalent summaries

37. Other equivalent summaries dll1 NULL

38. Other equivalent summaries dll1 NULL

39. Other equivalent summaries dll1 NULL

40. Other equivalent summaries dll1 NULL

41. Other equivalent summaries dll1 NULL

42. Other equivalent summaries NULL

43. Other equivalent summaries dll2 NULL

44. Other equivalent summaries dll2 NULL

45. Other equivalent summaries dll2 NULL

46. Other equivalent summaries dll2 NULL

47. Other equivalent summaries dll2 NULL

48. Other equivalent summaries NULL