1. Single-version locking scheduler Proving the single-version locking scheme correct is trivial, as the scheduler is a 2PL scheduler. 2. Multi-version pessimistic (locking) scheduler The multi-version pessimistic (locking) scheme is in fact a MV2PL scheduler. Holding a certify (commit) lock on a data item in MV2PL is exactly like having the NoMoreReadLocks bit set in the latest version of the data item in our implementation (see Section 4.2.1). Section 5.5.2 of [WV02] describes MV2PL in detail and proves it only admits 1SR multi-version histories. 3. Multi-version optimistic scheduler Let us now prove that the multi-version optimistic scheduler only admits 1SR multi-version histories. We use the notation and theorems from Section 5.2 of [BHG87]. The multi-version optimistic scheduler behaves like a MVTO scheduler, with the changes described below. Let transaction Tx be a committed transaction with a Begin timestamp of TxBegin and an End timestamp of TxEnd. Property 1: Timestamps are assigned in a monotonically increasing order, and each transaction has a unique begin and end timestamp, such that TxBegin < TxEnd. Property 2: A given version is valid for the interval specified by the begin and end timestamps. There is a total order << of versions for a given datum, as determined by the timestamp order of the nonoverlapping version validity intervals. Property 3: The transaction Tx reads the latest committed version as of TxRead (where TxBegin <= TxRead < TxEnd) and validates (that is, repeats) the read of the latest committed version as of TxEnd. The transaction fails if the two reads return different versions. Property 4: Updates or deletes to a version V first check the visibility of V. Checking the visibility of V is equivalent to reading V. Therefore, a write is always preceded by a read: if transaction Tx writes Vnew, then transaction Tx has first read Vold, where Vold << Vnew. Moreover, there exists no version V such that Vold << V << Vnew, otherwise Tx would have never committed: it would have failed during the Active phase when changing the end timestamp of Vold (see Section 3.1, paragraph “Update version”) 1.

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