1. Chapter 10 Global Properties

2. Unstable Predicate Detection A predicate is stable if, once it becomes true it remains true Snapshot algorithm is not useful for detection of global properties:  Not applicable for unstable predicates  Can not compute the least global state that satisfies a given predicate  Excessive overhead if frequency of snapshots is high

3. Predicates Any predicate B constructed from local predicates using boolean connectives can be written in a disjunctive normal form i.e. where q 1, q 2, …,q n are conjunctive predicates E.g. : x = y (where x and y are boolean) can be written as

4. Weak Conjunctive Predicate (WCP) A Weak Conjunctive Predicate (WCP) is true iff there exists a consistent global cut in which all the conjuncts are true Disjunctive predicates are easy to detect Given an algorithm for detecting WCP we can detect any predicate B constructed from local predicates using boolean connectives

5. WCP Algorithm outline Non-Checker process  Maintains a vector clock  Sends vector clock to checker process when predicate becomes true

6. WCP Algorithm outline Checker process  Maintains a separate queue for each non-checker process  Maintain a cut[1..N] (array of states of the processes)  If state cut[i] ! cut[j], then cut[i] = queue i.getNext()  Repeat above statement till all states in cut[ ] are concurrent  cut[ ] is the least CGS for which the predicate holds

7. WCP Detection - Checker Process

8. Overhead Analysis n: number of processes involved m: max number of messages sent or received by any process Space :  Each local snapshot : O(n)  At most O(mn) local snapshots  O(n 2 m) total space Time: n 2 m comparisons  O(n 2 m)

9. Is the time complexity optimal ? Lemma :  Let there be n elements in a set S. Any algorithm that determines whether all elements are incomparable must make at least n(n-1)/2 comparisons.

10. Is the time complexity optimal ? Theorem  Let S be any partially ordered finite set of size mn. We are given a decomposition of S into n sets P 0 … P n-1 such that P i is a chain of size m. Any algorithm that determines whether there exists an anti-chain of size n must make at least mn(n- 1)/2 comparisons Adversary algorithm

11. A Token based algorithm for WCP Monitor process runs on each node along with the application Monitor processes pass the token to each other Token stores candidate cut and information to determine if it is consistent Monitor Process Application Process Node P i

12. A Token based algorithm for WCP A token is sent to a process P i when current state from P i happened before some other state in the candidate cut Once the monitor process for P i has eliminated the current state  receive a new state from the application process  check for consistency conditions again. This process is repeated until  all states are eliminated from some process P i or  the WCP is detected.

13. A Token based algorithm for WCP Token consists of two vectors G and color  G represents the candidate global cut G[i] = k indicates that state (i,k) is part of the current cut Invariant: G[i] = k implies that any global cut C with state (i,s) 2 C and s < k cannot satisfy the WCP  color indicates which states have been eliminated If color[i]=red, then state (i,G[i]) has been eliminated and can never satisfy the global predicate

14. A Token based algorithm for WCP

15. Applications Distributed debugging Detect a bad condition  E.g. There is no leader,i.e., P 1 does not have a token and P 2 does not have a token and... P n does not have a token