Structural counter abstraction

Transcription

1 Srucural couner abracion Proving fair-erminaion of deph bounded yem Khiij Banal 1 wih Eric Kokinen 1, Thoma Wie 1, Damien Zufferey 2 1 New York Univeriy 2 IST Auria March 18, 2013 TACAS, Rome, Ialy

2 Inroducion Model: Deph-bounded yem Graph-rewrie baed raniion yem, which can be ued o model concurren heap-manipulaing algorihm, a well a diribued yem. Problem: Fair-erminaion problem Fairne: If a raniion i coninuouly enabled afer ome poin, i i aken infiniely ofen. Applicaion: Proving progre properie of concurren and diribued yem.

3 Treiber ack Top ne ne ack

4 Treiber ack puh pop Top ne ne ack

5 Treiber ack puh Top ne ne ack puh(, daa): do { pc1: = ->op; = new (, daa); : }while(!cas(->op,, ) );

6 Treiber ack pc1 Top ne ne ack puh(, daa): do { pc1: = ->op; = new (, daa); : }while(!cas(->op,, ) );

7 Treiber ack pc1 ne Top ne ne ack puh(, daa): do { pc1: = ->op; = new (, daa); : }while(!cas(->op,, ) );

8 Treiber ack pc1 ne Top ne ne ack puh(, daa): do { pc1: = ->op; = new (, daa); : }while(!cas(->op,, ) );

9 Treiber ack pc1 ne ne ne ack Top ne puh(, daa): do { pc1: = ->op; = new (, daa); : }while(!cas(->op,, ) );

10 Treiber ack pc1 ne ne ne ack Top ne puh(, daa): do { pc1: = ->op; = new (, daa); : }while(!cas(->op,, ) );

11 Treiber ack pc1 ne ne ack Top ne puh(, daa): do { pc1: = ->op; = new (, daa); : }while(!cas(->op,, ) );

12 Lock freedom a fair erminaion Treiber ack i lock-free. guaranee global progre: ome hread will finih individual hread migh arve Reduced o erminaion problem where arbirarily many bu finie number of hread are preen. A raniion which pawn procee a will, along wih a fairne conrain can be ued o encode hi.

13 Lock freedom a fair erminaion Treiber ack i lock-free. guaranee global progre: ome hread will finih individual hread migh arve Reduced o erminaion problem where arbirarily many bu finie number of hread are preen. A raniion which pawn procee a will, along wih a fairne conrain can be ued o encode hi. Challenge: Unbounded number of heap objec and hread objec.

14 Conribuion Work wih ymbolic graph which can model rucure ha arie commonly in hee yem, and required o be racked o prove erminaion. Conribuion: We inroduce a couner abracion derived from hee, hu called rucural couner abracion. I i ufficienly refined o be able o prove progre properie like lock-freedom of Treiber ack.

15 Relaed work Couner abracion for concurren yem A. Pnueli, J. Xu, and L. D. Zuck. Livene wih (0, 1, )-couner abracion. In CAV, G. Baler, M. Mazzucchi, T. Wahl, and D. Kroening. Symbolic couner abracion for concurren ofware. In CAV, 2009.

16 Relaed work Couner abracion for concurren yem A. Pnueli, J. Xu, and L. D. Zuck. Livene wih (0, 1, )-couner abracion. In CAV, G. Baler, M. Mazzucchi, T. Wahl, and D. Kroening. Symbolic couner abracion for concurren ofware. In CAV, Graph-baed analyi J. Berdine, B. Cook, D. Diefano, and P. W. O Hearn. Auomaic erminaion proof for program wih hape-hifing heap. In CAV, S. Gulwani, T. Lev-Ami, and M. Sagiv. A combinaion framework for racking pariion ize. In POPL, 2009.

17 Ouline Inroducion Model (need graph) Srucural couner abracion Implemenaion and concluion

18 Model Graph Tranformaion Syem Sae: graph. In our cae, ymbolic graph (on ne lide). Rule: rewrie one ubgraph wih anoher. op ack op ack Oher rule: Spawn CAS ucceed CAS fail pc1 Prepare rule

19 Need graph Top ack pc1 repreen arbrirary number of copie.

20 Need graph Top ack pc1

21 Top ack Need ubgraph repreen arbirary number of copie of he ubgraph pc1 Top ack pc1 pc1 pc1

22 Inducive invarian Top ack pc1 prepare Top ack pc1

23 Inducive invarian Top ack pc1 prepare cover Top ack pc1

24 Inducive invarian CAS ucceed G ucceed Top ack cover CAS fail G fail pc1 cover prepare cover Top ack pc1

25 Srucural couner abracion Inpu: Rewrie-rule, Inducive invarian a need graph Oupu: Couner yem Graph yem Couner yem Need graph Conrol locaion Node in need graph Couner Rule applicaion Couner updae Soundne. If he graph raniion yem ha a fair non-erminaing run, hen he couner yem will have a fair non-erminaing run.

26 l 1 y 1 y 3 Top ack y 4 y 6 y 7 y 2 y 5 pc1 y 8

27 prepare cover l 1 y 1 y 3 Top ack y 4 y 6 y 7 l 2 y 9 y 1 y2 y 3 y 4 ack Top y 6 y 7 y 2 pc1 y 5 y 8 y 10 y 5 pc1 y 8 prepare: (l 1, { y 9 = 1, y 10 = 1, y 5 = y 5 1, ideniy on re }, l 2 ) cover: (l 2, { y 1 = y 1 + y 9, y 9 = 0, y 2 = y 10 + y 2, y 10 = 0, ideniy on re }, l 1 )

28 Compuing Inducive Invarian Deph bounded yem: cla of well-rucured raniion yem [Meyer, 2008]. I ay if he lengh of he longe imple pah i bounded, hen yem i well-rucured wih he ordering given by ubgraph homomorphim. Analyi o compue over-approimaion of e of reachable ae of he WSTS [Ideal abracion, Zufferey, Wie, Henzinger, 2012]. Thi overapproimaion i a downward cloed e, alo inducive, given a finie union of ae repreened by he need graph. Many concurren and diribued proce can be modeled a deph-bounded procee for proving erminaion (Treiber ack wihou ne, ec.)

29 Implemenaion Inpu: Graph rewrie yem. 1. Picao 1 compue he inducive invarian a need graph[zwh 12]. 2. Picao eended o compue he couner abracion from he invarian[hi work]. 3. Couner program i fed o erminaion prover for couner yem, ARMC [Andrey Rybalchenko, Andrea Podelki]. We alo ue Z3 [Leonardo de Moura, Nikolaj Bjorner] and Prince[Philipp Rümmer] for variable eliminaion o opimize couner abracion. 1 hp://pub.i.ac.a/~zufferey/picao/

30 Eperimenal Reul Eample #loc #v # Î N Armc Toal Spli/merge Work ealing, 3 proceor Work ealing, parameerized Compue erver job queue Cha room min 6 min Map reduce Map reduce wih failure Treiber ack (coare-grained) Treiber ack (fine-grained) Herlihy/Wing queue Michael/Sco queue (dequeue only) Michael/Sco queue (enqueue only) Michael/Sco queue wk 3 wk Table : The column how he number of locaion, variable, and raniion in he couner abracion, and he running ime, in econd, for compuing he inducive invarian, conrucing he abracion, and for proving erminaion.

31 Relaed work R. Meyer. On boundedne in deph in he π-calculu. In Fifh Ifip Inernaional Conference On Theoreical Compuer Science Tc 2008, A. Pnueli, J. Xu, and L. D. Zuck. Livene wih (0, 1, )-couner abracion. In CAV, G. Baler, M. Mazzucchi, T. Wahl, and D. Kroening. Symbolic couner abracion for concurren ofware. In CAV, S. Gulwani, T. Lev-Ami, and M. Sagiv. A combinaion framework for racking pariion ize. In POPL, S. Johi and B. König. Applying he graph minor heorem o he verificaion of graph ranformaion yem. In CAV, A. Goman, B. Cook, M. J. Parkinon, and V. Vafeiadi. Proving ha non-blocking algorihm don block. In POPL, 2009.

32 Concluion Novel echnique for proving fair erminaion of DBS ha can be ued o prove progre properie of concurren daa rucure and diribued yem. An analyi ha i boh pracical and ufficienly precie buil on op of eiing erminaion prover for couner yem.