#### Nature acts twice

A convenient, alternative formulation can be given by allowing nature to act twice:

1. First, a nature action, , is chosen but is unknown to the robot.
2. Following this, a nature observation action is chosen to interfere with the robot's ability to sense .
Let denote a nature observation action, which is chosen from a nature observation action space, . A sensor mapping, , can now be defined that yields for each and . Thus, for each of the two kinds of nature actions, and , an observation, , is given. This yields an alternative way to express Formulation 9.5:

Formulation 9..6 (Nature Interferes with the Observation)
1. A nonempty, finite set called the action space.
2. A nonempty, finite set called the nature action space.
3. A nonempty, finite set called the observation space.
4. For each , a nonempty set called the nature observation action space.
5. A sensor mapping .
6. A function called the cost function.

This nicely unifies the nondeterministic and probabilistic models with a single function . To express a nondeterministic model, it is assumed that any is possible. Using ,

 such that (9.27)

For a probabilistic model, a distribution is specified (often, this may reduce to ). Suppose that when the domain of is restricted to some , then it forms an injective mapping from to . In other words, every nature observation action leads to a unique observation, assuming is fixed. Using and , is derived as

 (9.28)

If the injective assumption is lifted, then is replaced by a sum over all for which . In Formulation 9.6, the only difference between the nondeterministic and probabilistic models is the characterization of , which represents a kind of measurement interference. A strategy still takes the form . A hybrid model is even possible in which one nature action is modeled nondeterministically and the other probabilistically.

Steven M LaValle 2020-08-14