1. 1 The Computability of Relaxed Data Structures: Queues and Stacks as Examples The Computability of Relaxed Data Structures: Queues and Stacks as Examples Nir Shavit and Gadi Taubenfeld Version: August 2015 SIROCCO 2015Gadi Taubenfeld © 2015

2. 2 Semantics of concurrent data structures  Sequential specification -- set of legal runs/sequences.  Sequential specification -- set of legal runs/sequences.  Consistency condition -- e.g. linearizability, sequential consistency, …  Consistency condition -- e.g. linearizability, sequential consistency, … relaxed This paper SIROCCO 2015Gadi Taubenfeld © 2015

3. 3 Why to relax ?  Synchronization inherently limits parallelism.  Semantically weaker DS reduce the need from synchronization.  Thus, provides potential to achieve better performance and scalability.  Thus, provides potential to achieve better performance and scalability.  There are many published efficient implementations of relaxed data structures. (See related work.)  There are many published efficient implementations of relaxed data structures. (See related work.)  We are interested in computability not complexity.  We are interested in computability not complexity. SIROCCO 2015Gadi Taubenfeld © 2015

4. 4 Queue[a,b,c] enqueuedequeue peek ab c  * -- can insert/remove/return a value at arbitrary position  0 -- operation not supported SIROCCO 2015Gadi Taubenfeld © 2015

5. 5 Stack[a,b,c] pushpop top ab c SIROCCO 2015Gadi Taubenfeld © 2015

6. 6 Queue[a,b,c] enqueue dequeue peek  Queue[1,1,1] -- traditional FIFO queue  Queue[1,1,0] -- does not support peek  Queue[1,1,*] -- peek returns a random value  Stack[ 1,0,1] -- atomic read/write register  Queue[*,*,0] = Stack[*,*,0] -- multiset object SIROCCO 2015Gadi Taubenfeld © 2015

7. 7  Objects are wait-free & linearizable Consensus Number known results SIROCCO 2015Gadi Taubenfeld © 2015

8. 8 Relaxing the enqueue operation SIROCCO 2015Gadi Taubenfeld © 2015

9. 9 Relaxing the enqueue operation SIROCCO 2015Gadi Taubenfeld © 2015

10. 10 Relaxing the enqueue operation SIROCCO 2015Gadi Taubenfeld © 2015

11. 11 Relaxing the enqueue operation SIROCCO 2015Gadi Taubenfeld © 2015

12. 12 Relaxing the enqueue operation SIROCCO 2015Gadi Taubenfeld © 2015

13. 13 Relaxing the enqueue operation SIROCCO 2015Gadi Taubenfeld © 2015

14. 14 Relaxing the peek operation SIROCCO 2015Gadi Taubenfeld © 2015

15. 15 Relaxing the peek operation SIROCCO 2015Gadi Taubenfeld © 2015

16. 16 Relaxing the peek operation SIROCCO 2015Gadi Taubenfeld © 2015

17. 17 Relaxing the peek operation SIROCCO 2015Gadi Taubenfeld © 2015

18. 18 Relaxing the dequeue operation SIROCCO 2015Gadi Taubenfeld © 2015

19. 19 Relaxing the dequeue operation SIROCCO 2015Gadi Taubenfeld © 2015

20. 20 Relaxing the dequeue operation SIROCCO 2015Gadi Taubenfeld © 2015

21. 21 Not supporting the dequeue operation SIROCCO 2015Gadi Taubenfeld © 2015

22. 22 Not supporting the dequeue operation SIROCCO 2015Gadi Taubenfeld © 2015

23. 23 Not supporting the dequeue operation SIROCCO 2015Gadi Taubenfeld © 2015

24. 24 Atomic registers vs. relaxed queues SIROCCO 2015Gadi Taubenfeld © 2015

25. 25 Atomic registers vs. relaxed queues Can atomic registers implement … ? SIROCCO 2015Gadi Taubenfeld © 2015

26. 26 Atomic registers vs. relaxed queues No!!! Theorem: A queue[a,0,c] has no wait-free implementation from atomic registers, for every two positive integers a and c. Theorem: A queue[a,0,c] has no wait-free implementation from atomic registers, for every two positive integers a and c. SIROCCO 2015Gadi Taubenfeld © 2015

27. 27 Discussion  Each one of the infinitely many relaxed objects has one of the following consensus numbers: 1, 2, . Why only three?  Queue is more sensitive than stack to changes in its semantics.  Consider other types of relaxed data structures.  What is the internal structure among relaxed objects with the same consensus number? queue[1,1,0] + registers  queue[1,1,2] ? SIROCCO 2015Gadi Taubenfeld © 2015

28. 28 Thank you for listening SIROCCO 2015Gadi Taubenfeld © 2015