In Strips
(Fikes & Nilsson, 1971), each action a
is represented by three sets of facts: the precondition list Pre(a),
the add-list Add(a) and the delete-list Del(a), where Del(a) Í Pre(a). A state S is defined as a finite set of facts. An action a is applicable
to a state S if:
Pre(a) Í S (1)
The state resulting
from the application of an action a to state S is defined as:
S' = res(S,a) = S \ Del(a)
È Add(a) (2)
Inductively we can
define the state resulting from the application of a sequence of actions (a1,
a2, ..., aN) to a state S as:
S' = res(S, (a1,
a2, ..., aN)) = res( res(S,
(a1, a2, ..., aN-1)), aN) (3)
with the requirement that each
action ai is applicable to the state res(S, (a1,
a2, ..., ai-1)), for each i=1, 2, ..., N.
In the formalization used henceforth, the set of problem constants is assumed
to be finite and no function symbols are used, so the set of actions is finite.
A planning problem P
is a triplet P=(O, Initial, Goals), where O
is the set of ground actions, Initial is the initial state and Goals
is a set of facts. The task is to find a sequence of actions a1,
a2, ..., aN that can be applied to the
initial state, so that the state resulting from their application will be a
superset of Goals. The sequences of actions are called Plans. A
plan that can be applied to the initial state is called a valid plan. A
valid plan that achieves the Goals is called a solution of the
planning problem. A planning problem may have several or no solutions. In the
latter case the problem is described as unsolvable.
The next sub-section
gives a brief presentation of the Asp
heuristic, which was our motivation and helps to understand the following
concepts, whereas the subsequent sub-sections present the Grt heuristic in detail.
Ioannis
Refanidis
14-8-2001