ACL2-Certified AVL Trees

1. ACL2-Certified AVL Trees Ryan Ralston University of Oklahoma • NSF DUE-0633004 • Integrating Mechanized Logic into the SE Curriculum • Collaboratiive Project with Matthias Felleisen, Northeastern University

2. Formal Verification in Software Development Customer Specification Company Application Formal Verifier

3. Related Work • Isabelle implementation of verified AVL trees only verifies correctness of insertion and look-up for balance and order • Gamboa and Cowles verified properties of red-black trees

4. Defining AVL Structure in ACL2 • Single-rotation (zig), using (defstructure avl key left right height)‏ (defun easy-R (tr)‏ (let* ((L (lf (lf tr)))‏ (R (avl (key tr)‏ (rt (lf tr))‏ (rt tr)‏ (ht-incr (rt (lf tr)) (rt tr)))))‏ (avl (key (lf tr)) L R (ht-incr L R))))‏ ‏ • Double rotations (zig/zag defined similarly)‏ • Rebalancing operations select appropriate rotations

5. Insertion MaintainsCorrect Recorded Height easy-L easy-L easy-R rebal-L rot-L hard-L ht-incr insert hard-L-able->easy-L-able easy-R easy-L rebal-R rot-R easy-R hard-R ht-incr hard-R-able->easy-R

6. Deletion Does, Too rebal-L shrink ht-incr rebal-R delete rebal-L rebal-R raise-sacrum ht-incr

7. Deletion Preserves Keys(old keys stay in tree)‏ rebal-L rebal-R rebal-R ht-incr->ht=ht-meas shrink-not-key delete shrink->ht=ht-meas ht-incr->ht=ht-meas delete->ht=ht-meas shrink->ht=ht-meas

8. Deletion Conserves Keys(new tree has no new keys)‏ rebal-L rebal-R shrink rebal-R shrink-key ht-incr->ht=ht-meas delete ht-incr->ht=ht-meas shrink->ht=ht-meas delete->ht=ht-meas shrink->ht=ht-meas

9. Insertion Does Not Decrease Max Key in Tree ht-incr->ht=ht-meas insert->ht=ht-meas rebal-L insert rebal-R tree-max-bigger-than-all-keys insert-is-tree

10. DeletionDoes Not Increase Max Key in Tree delete->ht=ht-meas ht-incr->ht=ht-meas delete-lemma-lf rebal-L-is-tree rebal-L delete delete->ht=ht-meas ht-incr->ht=ht-meas delete-lemma-rt rebal-R-is-tree rebal-R

11. Insert Preserves Order ht-incr->ht=ht-meas insert-is-tree insert-lf insert->max rebal-R insert rebal-L insert-is-tree insert-rt insert->min insert->ht=ht-meas

12. Deletion Preserves Order delete-2-lemma-1 delete-2-lemma-2 ht-incr->ht=ht-meas delete-2-lemma-3 delete->ht=ht-meas rebal-L delete-3-lemma-1 delete delete-3-lemma-2 rebal-R delete-3-lemma-3 del->max delete-4-lemma-1 del->min delete-4-lemma-2 raise-sacrum delete-4-lemma-3

13. Insertion Preserves Balance insert-empty insert-root insert insert-left insert-inc-at-most-1 insert-right insert-inc-at-most-1

14. Noteworthy Facts • Requires a significant amount of code because of the number of cases it needs proven individually • The code does not build upon itself very well: double rotations theorems, for example, do not apply the single rotation theorems • My “handwritten” proof overlooked a detail I considered trivial, but ACL2 didn’t

15. Preservation/Conservation of Keys Theorems (defthm operation-preserves-and-conserves-keys (iff (in-tree? k tr) (in-tree? k (operation tr))))‏ (defthm operation-preserves-keys (implies (in-tree? k tr)‏ (in-tree? k (operation tr))))‏ (defthm operation-conserves-keys (implies (not (in-tree? k tr))‏ (not (in-tree? k (operation tr)))))‏

16. Working Backwards • Areas of Use include: tree max and min proofs on deletion. • Almost no unnecessary lemmas proven • The approach will work but can produce results such as: (defthm del-tree-max (implies (ht=ht-meas? tr)‏ (decreasing-max-p (del k tr) tr))‏ :hints (("Goal" :hands-off (rebal-L rebal-R ht-incr raise-sacrum decreasing-max-p))‏ ("Subgoal *1/7" :use ((:instance del-tree-max-lemma-5)))‏ ("Subgoal *1/5" :use ((:instance del-tree-max-lemma-4)))‏ ("Subgoal *1/3" :use ((:instance raise-sacrum-tree-max)))‏ ("Subgoal *1/2''" :use ((:instance avl-right-dec-max-p-tr)))‏ ("Subgoal *1/1'" :use ((:instance empty-tr1-is-dec-max-p (tr1 tr)‏ (tr2 tr))))))‏

17. Questions?