33 lines
No EOL
1.9 KiB
Markdown
33 lines
No EOL
1.9 KiB
Markdown
We have a set of n sequential processes $p_{1},...,p_n$ , that access $m$ concurrent objects $X_1,...,X_m$ by invoking operations on the form `Xi.op(args)(ret)`.
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When invoked by $p_j$, the invocation `Xi.op(args)(ret)` is modeled by two events: `inv[Xi.op(args) by pj]` and `res[Xi.op(ret) to pj]`.
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A **history** (or **trace**) is a pair $\hat{H}=(H, <_{H})$ where $H$ is a set of events and $<_{H}$ is a total order on them.
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The *semantics* (of systems and/or objects) will be given as the set of traces.
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A history is **sequential** if it is of the form `inv res inv res ... inv res inv inv inv ...`, where every res is the result of the immediately preceding inv. (The last invocations do not have a return).
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A sequential history can be represented as a sequence of operations.
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A history is **complete** if every inv is eventually followed by a corresponding res, it is **partial** otherwise.
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### Linearizability
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A complete history $\hat{H}$ is **linearizable** if there exists a sequential history $\hat{S}$ s.t.
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- $\forall X :\hat{S}|_{X} \in semantics(X)$
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- $\forall p:\hat{H}|_{p} = \hat{S}|p$
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- cannot swap actions performed by the same process
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- If $res[op] <_{H} inv[op']$, then $res[op] <_{S} inv[op']$
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- can rearrange events only if they overlap
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Given an history $\hat{K}$, we can define a binary relation on events $⟶_{K}$ s.t. (op, op’) ∈ ⟶K if and only if res[op] <K inv[op’]. We write op ⟶K op’ for denoting (op, op’) ∈ ⟶K. Hence, condition 3 of the previous Def. requires that ⟶H ⊆ ⟶S.
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![[Pasted image 20250318090733.png]]
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![[Pasted image 20250318090909.png]]But there is another linearization possible! I can also push a before if I pull it before c!
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#### Compositionality theorem
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$\hat{H}$ is linearizable if $\hat{H}|_{X}$ is linearizable, for all X in H
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(it is good to linearize complex traces)
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For all X, let $\hat{S}_{X}$ be a linearization of $\hat{H}_{X}$
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- $\hat{S}_{X} def$ |