CS294, YelickSelf Stabilizing, p1 CS Self-Stabilizing Systems

CS294, YelickSelf Stabilizing, p2 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

CS294, YelickSelf Stabilizing, p3 (Self) Stabilization History: Idea introduced by Dijkstra in 1973/74. Popularized by Lamport in Idea of stabilization: ability of a system to converge in finite number of steps from arbitrary states to desired state

CS294, YelickSelf Stabilizing, p4 Overview ConceptsTools AsynchronousCutSnapshot Asynchronous Fault-Tolerant Byzantine Faults Authorization, Xcast Asynchronous Fault-Tolerant Self- Stabilizing Local Checking, Counter Flushing

CS294, YelickSelf Stabilizing, p5 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

CS294, YelickSelf Stabilizing, p6 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

CS294, YelickSelf Stabilizing, p7 Stabilization (Refined)

CS294, YelickSelf Stabilizing, p8 Practical Issues Stabilizing protocols allow for –Corrupted state –Initialization errors –Not corruption of code Applications –Routing, Scheduling, Resource Allocation

CS294, YelickSelf Stabilizing, p9 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)

CS294, YelickSelf Stabilizing, p10 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

CS294, YelickSelf Stabilizing, p11 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?

CS294, YelickSelf Stabilizing, p12 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

CS294, YelickSelf Stabilizing, p13 Token Passing on a Line Top (Process n-1) x = 0 up = false 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

CS294, YelickSelf Stabilizing, p14 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

CS294, YelickSelf Stabilizing, p15 Token Passing Up Top (Process n-1) x = 0 up = false x = 1 up = true. x = 1 up = true Bottom (Process 0) if x[i] != x[i-1] then {x[i] := x[i-1]; up[i] := true} x = 1 up = true

CS294, YelickSelf Stabilizing, p16 Token Passing Down Top (Process n-1) x = 0 up = false x = 1 up = true. x = 1 up = true Bottom (Process 0) if x[n-2] != x[n-1] then x[n-1] := x[n-2] x = 1

CS294, YelickSelf Stabilizing, p17 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

CS294, YelickSelf Stabilizing, p18 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

CS294, YelickSelf Stabilizing, p19 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

CS294, YelickSelf Stabilizing, p20 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

CS294, YelickSelf Stabilizing, p21 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

CS294, YelickSelf Stabilizing, p22 Token Ring Execution x-1x x x x 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 token p0

CS294, YelickSelf Stabilizing, p23 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

CS294, YelickSelf Stabilizing, p24 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

CS294, YelickSelf Stabilizing, p25 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

CS294, YelickSelf Stabilizing, p26 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

CS294, YelickSelf Stabilizing, p27 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