1. Expand, Enlarge, and Check for Branching Vector Addition Systems Rupak Majumdar Zilong Wang MPI-SWS

2. Branching Vector Addition Systems(BVAS) A generalization of vector addition systems (VAS) A very expressive concurrency model – spawn and wait – asynchronous programming Program safety  coverability problem for BVAS

3. Coverability Results CoverabilityVASBVAS Theory Practice EXPSPACE-complete [Lipton 76, Rackoff 78] Nondeterministically guess a doubly exponential covering path 2EXPTIME-complete [Demri et al. 09] Nondeterministically guess a doubly exponential covering tree Backward reachability [Abdulla et al. 96] [2EXPTIME: Bozzelli & Ganty 11] Expand, Enlarge, and Check (EEC) [Geeraerts et al. 04] [complexity unknown]

4. Coverability Results CoverabilityVASBVAS Theory Practice EXPSPACE-complete [Lipton 76, Rackoff 78] Nondeterministically guess a doubly exponential covering path 2EXPTIME-complete [Demri et al. 09] Nondeterministically guess a doubly exponential covering tree Backward reachability [Abdulla et al. 96] [2EXPTIME: Bozzelli & Ganty 11] EEC EEC [2EXPTIME] [EXPSPACE]

5. Outline Recap of EEC for VAS Complexity analysis of EEC for VAS Generalized to EEC for BVAS Experimental results of EEC for BVAS

6. Vector Addition System (VAS) VAS is the dimension of vectors is the initial vector is a finite set of unary rules

7. Derivation initial vector dim unary rules (1, 2) (2, -1) (3, 1) (-1, -1) (2, 0) (1, 2) + (2, -1) = (3, 1) (3, 1) + (-1, -1) = (2, 0) (-1, -1) (1, -1) derives

8. Coverability Problem Given a VAS and a target, is there a covering derivation of in ? is a covering (derivation) of if derives a vector such that

9. EEC for VAS

10. Truncated and Extended Derivations w.r.t a bound Normal Truncated ( = 2) Extended ( = 2) If a number >, truncate it to If a number >, extend it to

11. EEC for VAS No Cover Uncover Yes

12. Soundness and Completeness [Geeraerts et al. 04]

13. Complexity Analysis

14. EEC for VAS How many iterations are required for termination? Finite graph reachability. Size of the graph is No Cover Uncover Yes

15. Definitions Given a VAS and a vector, define input size : number of bits required to encode and in binary : the absolute value of the smallest negative integer in (i.e. the maximal decrease in one step of a derivation) : the greatest integer in

16. Lemmas Lemma 1 [Rackoff 78, Demri et al. 09]: If there is a covering derivation of, there is one whose length is at most Lemma 2: For all, if there is a covering extended derivation of, there is one whose length is at most

17. Pf: Let We show: Theorem 1: EEC for VAS terminates in iterations Theorem 1.If is coverable, there is a covering truncated derivation 2.If is uncoverable, there is no covering extended derivation

18. Proof of Claim 1 By Lemma 1: If is coverable, there is a covering truncated derivation No truncation in

19. Proof of Claim 1 By Lemma 1: There is a truncation in If is coverable, there is a covering truncated derivation

20. Pf: Let We show: Theorem 1: EEC for VAS terminates in iterations Theorem 1.If is coverable, there is a covering truncated derivation 2.If is uncoverable, there is no covering extended derivation ✓

21. Proof of Claim 2 By Lemma 2: If is uncoverable, there is no covering extended derivation No extension in

22. Proof of Claim 2 By Lemma 2: There is an extension in If is uncoverable, there is no covering extended derivation

23. Pf: Let We show: Theorem 1: EEC for VAS terminates in iterations Theorem 1.If is coverable, there is a covering truncated derivation 2.If is uncoverable, there is no covering extended derivation ✓ ✓

24. Theorem 1: EEC for VAS terminates in iterations Pf: By Theorem 1, = Each iteration solves two finite graph reachability problems Each graph has at most or nodes, which is Finite graph reachability problem is in NLOGSPACE Theorem 2: EEC for VAS is in EXPSPACE

25. Branching Vector Addition System

26. BVAS  is the dimension of vectors  is a finite set of axioms  is a finite set of unary rules  is a finite set of binary rules

27. Derivation (0,3)+(4,2)+(0,-4)=(4, 1) (0, 1) (5, -2) derives (5, 1) dim axiomsunary rules binary rule (0, 1) (0, 2) (0, 3) (3, 2) (1, 0) (4, 2) (0, -4) (1, 0) (5, 1)

28. Coverability Problem Given a BVAS and a target, is there a covering derivation of in ? is a covering (derivation) of if derives a vector such that

29. EEC for BVAS No Cover Uncover Yes These are trees

30. Theorems about EEC for BVAS Theorem 4: EEC for BVAS terminates in iterations Theorem 5: EEC for BVAS is in 2EXPTIME

31. Implementation and Experimental Results

32. Single-wait Programs [Bouajjani & Emmi 12] A function can 1.call unboundedly many asynchronous functions running in parallel 2.wait till the first return value comes back State reachability  BVAS coverability

33. DNS lookup dns_server(Name) {... // do something else do { post r0 Result=ret); } while(*)... // do something else ewait r0; assert(is_valid_ip(Result));... // do something else } lookup(Name) { // lookup() returns ip... while(true) { allocate(Buf); post r1 Buf=ret);//server1 returns err or ip post r1 Buf=ret);//server2 returns err or ip... // do something else ewait r1; if (Buf == err) { free(Buf); } return Buf; } Can this assertion fail? continue; Asynchronously make unboundedly many lookups wait till the first return value that comes back. Asynchronously make unboundedly many lookups wait till the first return value that comes back. Ask two remote servers for an ip address wait for the first return value Ask two remote servers for an ip address wait for the first return value

34. Experimental Results #server#dimension#axiom#urule#bruleresulttime 2192091536950Uncover31.25s 322231483211664Uncover79.16s 425262264018326Uncover151.46s 528293307027392Uncover279.56s 631324663839366Uncover463.71s 6(buggy)31324007732805Cover63.58s

35. Summary CoverabilityVASBVAS Theory Practice EXPSPACE-complete [Lipton 76, Rackoff 78] Nondeterministically guess a doubly exponential covering path 2EXPTIME-complete [Demri et al. 09] Nondeterministically guess a doubly exponential covering tree Backward reachability [Abdulla et al. 96] [2EXPTIME: Bozelli & Ganty 11] EEC EEC [2EXPTIME] [EXPSPACE]

36. Questions? www.mpi-sws.org/~zilong