We introduced definitions for consensus, Byzantine Broadcast (BB) and Byzantine Agreement (BA), in an earlier post. In this post, we will discuss how consensus protocols are used in State Machine Replication (SMR). We will compare and contrast this setting to that of traditional BB and BA.

State Machine Replication is a fundamental approach in distributed computing for building fault tolerant systems.

State machine. A state machine, at any point, stores a state of the system. It receives a set of inputs (also referred to as commands). The state machine applies these inputs in a sequential order using a transition function to generate an output and an updated state. A succinct description of the state machine is as follows:

state = init
log = []
while true:
on receiving cmd from a client:
log.append(cmd)
state, output = apply(cmd, state)
send output to the client


Here, the state machine is initialized to an initial state init. When it receives an input cmd from a client, it first adds the input to a log. It then executes the cmd by applying the transition function apply to the state. As a result, it obtains an updated state and a result output. The result is then sent back to the client.

An example state machine is the Bitcoin ledger. The state consists of the set of public keys along with the associated Bitcoins (see UTXO). The input (or cmd) to the state machine is a transaction between parties (see Bitcoin core api). The log corresponds to the Bitcoin ledger. And the transition function apply is the function that determines whether a transaction is valid, and if it is, performs the desired bitcoin script operation (see script).

Fault Tolerant State machine replication (FT-SMR). In a client-server setting, the servers simulate a state machine in a fault tolerant manner. Clients send commands as if there is a single state machine. Maintaining a single server is prone to crashes or Byzantine faults. Thus, instead of maintaining a single server, an SMR system uses multiple server replicas, some of which can be faulty. The group of servers presents the same interface as that of a single server to the client.

The server replicas all initially start with the same state. However, when they receive concurrent requests from a client, honest replicas first need to agree on the sequence of client commands received by them. This problem is called log replication, and it is a multi-shot version of consensus. After the sequence is agreed upon, the replicas apply the commands in the log, one by one, using the apply transition. Assuming the transition is deterministic, all honest server replicas maintain an identical state at all times.

Thus, a Fault Tolerant SMR system needs to perform log replication efficiently and then execute the commands on the log. Specifically, it needs to guarantee the following:

(Safety) Any two honest replicas store the same sequence of commands in their logs.

(Liveness) Honest server replicas will eventually execute a command proposed by a client.

The requirements for a FT-SMR seem similar to those for BB and BA. Safety is akin to the agreement property, whereas liveness is similar to the termination property. However, there are a few differences between them:

1. Consensus on a sequence of values. FT-SMR needs log replication, or multi-shot consensus. Conceptually, one can sequentially compose single-shot consensus protocols. In practice, SMR protocols may optimize without necessarily relying on a sequential composition (more discussed below).

2. Who are the learners? In BB and BA, the parties executing the protocol are the ones learning the result. In FT-SMR, the replicas engage in the consensus protocol but eventually need to convince clients of the result. In the presence of Byzantine faults, the clients in an FT-SMR protocol may communicate with multiple replicas before learning about the commit. If there are $f$ Byzantine faults, the client needs to communicate with at least $f+1$ replicas to know that it has communicated with at least one honest replica.

3. Fault tolerance. An essential consequence of clients not participating in the protocol is the fault tolerance one can obtain. With BB, we know of protocols such as Dolev-Strong that can tolerate $f < n-1$ faults among $n$ replicas. SMR protocols that obtain Safety and Liveness cannot tolerate more than a minority corruption.

4. External validity. Finally, the definition does not explicitly state a validity property as in BB and BA. Fault Tolerant SMR protocols generally satisfy external validity, i.e., a command is said to be valid so far as the client signs the value.

### Optimizing for a sequence of values

Since FT-SMR protocols agree on a sequence of values, practical approaches for SMR (such as PBFT, Paxos, etc.) use a steady-state-and-view-change approach to architect log replication. In the steady-state, there is a designated leader that drives consensus. Typically, the leader does not change until it fails (e.g., due to network delays) or if Byzantine behavior is detected. If the leader fails, the replicas vote to de-throne the leader and elect a new one. The process of choosing a new leader is called view-change. The presence of a single leader for more extended periods yields simplicity and efficiency when the leader is honest. However, it also reduces the amount of decentralization and can cause delays if Byzantine replicas are elected as leaders.