The discrete feasible planning model will be defined using state-space
models, which will appear repeatedly throughout this book. Most of
these will be natural extensions of the model presented in this
section. The basic idea is that each distinct situation for the world
is called a *state*, denoted by , and the set of all possible
states is called a *state space*, . For
discrete planning, it will be important that this
set is countable; in most cases it will be finite. In a given
application, the state space should be defined carefully so that
irrelevant information is not encoded into a state (e.g., a planning
problem that involves moving a robot in France should not encode
information about whether certain light bulbs are on in China). The
inclusion of irrelevant information can easily convert a problem that
is amenable to efficient algorithmic solutions into one that is
intractable. On the other hand, it is important that is large
enough to include all information that is relevant to solve the task.

The world may be transformed through the application of *actions*
that are chosen by the planner. Each action, , when applied from
the current state, , produces a new state, , as specified
by a *state transition function*, . It is convenient to use
to express a *state transition equation*,

Let denote the *action space* for each state , which
represents the set of all actions that could be applied from . For
distinct
, and are not
necessarily disjoint; the same action may be applicable in multiple
states. Therefore, it is convenient to define the set of all
possible actions over all states:

(2.2) |

As part of the planning problem, a set
of *goal states* is defined. The task of a planning algorithm is to find
a finite sequence of actions that when applied, transforms the
initial state to some state in . The model is
summarized as:

It is often convenient to express Formulation 2.1 as a
directed *state transition graph*. The set of vertices is the
state space . A directed edge from to exists
in the graph if and only if there exists an action
such
that
. The initial state and goal set are designated as
special vertices in the graph, which completes the representation of
Formulation 2.1 in graph form.

Steven M LaValle 2020-08-14