Concurrent specifications beyond linearizability

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1 Concurrent specifications beyond linearizability Éric Goubault Jérémy Ledent Samuel Mimram École Polytechnique, France OPODIS 2018, Hong Kong December 19, / 14

2 Objects Processes communicate through shared objects. For example: 2 / 14

3 Objects Processes communicate through shared objects. For example: Hardware: Read/Write registers, test&set, CAS,... 2 / 14

4 Objects Processes communicate through shared objects. For example: Hardware: Read/Write registers, test&set, CAS,... Data structures: lists, queues, hashmaps,... 2 / 14

5 Objects Processes communicate through shared objects. For example: Hardware: Read/Write registers, test&set, CAS,... Data structures: lists, queues, hashmaps,... Message passing interfaces 2 / 14

6 Objects Processes communicate through shared objects. For example: Hardware: Read/Write registers, test&set, CAS,... Data structures: lists, queues, hashmaps,... Message passing interfaces Immediate-snapshot, consensus, set-agreement,... 2 / 14

7 Objects Processes communicate through shared objects. For example: Hardware: Read/Write registers, test&set, CAS,... Data structures: lists, queues, hashmaps,... Message passing interfaces Immediate-snapshot, consensus, set-agreement,... Goal: can we implement object B using objects A 1,..., A k? 2 / 14

8 Objects Processes communicate through shared objects. For example: Hardware: Read/Write registers, test&set, CAS,... Data structures: lists, queues, hashmaps,... Message passing interfaces Immediate-snapshot, consensus, set-agreement,... Goal: can we implement object B using objects A 1,..., A k? We need to specify the behavior of the objects. 2 / 14

9 Concurrent specifications Idea: the specification of an object is the set of all the correct execution traces (Lamport, 1986). 3 / 14

10 Concurrent specifications Idea: the specification of an object is the set of all the correct execution traces (Lamport, 1986). P 0 push(0) ok pop() 2 pop() 0 P 1 P 2 push(2) ok 3 / 14

11 Concurrent specifications Idea: the specification of an object is the set of all the correct execution traces (Lamport, 1986). P 0 push(0) ok pop() 2 pop() 0 P 1 P 2 push(2) ok T = i push,0 0 r ok 0 i push,2 2 i pop 1 r 2 1 i pop 0 r ok 2 r 0 0 Trace formalism: Time is abstracted away. Alternation of invocations and responses on each process. 3 / 14

12 Concurrent specifications Idea: the specification of an object is the set of all the correct execution traces (Lamport, 1986). P 0 push(0) ok pop() 2 pop() 0 P 1 P 2 push(2) ok 3 / 14

13 Concurrent specifications Idea: the specification of an object is the set of all the correct execution traces (Lamport, 1986). P 0 push(0) ok pop() 2 pop() 0 P 1 P 2 push(2) ok Write T for the set of all execution traces. A concurrent specification is a subset σ T. 3 / 14

14 Concurrent specifications Idea: the specification of an object is the set of all the correct execution traces (Lamport, 1986). P 0 push(0) ok pop() 2 pop() 0 P 1 P 2 push(2) ok Write T for the set of all execution traces. A concurrent specification is a subset σ T. A program implements a specification σ if all the traces that it can produce belong to σ. 3 / 14

15 Linearizability (Herlihy & Wing, 1990) Lin SeqSpec ConcSpec Input: a sequential specification σ (e.g. list, queue,...). Output: a concurrent specification Lin(σ). 4 / 14

16 Linearizability (Herlihy & Wing, 1990) Lin SeqSpec ConcSpec Input: a sequential specification σ (e.g. list, queue,...). Output: a concurrent specification Lin(σ). P 0 P 1 P 2 4 / 14

17 Linearizability (Herlihy & Wing, 1990) Lin SeqSpec ConcSpec Input: a sequential specification σ (e.g. list, queue,...). Output: a concurrent specification Lin(σ). P 0 P 1 P 2 4 / 14

18 Linearizability (Herlihy & Wing, 1990) Lin SeqSpec ConcSpec Input: a sequential specification σ (e.g. list, queue,...). Output: a concurrent specification Lin(σ). P 0 P 1 P 2 Lin(σ) = {T concurrent trace T is linearizable w.r.t. σ} 4 / 14

19 Linearizability (Herlihy & Wing, 1990) Lin SeqSpec ConcSpec Input: a sequential specification σ (e.g. list, queue,...). Output: a concurrent specification Lin(σ). P 0 P 1 P 2 Lin(σ) = {T concurrent trace T is linearizable w.r.t. σ} Some objects are not linearizable! Their specification cannot be expressed as Lin(σ), for any σ. 4 / 14

20 Concurrent variants of linearizability Set-linearizability (Neiger, 1994) P 0 P 1 P 2 Can specify: exchanger, immediate snapshot, set agreement. 5 / 14

21 Concurrent variants of linearizability Set-linearizability (Neiger, 1994) P 0 P 1 P 2 Can specify: exchanger, immediate snapshot, set agreement. Cannot specify: validity, write-snapshot. 5 / 14

22 Concurrent variants of linearizability Set-linearizability (Neiger, 1994) P 0 P 1 P 2 Can specify: exchanger, immediate snapshot, set agreement. Cannot specify: validity, write-snapshot. Interval-linearizability (Castañeda, Rajsbaum, Raynal, 2015) P 0 P 1 P 2 5 / 14

23 Concurrent variants of linearizability Set-linearizability (Neiger, 1994) P 0 P 1 P 2 Can specify: exchanger, immediate snapshot, set agreement. Cannot specify: validity, write-snapshot. Interval-linearizability (Castañeda, Rajsbaum, Raynal, 2015) P 0 P 1 P 2 Can specify every task! 5 / 14

24 Overview Concurrent specifications 6 / 14

25 Overview Concurrent specifications Linearizability stack queue test&set 6 / 14

26 Overview Concurrent specifications Linearizability immediate snapshot stack exchanger queue test&set set-agreement 6 / 14

27 Overview Concurrent specifications Set-linearizability Linearizability immediate snapshot stack exchanger queue test&set set-agreement 6 / 14

28 Overview Concurrent specifications Set-linearizability write-snapshot Linearizability validity immediate snapshot stack exchanger queue test&set set-agreement 6 / 14

29 Overview Concurrent specifications Interval-linearizability Set-linearizability write-snapshot Linearizability validity immediate snapshot stack exchanger queue test&set set-agreement 6 / 14

30 Overview Concurrent specifications Interval-linearizability Set-linearizability write-snapshot Linearizability validity immediate snapshot stack exchanger queue? test&set set-agreement 6 / 14

31 Overview Concurrent specifications Prefix-closed concurrent specifications Interval-linearizability Set-linearizability write-snapshot Linearizability validity immediate snapshot stack exchanger queue? test&set set-agreement 6 / 14

32 Overview Concurrent specifications Prefix-closed concurrent specifications This talk: add a few more "desirable" properties Interval-linearizability Set-linearizability write-snapshot Linearizability validity immediate snapshot stack exchanger queue? test&set set-agreement 6 / 14

33 Overview Concurrent specifications Prefix-closed concurrent specifications Interval-linearizability Set-linearizability write-snapshot Linearizability validity immediate snapshot stack exchanger queue test&set set-agreement 6 / 14

34 Relevant concurrent specifications We write ConcSpec for the set of concurrent specifications σ T satisfying the following properties. (1) prefix-closure: if t t σ then t σ, (2) non-emptiness: ε σ, (3) receptivity: if t σ and t has no pending invocation of process i, then t i x i σ for every input value x, 7 / 14

35 Relevant concurrent specifications We write ConcSpec for the set of concurrent specifications σ T satisfying the following properties. (1) prefix-closure: if t t σ then t σ, (2) non-emptiness: ε σ, (3) receptivity: if t σ and t has no pending invocation of process i, then t i x i σ for every input value x, (4) totality: if t σ and t has a pending invocation of process i, then there exists an output x such that t ri x σ, 7 / 14

36 Relevant concurrent specifications We write ConcSpec for the set of concurrent specifications σ T satisfying the following properties. (1) prefix-closure: if t t σ then t σ, (2) non-emptiness: ε σ, (3) receptivity: if t σ and t has no pending invocation of process i, then t i x i σ for every input value x, (4) totality: if t σ and t has a pending invocation of process i, then there exists an output x such that t ri x σ, (5) σ has the expansion property. 7 / 14

37 Expansion of intervals A concurrent specification satisfies the expansion property if: 8 / 14

38 Expansion of intervals A concurrent specification satisfies the expansion property if: For any correct execution trace, P 0 P 1 P 2 a b c d e f g h i j k l 8 / 14

39 Expansion of intervals A concurrent specification satisfies the expansion property if: For any correct execution trace, P 0 P 1 P 2 a b c d e f g h i j k l if we expand the intervals, P 0 P 1 P 2 a c d e b g f j h k i l then the resulting trace is still correct. 8 / 14

40 Example: the Exchanger object Similar to the one available in Java 1 : A synchronization point at which threads can pair and swap elements within pairs. Here, we consider a wait-free variant. 1 java.util.concurrent.exchanger<v> 9 / 14

41 Example: the Exchanger object Similar to the one available in Java 1 : A synchronization point at which threads can pair and swap elements within pairs. Here, we consider a wait-free variant. A typical execution of the exchanger looks like this: P 0 exchange(0) 2 P 1 exchange(2) 0 P 2 exchange(42) exchange( a ) Fail Fail exchange( b ) c exchange( c ) b 1 java.util.concurrent.exchanger<v> 9 / 14

42 Example: the Exchanger object (2) The following execution is correct: exchange(0) P 0 P 1 P 2 Fail exchange(1) Fail exchange(2) Fail 10 / 14

43 Example: the Exchanger object (2) The following execution is correct: exchange(0) P 0 P 1 P 2 Fail exchange(1) Fail exchange(2) Fail Hence, according to the expansion property, exchange(0) P 0 exchange(1) P 1 exchange(2) P 2 Fail Fail Fail should be considered correct too! 10 / 14

44 Expansion is a desirable property We fix a set {A 1,..., A k } of shared objects, along with their concurrent specifications. 11 / 14

45 Expansion is a desirable property We fix a set {A 1,..., A k } of shared objects, along with their concurrent specifications. A program P using these objects can: call the objects, do local computations, use branching, loops. 11 / 14

46 Expansion is a desirable property We fix a set {A 1,..., A k } of shared objects, along with their concurrent specifications. A program P using these objects can: call the objects, do local computations, use branching, loops. Given a program P, we can define its semantics P, which is the set of execution traces that P can produce. 11 / 14

47 Expansion is a desirable property We fix a set {A 1,..., A k } of shared objects, along with their concurrent specifications. A program P using these objects can: call the objects, do local computations, use branching, loops. Given a program P, we can define its semantics P, which is the set of execution traces that P can produce. Theorem The semantics P of any program P has the expansion property. Moreover, if P is wait-free, then P ConcSpec. 11 / 14

48 Linearizability gives expansion for free Linearizability-based techniques always produce specifications which satisfy the expansion property. Theorem For every sequential specification σ, Lin(σ) ConcSpec. 12 / 14

49 Linearizability gives expansion for free Linearizability-based techniques always produce specifications which satisfy the expansion property. Theorem For every sequential specification σ, Lin(σ) ConcSpec. Proof. If some execution trace is linearizable, P 0 P 1 P 2 12 / 14

50 Linearizability gives expansion for free Linearizability-based techniques always produce specifications which satisfy the expansion property. Theorem For every sequential specification σ, Lin(σ) ConcSpec. Proof. If some execution trace is linearizable, P 0 P 1 P 2 12 / 14

51 Linearizability gives expansion for free Linearizability-based techniques always produce specifications which satisfy the expansion property. Theorem For every sequential specification σ, Lin(σ) ConcSpec. Proof. If some execution trace is linearizable, P 0 P 1 P 2 Then any trace obtained by expanding it is still linearizable. P 0 P 1 P 2 12 / 14

52 Linearizability gives expansion for free Linearizability-based techniques always produce specifications which satisfy the expansion property. Theorem For every sequential specification σ, Lin(σ) ConcSpec. Proof. If some execution trace is linearizable, P 0 P 1 P 2 Then any trace obtained by expanding it is still linearizable. P 0 P 1 P 2 12 / 14

53 A Galois connection SeqSpec ConcSpec Lin 13 / 14

54 A Galois connection U SeqSpec ConcSpec Lin 13 / 14

55 A Galois connection U SeqSpec Lin ConcSpec Theorem The maps Lin and U form a Galois connection: for every σ SeqSpec and τ ConcSpec, Lin(σ) τ σ U(τ) 13 / 14

56 Applications By the properties of Galois connections, Lin(U(Lin(σ))) = Lin(σ) This yields a simple criterion to check whether a given specification τ is linearizable: check whether Lin(U(τ)) = τ. 14 / 14

57 Applications By the properties of Galois connections, Lin(U(Lin(σ))) = Lin(σ) This yields a simple criterion to check whether a given specification τ is linearizable: check whether Lin(U(τ)) = τ. The Galois connection for interval linearizability has the following corollary: Theorem ConcSpec is the set of interval-linearizable specifications. 14 / 14

58 Thanks! 14 / 14

The Wait-Free Hierarchy

The Wait-Free Hierarchy Jennifer L. Welch References 1 M. Herlihy, Wait-Free Synchronization, ACM TOPLAS, 13(1):124-149 (1991) M. Fischer, N. Lynch, and M. Paterson, Impossibility of Distributed Consensus with One Faulty Process,