CS 294-8 Self-Stabilizing Systems cs.berkeley/~yelick/294

1. CS 294-8Self-Stabilizing Systems http://www.cs.berkeley.edu/~yelick/294

2. Administrivia • No seminar or class Thursday • Sign up for project meetings tomorrow, Thursday, or Tuesday (?) • Poster session Wednesday, 12/13, in the Woz, 2-5pm • Final papers due Friday, 12/15

3. (Self) Stabilization • History: Idea introduced by Dijkstra in 1973/74. Popularized by Lamport in 1983. • Idea of stabilization: ability of a system to converge in finite number of steps from arbitrary states to desired state

4. Stabilization • Motivation: fault tolerance • Especially transient faults • Also useful for others (crashes, Byzantine) • Where: Stabilization ideas appear in physics, control theory, mathematical analysis, and systems science

5. Stabilization • Definition: Let P be a state predicate of a system S. S is stabilizing to P iff it satisfies the following: • Closure: P is closer in S: any computation that starts in a state in P leads to states that are in P • Convergence: every computation of S has a finite prefix such that the following is in P

6. Stabilization (Refined)

7. Practical Issues • Stabilizing protocols allow for • Corrupted state • Initialization errors • Not corruption of code • Applications • Routing, Scheduling, Resource Allocation

8. Dijkstra’s Model • The concurrency model is unrealistic, but useful for illustration • Processors are organized in a sparse, connected graph • At each “step” a processor looks at its own step and neighbors, and changes its own state • A “demon” selects the processor to executed (fairly)

9. Impossibility of Stabilization • If the processors are truly identical (symmetric), then stabilization is impossible • Consider an N processor system, with N a non-prime, say N = 2m • Consider an initial state that is cyclically symmetric, e.g., • s, t, s, t, s, t • pi’s state is s for i even, and t for i odd • Then the scheduling “demon” can schedule all even processors (which will all move to s’) and then all odd (move to t’), so no progress will be made

10. Implications of Impossibility • How important is this result? • Burns and Pachl show that with a prime number of processors, self-stabilization with symmetric processors is possible • More importantly, how realistic is the symmetry assumption?

11. Mutual Exclusion on a Line • The following simple example is a solution to the mutual exclusion problem: • n processors are connected in a line • Each talks to 2 neighbors (1 on the ends) • State: • Each process has 2 variables • up: token is above if true, below if false • x: a bit used for token passing

12. Token Passing on a Line • Top (Process n-1) x = 0 up = false x = 0 up = false x = 0 up = false x = 1 up = true x = 1 up = true . . . x = 1 up = true • Bottom (Process 0) • Logical token: • Token is at one of 2 procs where up differs • If x’s differ, upper proc, if same, lower proc

13. Token Passing Program • Bottom-move-up[0] • If x[0] = x[1] and up[1] = false then x[0] := ~x[0] • Top-move-down[n-1] • If x[n-2] != x[n-1] then x[n-1] := x[n-2] • Middle-move-up[i] • if x[i] != x[i-1] then {x[i] := x[i-1]; up[i] := true} • Middle-move-down[i] • if (x[i] = x[i+1] and up[i] = true and up[i+1]=true then up[i] = false • This is Dijkstra’s second, “4-state” algorithm

14. Token Passing Up • Top (Process n-1) x = 0 up = false x = 0 up = false x = 0 up = false x = 1 up = true x = 1 up = true . . . x = 1 up = true • Bottom (Process 0) x = 1 up = true if x[i] != x[i-1] then {x[i] := x[i-1]; up[i] := true}

15. Token Passing Down • Top (Process n-1) x = 0 up = false x = 1 up = true x = 1 up = true x = 1 up = true x = 1 up = true . . . x = 1 up = true • Bottom (Process 0) x = 1 if x[n-2] != x[n-1] then x[n-1] := x[n-2]

16. Proof Idea for Correct States • If the initialization of states is correct • One can divide the processor line in two parts based on “up” • Two processors, i and i-1 in between • All processors above i have same x value as x[i]; all below i-1 same as x[i-1] • An action in the program is enabled only when the token is held • Only 1 action is enabled (and only 1 process holds the token at any given time) • The above can be checked by examining the predicates on the rules

17. Locally Checkable Properties • In any good state, the following hold: • If up[i-1] = up[i], then x[i-1]=x[i] • If up[i] = true then up[i-1]=true • These are enough to show that only 1 processor is enabled • These are locally checkable • a local set (pair) of processors can detect an incorrect state

18. General Stabilization Technique 1 • Varghese proposed local checking and correction as a general technique • Turn local checks into local correction • Consider processors as tree (line is special case) • Consider I-1 to be I’s parent • For each node I (I != 0), add Correction action: check the local predicate between I and its parent, correct I’s state if necessary • Correction affects only child, not parent

19. Practical Issues • Dijkstra’s algorithm works without the explicit correction step • For more complex protocols, correction is used • Although Dijkstra’s algorithm is self-stabilizing, it goes through states where mutual exclusion is not guaranteed

20. Token Passing on Ring • Processor 0 and n-1 are neighbors • Initially, count = 0, except for processor 0 where count = 1 • Zero-move • If count[0] = count[n-1] then count[0] = count[0]+1 mod (n+1) • Other-move • If count[i] != count[i-1] then count[i] := count[i-1] • Note: this is Dijkstra’s first, k-state algorithm

21. Token Ring Execution Good States: • For I = 1…n=1, either count[I-1]=count[I] or count[I-1] = count[I]+1 • Either count[0] = count[n-1] or count[0] = count[n-1]+1 x x-1 x p0 x-1 x x-1 x x-1 x-1 x-1 x-1 x-1 token

22. Proof Idea • The following can be shown • In any execution, P0 will eventually increment its counter (because all other processor decrease # of counter values) • In any execution P0 will eventually reach a “fresh” counter value • Any state in which P0 has a fresh counter value m is eventually followed by a state in which all processes have m

23. General Stabilization Technique 2 • Varghese proposes counter flushing as a general technique for stabilization • Starting with some sender (P0) sending to others, which messages in rounds • Make stabilizing by numbering messages with counters (max ctr > N) • Sender must eventually get “fresh” value

24. Compilers and Stabilization • Two useful properties for compilers (according to Schneider): • Self-stabilizing source code should produce self-stabilizing object • Compiler should produce a self-stabilizing version of our program even if the source code is not

25. Compilers Con’t • Fundamental difference between symmetric and asymmetric rings • Self-stabilization is “unstable” across architectures • There is a class of programs for which a compiler can be written to “force” stabilization

26. Summary • Self-stabilizing algorithms • Overlooked for 10 years • Revived in distributed algorithms community • Algorithms for: MST, Communication, … • Relevance to practice • Tolerating transient faults is important • Do these ideas appear in real systems? • See http://www.cs.uiowa.edu/ftp/selfstab/bibliography/stabib.html