A [state machine](http://en.wikipedia.org/wiki/Finite-state_machine) is defined as follows:

- `Input` - a set of inputs
- `Output` - a set of outputs
- `State` - a set of states
- `S0 ∈ S` - an initial state
- `T : Input * State -> Output * State` - a transition function

If you model your services (aggregates, projections, process managers, sagas, whatever) as state machines, one issue to address is management of `State`. There must be a mechanism to provide `State` to the state machine, and to persist resulting `State` for subsequent retrieval. One way to address this is by storing `State` is a key-value store. Another way is to use a SQL database. Yet another way is event sourcing. The benefit of even sourcing is that you never need to store `State` itself. Instead, you rely on the `Output` of a service to reconstitute state. In order to do that, the state machine transition function needs to be factored into two functions as follows:

```
exec  : Input * State -> Output
apply : Output * State -> State
```

These two functions can be combined to yield the original transition function with the added benefit that the `apply` function can be used to reconstitute state based on past outputs. This can be done as follows:

```
state = fold apply S0 outputs
```

Where `fold` is a [left fold](http://en.wikipedia.org/wiki/Fold_\(higher-order_function\)) and `outputs` is a set of outputs retrieved from an event store (such as @GetEventStore). 

In order for this to be correct, the apply functions needs to be deterministic. This is where the event semantic becomes helpful.

Alternatively, the transition function can be factored a slightly different way to support command sourcing as follows:

```
exec  : Input * State -> Output
apply : Input * State -> State
```

These functions can also be combined to yield the state machine transition function. The difference from event sourcing, is that we rely on past inputs rather than past outputs to reconstitute state. Note also that the `apply` functions has to be deterministic. Also, with command sourcing, the `Input`s must be stored in a durable log (such as @GetEventStore). 

An example in F# [here](https://github.com/eulerfx/DDDInventoryItemFSharp/blob/master/DDDInventoryItemFSharp/Aggregate.fs).