# What is the correct name for state explosion from sensor discretization?

The position of a robot on a map contains of an x/y value, for example $$position(x=100.23,y=400.78)$$. The internal representation of the variable is a 32bit float which is equal to 4 byte in the RAM memory. For storing the absolute position of the robot (x,y) only $$4+4=8$$ bytes are needed. During the robot movements, the position is updated continuously.

The problem is, that a 32 bit float variable creates a state space of $$2^{32}=4294967296$$. Which means there are endless amount of possible positions in which the robot can be. A robot control system maps the sensor readings to an action. If the input space is large, then the control system gets more complicated.

What is the term used in the literature for describing the problem of exploding state space of sensor variables? Can it be handled with discretization?

## 1 Answer

I would refer to your problem as having a continuous state space. By using a 32-bit float variable you discritize it. However, creating states for every possible value of a 32-bit float variable is probably too much. You should decide on:

• the variable range: what is the real range of the position variables (e.g. from 0 m to 10 m),
• and what is the resolution you require for your problem (e.g. 0.01 m or 0.1 m).

Note that you should take into account:

• the sensor range resolution,
• the required range and resolution for the problem.

Then, based on the number of states you could decide to discretize the states or to use a Monte Carlo approach.

See for example the work of Brechtel et al. (2013).