In this chapter, we introduce another useful tool for monitoring distributed computations. A distributed computation is generally monitored to detect if the system has reached a global state satisfying a certain property. For example, a token ring system may be monitored for the loss of the token. A distributed database system may be monitored for deadlocks. The global snapshot algorithm discussed in Chapter 9 can be used to detect a stable predicate in a distributed computation. To define stable predicates, we use the notion of the reachability of one global state from another. For two consistent global states *G* and *H,* we say that *G* ≤ *H* if *H* is reachable from G. A predicate *B* is *stable* iff

∀*G*, *H* : *G* ≤ *H* : *B*(*G*) ⇒ *B*(*H*)

In other words, a property *B* is stable if once it becomes true, it stays true. Some examples of stable properties are deadlock, termination, and loss of a token. Once a system has deadlocked or terminated, it remains in that state. A simple algorithm to detect a stable property is as follows. Compute a consistent global state. If the property *B* is true in that global state, then we are done. Otherwise, we repeat the process after some period of time. It is easily seen that if the stable property ever becomes true, the algorithm will detect it. Conversely, if the algorithm detects that some stable property *B* is true, then the property must have become true in the past (and is therefore also true currently).

Formally, if ...

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