1 / 21

Proving L inearizability via Refinement Heike Wehrheim University of Paderborn

Proving L inearizability via Refinement Heike Wehrheim University of Paderborn Joint work with John Derrick (U. of Sheffield), Gerhard Schellhorn (U. of Augsburg). Motivation. Concurrent objects = data structures accessed by concurrent processes

vance
Télécharger la présentation

Proving L inearizability via Refinement Heike Wehrheim University of Paderborn

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ProvingLinearizability via Refinement Heike Wehrheim University of Paderborn Joint workwith John Derrick (U. of Sheffield), Gerhard Schellhorn (U. of Augsburg)

  2. Motivation Concurrentobjects = datastructuresaccessedbyconcurrentprocesses Examples: stacks, sets, queues, … Implementedby: linkedlists, hashtables, … Questions: • implementationcorrect? • whatis „correctness“?

  3. Set – Abstract Data Type add(4,true) contains(3,true) remove(1,true) Atomicoperations: add, remove, contains {1,3} {1,3,4} {1,3,4} {3,4}

  4. Set - Implementation -1 2 4 6 1 head tail • Nodes: • val (values) • next (pointer) • lock (lockedor not) • marked(deletedor not) (algorithm due to Heller et al., HHLMSS) Linkedlist

  5. Implementation ofadd -1 2 4 6 1 head 3 tail add(e) n1, n3 : locate(e); if n3.val != e then n2 := newNode(e); n2.next := n3; n1.next := n2; result := true else result := false; n1.unlock(); n2.unlock(); returnresult n2 add(3) n1 n3 locate: lockstwoadjacentnodes

  6. In addition … removesimilar • locate, and • thenfirstmarking, thenphysicallyremovingnode Concurrency: - severalprocessesaccessingthelinkedlistatthe same time

  7. Correctness? Implementation shouldbehaveasspecified in theabstractdata type, e.g., • addreallyadds an element, removereallyremovesit etc. • whatifremove(4,), add(4,), contains(4,) areexecutedconcurrently?

  8. Linearizability „Linearizability provides the illusion that each operation applied by concurrent processes takes effect instantaneously at some point between its invocation and response.“ Herlihy, Wing, 1990

  9. Linearizationpoints Linearizationpoint: add(3,true) Linearizationpoint: remove(3,true) invoke: add(3,) return: add(3,true) return: remove(3,true) invoke: remove(3,) Not linearizable return: add(3,true) invoke: add(3,) invoke: remove(3,) return: remove(3,false)

  10. First Idea add(e) n1, n3 := locate(e); if n3.val != e then n2 := newNode(e); n2.next := n3; n1.next := n2; result := true else result := false; n1.unlock(); n2.unlock(); returnresult Find linearizationpoint (LP) foreveryoperation Show sequenceofLPstomakeup a valid executionsequenceoftheabstractdata type (proofbyrefinement)

  11. Refinementgraphically Abstract Concrete P1 LP P2 LP P3 LP

  12. Task Given: • Abstract Set: AState, AOp1, AOp2 • ConcreteSet: CState, COp1, COp2, COp3 Show refinementbyprovingconcretesettosimulateabstractset

  13. Refinementprovenbysimulation R: abstractionrelationrelatingabstractandconcretestatespace AOp(true) R R R R R R COp3 InvOp COp1 COp2 RetOp(true) Find matchingstepsofconcreteoperations in abstractspecification LP

  14. Difficulty What´stheconcretestatespace? • head, tail, linkedlistandlocal variables of all processes? • Whatare “all processes“? Oursolution: • Localproofobligations • Consideringatmosttwoprocessesat a time However, still provinglinearizabilityfor an arbitrarynumberofconcurrentprocesses

  15. Simulation conditions AOp as as´ R ls: localstateofoneprocess R COp gs´,ls´ gs,ls Concretestatesplitintoglobal stateGS (head, tailandlinkedlist) andlocalstateLS (n1, n2, …) 8as : AS, gs, gs´ : GS, ls,ls´ : LS: R(as,gs,ls) ÆCOp(gs,ls,gs´,ls´) ) R(as,gs´,ls´) ÆAOp(as,as´) Æ R(as´,gs´,ls´)

  16. Non-interference 8as : AS, gs,gs´: GS, lsp,lsp`,lsq : LS: R(as,gs,lsp) ÆR(as,gs,lsq)ÆCOp(gs,lsp,gs´,lsp´) ÆAOp(as,as´) )R(as´,gs´,lsq) Stepsofoneprocess do not interferewithsimulationproofsofotherprocesses

  17. In addition Provedisjointnessoflocalstatestobekept Prove an invarianttobemaintained (e.g., tailalwaysreachablefromhead)

  18. Operation contains contains(e): curr := head; while (curr.val < e) curr := curr.next; ifcurr.marked thenresult := false; elseresult := (curr.cal == e); returnresult Usesnolockingat all! Whereisthe LP ofcontains?

  19. Example contains(4) started -1 2 4 6 1 curr LP ofcontainscannotbestaticallydetermined; it‘ssometimessetbyotherprocesses! remove(4) -1 2 4 6 1 add(4) 4 -1 2 4 6 1

  20. Solution Potential LPs • containspotentiallylinearizesseveraltimes • refinementcondition: • every potential LP hastohave a matchingabstractstep • The thusconstructedsequenceofabstractoperationsmightcontainseveral potential LPs; the last oneistheactualone ) all operationsarelinearizable

  21. Conclusion All proofsmechanised: KIV, interactiveprover (U of Augsburg) Results: • Localproofobligations hold for • Set with lock coupling • Lazyset • Stack • Localproofobligationsactuallyguaranteelinearizability

More Related