1. A Resource-based Logic for Termination and Non-Termination Proofs Ton Chanh Le Cristian Gherghina Aquinas Hobor Wei-Ngan Chin National University of Singapore 1ICFEM 2014

2. A Resource-based Logic for Termination and Non-Termination Proofs ICFEM 20142

3. Hoare Logic for Partial Correctness Proving Partial Correctness ICFEM 20143

4. Hoare Logic for Total Correctness Proving Termination ICFEM 20144

5. Hoare Logic for Total Correctness Proving Non-Termination ICFEM 20145

6. (Non-)Termination Specification “So-called partial correctness is inadequate: if a program is intended to terminate, that fact must be part of its specification.” – Cliff Jones Non-termination specification is also important for More comprehensive specifications to understand program better A clearer distinction between expected non-termination and failure of termination proofs ICFEM 20146

7. Research Questions requires x ≥ 0  y < 0 variance x ensures true; requires x ≥ 0  y ≥ 0 ensures false; ICFEM 20147 Is the separation between termination and non- termination specifications good? while (x ≥ 0) { x = x + y; }

8. Research Questions requires x ≥ 0  y < 0 variance x ensures true; requires x ≥ 0  y ≥ 0 ensures false; ICFEM 20148 How can the termination and non-termination specifications be unified? while (x ≥ 0) { x = x + y; }

9. A Unified Specification Logic A unified logical foundation for both termination and non-termination reasoning Integration of termination and non-termination specifications into expressive logics for functional correctness and safety verification ICFEM 20149

10. Temporal Predicates Term M : termination Loop : definite non-termination MayLoop : possible non-termination ICFEM 201410

11. Example while (x ≥ 0) { x = x + y; } requires (x ≥ 0  y < 0  Term[x])  (x ≥ 0  y ≥ 0  Loop) ensures (x ≥ 0  y ≥ 0  false); ICFEM 201411

12. Example while (x ≥ 0) { x = x + y; } case { x requires Term ensures true; x ≥ 0 -> case { y ≥ 0 -> requires Loop ensures false; y requires Term[x] ensures true; } ICFEM 201412

13. A Resource-based Logic for Termination and Non-Termination Proofs ICFEM 201413

14. Foundation for Specification Logic Solution: Consider non-termination and termination as resources. Our proposal: A new logic for consumable resources which captures the concept of resource capacity; tracking both minimum and maximum of resource usage ICFEM 201414

15. Resource Assertions ICFEM 201415

16. Temporal Entailment ICFEM 201416

17. Flow-Insensitive Temporal Entailment ICFEM 201417

18. HipTNT: A Termination Verification System ICFEM 201418 http://loris-7.ddns.comp.nus.edu.sg/~project/hiptnt

19. HipTNT+: A Termination Inference System void loop (int x, int y) { if (x < 0) return; else loop(x + y, y); } ICFEM 201419 http://loris-7.ddns.comp.nus.edu.sg/~project/hiptnt

20. HipTNT+: A Termination Inference System void loop (int x, int y) { if (x < 0) return; else loop(x + y, y - 1); } ICFEM 201420 http://loris-7.ddns.comp.nus.edu.sg/~project/hiptnt

21. HipTNT+: A Termination Inference System int Ack(int m, int n) { if (m == 0) return n + 1; else if (n == 0) return Ack(m-1, 1); else return Ack(m-1, Ack(m, n-1)); } ICFEM 201421 http://loris-7.ddns.comp.nus.edu.sg/~project/hiptnt

22. HipTNT+: A Termination Inference System int Ack(int m, int n) case { m requires Loop ensures false; m = 0 -> requires Term ensures res = n + 1; m > 0  n requires Loop ensures false; m > 0  n ≥ 0 -> requires Term[m, n] ensures res ≥ n + 1; } ICFEM 201422 http://loris-7.ddns.comp.nus.edu.sg/~project/hiptnt

23. Conclusions “Termination and Non-Termination as Resources” provides a unified framework to combine both partial correctness and (non-)termination verification The termination inference can benefit from mechanisms developed for safety verification, such as shape inference ICFEM 201423