Linearizability

Distributed/networked systems employ information replication to accomplish availability in addition to scalability. Consistency is concerned amongst the query of what should tumble out if a customer modifies but about information items in addition to concurrently but about other customer reads or modifies the same items perchance at a dissimilar replica.

Linearizability is a rigid degree of consistency. (That is why it is too called equally strong-consistency.) For a organization to satisfy linearizability, 

• each performance must seem (from customer perspective) to occur at an instantaneous indicate betwixt its begin fourth dimension (when the customer submits it) in addition to terminate fourth dimension (when the customer receives the response), and 
• execution at these instantaneous points should degree a valid sequential execution (i.e., it should live equally if operations are executed ane at a fourth dimension ---without concurrency, similar they are existence executed past times a unmarried node) 

Let's simplify things further. In practice, consistency is ofttimes defined for systems that lead keep ii really specific operations: read in addition to write. If the write operations are destructive, rather than cumulative write/update, defining linearizability becomes fifty-fifty are to a greater extent than straightforward.  

Linearizability (i.e., rigid consistency) ensures that a read performance returns the value that was in conclusion written for a given object. 

But nosotros don't know when the write (or read) genuinely takes outcome (that is hidden from us). We solely know that it has to convey outcome atomically betwixt invocation in addition to answer of the corresponding operation. And that yet makes things interesting/nontrivial for linearizability checking.

Examples

I volition purpose the same color to tally asking answer pairs. 

PUT-req(k1,a), PUTresp(k1,a)

Let's begin amongst a basic example. This is a customer sentiment of the system. The PUT execution should convey "effect" inwards betwixt the invocation PUT-req(k1,a) (which requests the organization to PUT the value a to the substitution k1) in addition to the answer received from the system PUTresp(k1,a) (which acknowledges that the PUT asking has been completed for value a on substitution k1).

PUTreq(k1,a), PUTresp(k1,a), PUTreq(k1,b), GETreq(k1), GETresp(k1,a)

Is this linearizable?

Yes, because PUTreq(k1,b) has non been responded yet, in addition to it is possible that it powerfulness non lead keep taken outcome when GET is genuinely executed (when GET takes outcome on the organization side).

PUTreq(k1,a), PUTresp(k1,a), PUTreq(k1,b), GETreq(k1), GETresp(k1,b)

Is this linearizable?

Yes, because it is possible that the PUTreq(k1,b) powerfulness lead keep taken outcome earlier GET is executed.

Notice that for the same prefix of visible operations, both GETresp(a) in addition to GETresp(b) were valid responses inwards a linearizable system. 

PUTreq(k1,a), PUTresp(k1,a), PUTreq(k1,b), GETreq(k1), GETresp(k1,NULL)

Is this linearizable?

No, because nosotros know PUT(k1,a) has definitely taken effect. We lead keep seen the response. So, GET cannot furnish NULL (no value). There has non been a delete inwards betwixt either.

PUTreq(k1,a), PUTresp(k1,a), PUTreq(k1,b), GETreq(k1), GETresp(k1,b), GETreq(k1), GETresp(k1,a)

Is this linearizable?

No. The offset GET response has established that the organization had seen the effects of PUT(k1,b), in addition to nosotros cannot revert dorsum on that inwards a linearizable system.

PUTreq(k1,a), GETreq(k1), GETresp(k1,a), PUTreq(k1,b), PUTresp(k1,b), PUTresp(k1,a),  GETreq(k1), GETresp(k1,?)

What should live the value returned inwards the last GETresp(k1,?) in a linearizable system? The in conclusion PUT answer returned (from customer perspective) is PUTresp(k1,a). But nosotros don't evaluate linearizability based on what is the in conclusion PUT answer seen. We need to evaluate it inwards damage of which pose has "taken effect" almost recently. Is it possible for PUT(a) to tumble out (take effect) afterwards PUT(b)? No, that is ruled out past times the GETresp(k1,a) that the customer has seen (similar to the previous example). So the latest PUT to convey outcome inwards a sequential execution would live PUT(b), in addition to the linearizable organization should return GETresp(k1,b).

What if that offset GET was non there? Without that observation point, which pinned downwards a result, would in that place live a constraint inwards the end GET? What would live acceptable results a linearizable organization could furnish inwards that case? 

PUTreq(k1,a), PUTreq(k1,b), PUTresp(k1,b), PUTresp(k1,a),  GETreq(k1), GETresp(k1,?)

Returning a would live acceptable, because it is possible that PUT of a powerfulness lead keep taken outcome afterwards PUT of b. Returning b would too live acceptable, because it is possible that on the organization side PUT of b might lead keep taken outcome afterwards PUT of a. Returning NULL would non live possible because nosotros lead keep a value (even though nosotros don't know whether it is a or b that is the latest) that has been PUT for the key.

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