Let us define $op ->_{proc} op'$ to hold whenever there exists a process p that issues both operations with `res[op]` happening before `inv[op']`.

### Sequential consistency
**Def:** a complete history is sequentially consistent if there exists a sequential history $𝑆$ s.t.
![](images/Pasted%20image%2020250324082534.png)

![](images/Pasted%20image%2020250324082545.png)

>[!warning]
>The problem with sequential consistency is that it is NOT compositional.

Consider for example the following two processes:
```
p1: Q.enq(a); Q'.enq(b'); Q'deq()->b'
p2: Q'.enq(a'); Q.enq(b); Q.deq()->b
```
In isolation, both processes are sequentially consistent.

However, no total order on the previous 6 operations respects the semantics of a queue:
	- if p1 receives b' from Q'.deq, we have that Q'.enq(a'), must arrive after Q'.enq(b')
	- to respect $\to_{proc}$, also the remaining behavior of p2 must arrive after
	- hence, Q.enq(a) arrived before Q.enq(b) and so it is not possible for p2 to receive b from its Q.deq.

Hence, we have two histories that are sequentially consistent but whose composition cannot be sequentially consistent $\to$ **no compositionality!**

### Serializability
(typical notion in databases)

- instead of processes, we have transactions
- consequently, we have also two other kinds of events: `abort(t)` and `commit(t)`
	- in every history, we have at most one of these events for every transaction
	- if the history is complete, we must have exactly one of these events for transaction
- a sequential history is formed by committed transactions only

**Def:** a complete history $\hat{H}$ is **serializable** if there exists a sequential history $\hat{S}$ s.t.
1. $\forall X : \hat{S}|_X \in semantics(X)$
2. $S = \{e \in H : e \in t \in committedTrans(\hat{H})\}$
3. $\to_{trans} \subseteq \to_{S}$ where $\to_{trans}$ is defined like $\to_{proc}$ in sequential consistency.

It is a more general notion than linearizability, **but it is not compositional** (consider the previous example but with transactions instead of processes).