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Proof of liveness: an example

Proof of liveness: an example. 0. Clock phase synchronization System of n clocks ticking at the same rate. Each clock is 3-valued, i,e it ticks as 0, 1, 2, 0, 1, 2… A failure may arbitrarily alter the clock phases. The clocks need to return to the same phase. 1. 2. 3. n-1.

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Proof of liveness: an example

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  1. Proof of liveness: an example 0 Clock phase synchronization System of n clocks ticking at the same rate. Each clock is 3-valued, i,e it ticks as 0, 1, 2, 0, 1, 2… A failure may arbitrarily alter the clock phases. The clocks need to return to the same phase. . 1 2 3 n-1

  2. Clock phase synchronization {Program for each clock} (c[k] = phase of clock k, initially arbitrary) do j: j  N(i) :: c[j] = c[i] +1 mod 3  c[i] := c[i] + 2 mod 3 [] j: j N(i) :: c[j] ≠ c[i] +1 mod 3  c[i] := c[i] + 1 mod 3 od Show that eventually all clocks will return to the same phase (convergence), and continue to be in the same phase (closure) Proof of liveness: an example k: c[k]  {0.1.2} 0 1 2 3 n-1

  3. Let D = d[0] + d[1] + d[2] + … + d[n-1] d[i] = 0 if no arrow points towards clock i; = i + 1 if a pointing towards clock i; n - i if a  pointing towards clock i; = 1 if both  and point towards clock i. By definition, D ≥ 0. Also, D decreases after every step in the system. So the number of arrows must reduce to 0. Proof of convergence 0 1 2 n-1 0 2 2 2 0 1 1 0 1 1 2 2 2 2 2 Understand the game of arrows

  4. Primary standard = rotation of earth De facto primary standard = atomic clock (1 atomic second = 9,192,631,770 orbital transitions of Cesium 133 atom. 86400 atomic sec = 1 solar day – 3 ms (requires leap second correction each year) Coordinated Universal Time (UTC) = GMT ± number of hours in your time zone Time and Clock

  5. Global positioning system: GPS Location and precise time computed by triangulation Right now GPS time is nearly 14 seconds ahead of UTC, since It does not use leap sec. correction Per the theory of relativity, an additional correction is needed. Locally compensate by the Receivers. A system of 32 satellites broadcast accurate spatial corordinates and time maintained by atomic clocks

  6. Simultaneous? Happening at the same time? NO. There is nothing called simultaneous in the physical world. What does “concurrent” mean? Alice Explosion 2 Explosion 1 Bob

  7. Sequential and Concurrent events Sequential = Totally ordered in time. Total ordering is feasible in a single process that has only one clock. This is not true in a distributed system. Two issues are important here: • How to synchronize physical clocks ? (Unfortunately, clocks can never be perfectly synchronized) • Can we define sequential and concurrent events without using physical clocks?

  8. Causality Causality helps identify sequential and concurrent events without using physical clocks. Joke  Re: joke ( implies causally ordered before or happened before) Message sent  message received Local ordering: a  b  c (based on the local clock)

  9. Defining causal relationship Rule 1. If a, b are two events in a single process P, and the time of a is less than the time of b then a b. Rule 2. If a = sending a message, and b = receipt of that message, then a  b. Rule 3.a  bb  c a  c

  10. a  d since (a  bb  cc  d) e  d since (e  ff  d) (Note that  defines a PARTIAL order). Is g f or f g? NO.They are concurrent. . Example of causality Concurrency = absence of causal order

  11. LC is a counter. Its value respects causal ordering as follows a  b  LC(a) < LC(b) Note that LC(a) < LC(b) does NOT imply a  b. Each process maintains its logical clock as follows: LC1. Each time a local event takes place, increment LC. LC2. Append the value of LC to outgoing messages. LC3. When receiving a message, set LC to 1 + max (local LC, message LC) Logical clocks

  12. Total order is important for some applications like scheduling (first-come first served). But total order does not exist! What can we do? Strengthen the causal order to define a total order (<<) among events. Use LC to define total order (in case two LC’s are equal, process id’s will be used to break the tie). Let a, b be events in processes i and j respectively. Then a << b iff -- LC(a) < LC(b) OR -- LC(a) = LC(b) and i < j a  b  a << b, but the converse is not true. Total order in a distributed system The value of LC of an event is called its timestamp.

  13. Causality detection can be an important issue in applications like group communication. Logical clocks do not detect causal ordering. Vector clocksdo. a  b  VC(a) < VC(b) Vector clock C may receive Re:joke before joke, which is bad!

  14. {Actions of process j} 1. Increment VC[j] for each local event. 2. Append the local VCto every outgoing message. 3. When a message with a vector timestamp Tarrives from i, first increment the jth component VC[j] of the local vector clock, and then update it as follows: k: 0 ≤ k ≤N-1:: VC[k] := max (T[k], VC[k]). Implementing VC jth component of VC

  15. Example [3, 3, 4, 5, 3, 2, 1, 4] < [3, 3, 4, 5, 3, 2, 2, 5] But, [3, 3, 4, 5, 3, 2, 1, 4] and [3, 3, 4, 5, 3, 2, 2, 3] are not comparable Vector clocks Let a, b be two events. Define. VC(a) < VC(b) iff i : 0 ≤ i ≤ N-1 : VC(a)[i] ≤ VC(b)[i], and  j : 0 ≤ j ≤ N-1 : VC(a)[j] < VC(b)[j], VC(a) < VC(b)  a  b Causality detection

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