State Machine: State Explosion

The main problem that’s stopping widespread usage of state machines is the fact that beyond very simple examples, state machines often end up with a large number of states, a lot of them with identical transitions. Statecharts solve this state explosion problem.

A classic example is that of a user interface element that might be modeled as being valid or invalid and have a state machine that describes this behaviour. Here is an example of such a simple state machine.

Simple state machine with two states, valid and invalid

The machine starts in the Valid state, and upon receiving the events i or v, transitions to the invalid or valid states, respectively.

Now let’s say we wanted to allow the component to be enabled. The e event enables it, and the d event disables it. The state machine would need to know that a component is enabled or disabled . With traditional state machines you end up with four states: valid/enabled, invalid/enabled, valid/disabled, and invalid/disabled.

Somewhat simple state machine with four states

To begin with, this doesn’t look too bad, because we’ve omitted the transitions. We need two sets of transitions for the existing i and v events, and we need similar transitions for the e and d events for enable and disable. In total we can see that in order to support four states, we need eight transitions:

Not so simple state machine with four states and eight transitions

Four states isn’t such a big problem, see what happens when we add another feature that we want to model. Let’s say we add a dirty bit to indicate that the user has made a change to the field, then we end up with state names like valid/enabled/unchanged and a total of 8 states:

Not so simple state machine with eight states

Some states might also not make sense to model; perhaps an unchanged field should not be considered invalid. This simplifies the diagram somewhat, but not enough. The sheer volume of states is already overwhelming.

However, the problem is exacerbated by the transitions to go with each new set of states:

Complex state machine with eight states and twelve transitions

What we are witnessing is referred to as the “state explosion” of state machines, because adding a new aspect to the state machine multiplies the number of states that need to be modeled, and creates a disproportionately high number of transitions.

The number of states needed to represent all possible combinations of variables is the cartesian product of the sets of variables.

Statecharts to the rescue

Statecharts address this problem by way of parallel states, hierarchies, and guards to name a few.

Let’s explore how our simple example could benefit from each of these three techniques.

Parallel states

We will start with the simple “valid / invalid” state machine:

Simple state machine with two states, valid and invalid

Let’s say that the enabled/disabled trait and changed/unchanged trait are completely independent of one another, in other words, that any combination of these variables make sense. If that’s the case, then the independent variables can be modeled using a parallel state:

Simple state machine with six states, and six transitions

Each region (demarcated by the dashed lines) are completely independent of each other. Adding more regions does not cause an explosion of states; the statechart grows more or less linearly as more features are added.


In statecharts, states can be organized hierarchically. When we want to model a new trait, we can attempt to understand how this new trait depends upon the existing states. Perhaps the new trait only makes sense sometimes and not always.

Let’s say that we decide that the valid/invalid trait only makes sense if the field has been modified. In other words, if the field is unchanged we don’t care about the validity of the field. In order to model this, we can move the valid/invalid states as substates of changed:

Simple state machine with valid/invalid as substate of changed

We still have the same number of fields, and the same number of transitions, but introduced a strict dependency, namely that some states only make sense in the context of another state. By leveraging hierarchies, the state explosion problem goes away too.


A guard is a sort of pre condition to a transition, in a way it stops a transition from happening if some condition is not true.

To illustrate how guards can help, let’s go back to the parallel version of our statechart, but add the constraint that we can not go from changed to unchanged if we are in the invalid state. In other words, we have to be valid in order to go to unchanged.

We can do this by guarding the transition from changed to unchanged, adding the condition in valid (no pun intended) to the transition:

Simple state machine with six states, and six transitions

Again, we have the same number of states, even when states only partially depend on one another.

See also