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At this point we would like to define a history H to be serializable if it is equivalent to some serial history H,. This would be a perfectly reasonable defi- 32 CHAPTER 2 / SERIALIZABILITY THEORY nition if H were a collzplete history. Otherwise there are problems. First, there is an artificia1 problem in that a partial history can never be equivalent to a serial one. This is because serial histories must be complete by definition, and two histories can be equivalent only if they contain the same set of operations.

Ti,,t is a topological sort of SG(H). Let H, be the serial history Ti, TiL . . Ti,,,. We claim that C(H) E H,. To see this, let pi E Ti and qj E Tj, where T,, Tj are committed in H. Suppose pi, qj conflict and pi

The value of x returned by Read(x) depends on whether this operation precedes or follows Write(x). We want to formalize the definition of a transaction as a partial ordering of operations. In mathematics, it is common practice to write a partial order as an ordered pair (C, <), where C is the set of elements being ordered and < is the ordering relation. ), where C, is the set of operations of 7-,and

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