Solving a simple linearizability problem

Let us tackle the following problem:

Give a two-thread history that is sequentially consistent, but not linearizable. Use a single atomic register r that initially contains the value zero and supports the operations read() and write(). (HS20)

To be able to solve this, we must recall what it means for a history to be sequentially consistent and what it means to be linearizable.

1. A history H is sequentially consistent if it can be extended to a history G that is equivalent to a legal sequential history S by

• appending zero or more responses to pending invocations that took effect and
• discarding zero or more pending invocations that did not take effect

2. A history H is linearizable if it can be extended to a history G that is equivalent to a legal sequential history S with $\to_G \subset \to_S$ by:

• appending zero or more responses to pending invocations that took effect and
• discarding zero or more pending invocations that did not take effect

These two definitions are very similar. The additional constraint that linearizability brings to the table is that $\to_G \subset \to_S$, which means that we cannot reorder operations done by different threads. For sequential consistency, we can reorder operations by different threads, but not those in the same thread (so program order has to be respected).

Alright. We thus want to find a history where we have to reorder stuff for it to be equivalent to a legal sequential history. This way, we keep sequential consistency, but we don’t have linearizability.

A possible solution could be

A: r.write(1)
A: void
B: r.write(2)
B: void
A: r.read()
A: 1

since we can reorder this to

B: r.write(2)
B: void
A: r.write(1)
A: void
A: r.read()
A: 1

which is then a legal sequential history. Because we have to reorder invocations across threads to make this work, the history is not linearizable, but it is sequentially consistent, just like the exercise demands.