A paper a day keeps the doctor away: Brewer's Conjecture and the Feasibility of Consistent, Available, Partition-Tolerant Web Services

Sixteen year ago, Eric Brewer introduced what is now known as the CAP theorem, which states that for a web service it is impossible to guarantee consistency, availability, and partition tolerance.  The conjecture was based on Brewer's experiences at Inktomi--a search engine company he cofounded, and was published without proof.  Gilbert and Lynch presented one in their paper: "Brewer's Conjecture and the Feasibility of Consistent, Available, Partition-Tolerant Web Services."

The paper is a good theoretical read, and the proofs the authors present are very tractable. They first begin by  formalizing the concepts of consistency (the authors use atomic in the paper), availability, and partition tolerance. For a consistent service, there is a total order on all operations such that each operation looks as if it were completed at a single instant. For availability, every request received by a non-failing node in the system must result in a response. Finally for partition tolerance the network is  allowed to lose arbitrarily many messages sent from one node to another.

The authors use these definitions to present their first impossibility result:

"It is impossible in the asynchronous network model to implement a read/write data object that guarantees the following properties:

• Availability 
• Atomic consistency

 in all fair executions (including those in which messages are lost). "

They prove the assertion by contradiction. The proof uses  two nodes/partitions in the system $A$ and $B$, where all the messages between $A$ and $B$ are lost. The proof assumes two operations $\alpha$ and $\beta$ that execute separately on $A$ and $B$, and are ordered such that $\beta$ occurs after $\alpha$ has ended.  $\alpha$ executes a write on partition $A$, $\beta$ executes a read from partition $B$ with all messages between $A$ and $B$ lost. Each operation on its own returns consistent results, while combined together as a new operation $\alpha+\beta$, return inconsistent data, proving the theorem.

The authors extend the result through a similar method of argument to all types of executions, since nodes $A$ and $B$ can't tell if the messages between them are lost in an asynchronous network (without the concept of clocks or time). The authors provide some example systems for asynchronous networks that provide two of the three guarantees (C,A, and P).

For partially synchronous systems, where every node has a clock, and all clocks increase at the same rate, but are not synchronized, the authors present another impossibility result:

"It is impossible in the partially synchronous network model to implement a read/write data object that guarantees the following properties:

• Availability
• Atomic consistency

in all executions (even those in which messages are lost)"

The proof is similar to the original impossibility result, with execution $\beta$ sufficiently delayed for the messages not to reach partition $B$.

The authors close by providing a weaker consistency condition that allows stale data to be returned when there are partitions through the use of a centralized node, and the formal requirements it places on the quality of the stale data returned .