For completeness,
we present a formal semantics for PPDDL planning problems in terms of
a mapping to a probabilistic transition system with rewards. A
planning problem defines a set of state variables *V*, possibly
containing both Boolean and numeric state variables, although we only
consider planning problems without any numeric state variables in this
section. An assignment of values to state variables defines a state,
and the state space *S*
of the planning problem is the set of states
representing all possible assignments of values to variables. In
addition to *V*, a planning problem defines an initial-state
distribution
*p*_{0} : *S* → [0, 1]
with
*p*_{0}(*s*) = 1
(that is, *p*_{0}
is a probability distribution over states), a formula
φ_{G}
over *V*
characterizing a set of goal states
*G* = {*s* | *s* φ_{G}}, a one-time reward *r*_{G}
associated with entering
a goal state, and a set of actions *A*
instantiated from PPDDL action
schemata. For goal-directed planning problems, without explicit
rewards, we use *r*_{G} = 1.

2005-12-06