Sequentially consistent but not linearizable execution - concurrency

Question

I'm trying to understand the difference between linearizability and sequential consistency. More specifically, I would like to have an example of an execution which is sequentially consistent but not linearizable.
My professor gave me the following example of such execution:
Alice and bob write checks to each other.
Alice’s Statement:
-10 Check Alice -> Bob
0 Check Bob -> Alice
Bob’s Statement
-10 Check Bob -> Alice
0 Check Alice -> Bob
Both overdraft.
It is sequential: each client sees a consistent order
It is not linearizable: no globally linear story
But I didn't get it. Line
n Check A -> B
is supposed to be interpreted as "A writes a check to B and its account after the operation is n".
I don't understand why the operation shouldn't be linearisable: both Alice and Bob end up with 0 in the end which is a consistent value, so maybe I didn't get the definition of 'linearisability' properly.

Answer 1

First off, what your professor gave you is not the history/execution explicitly, but the projection of the history on both of its threads.
A history H is linearizable if by leaving away pending invocations and/or adding responses to pending invocations (that you didn't leave away) you can get a history that is equivalent to a sequential history S that doesn't contradict any precedences implied by H.
In other words, the reason your example is not linearizable is because the operations (financial transactions) can't be assigned a single point in time. First money is deducted, and then later added, and this behaviour is observed by the threads/individual bank statements.
If the bank statements were
Alice’s Statement:
-10 Check Alice -> Bob
0 Check Bob -> Alice
Bob’s Statement
10 Check Alice -> Bob
0 Check Bob -> Alice
then we would have a history S as follows:
Alice: Send Bob 10
Alice: Send completed
Bob: Send Alice 10
Bob: Send completed
But in your example the history could be
Alice: Send Bob 10 (i.e. money is gone)
Bob: Send Alice 10 (i.e. money is gone)
Bob: Send completed (i.e. money arrived)
Alice: Send completed (i.e. money arrived)
(or any combination of lines 1/2 switched and 3/4 switched) and you can't reorder that sequentially (i.e. pairs of started/completed together) without changing the account balance observed by each thread in between.