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5.1  Local Optimal States

During the search phase, Grt always selects to expand the most promising state, according to its heuristic. If the various facts of a problem were independent or even if Grt always managed to track their interactions through the related facts, this strategy would be optimal. However, this is not always the case and some times the search is led to local optimal states. Therefore, the planner should temporarily backtrack to less promising states, before selecting the most promising ones. Figure 2 presents an example situation:

 

 

Initial state

 

 

Goal state

2

K

 

 

 

2

 

 

K

1

 

 

 

 

1

 

 

 

0

R

 

 

 

0

 

 

R

 

0

1

2

 

 

0

1

2

Figure 2: A 3x3 grid problem.

The problem refers to a grid-like domain (McDermott, 1999), where K is a key and R is a robot. The robot can only proceed to adjacent positions. The valid actions are get and leave the key and move the robot. Table 3 shows part of the Greedy Regression Table for the problem of Figure 2.

According to this Table, the distance between the initial and the goal state is 10. There are two applicable to the initial state actions, moving R to n1_0 and moving R to n0_1. After moving R to n1_0 the resulting state has a distance from the goals equal to 9, whereas after moving R to n0_1 the resulting state has a distance from the goals equal to 11. So the planner decides to move R to n1_0 and subsequently to n2_0. However, it is obvious that the optimal first movements are moving the robot to n0_1, next to n0_2, getting the key etc.

 

Fact

Distance

from Goals

Related Facts

 

 

 

(at R n2_0)

0

( )

(at K n2_2)

0

( )

(at R n1_0)

1

( )

(at R n0_0)

2

( )

(at R n0_1)

3

( )

(at R n2_1)

1

( )

(at R n2_2)

2

( )

(in R K)

3

( (at R n2_2) )

(at R n1_2)

3

( )

(at K n1_2)

7

( (at R n1_2) )

(at R n0_2)

4

( )

(at K n0_2)

8

( (at R n0_2) )

Table 3: Part of the Greedy Regression Table for the 3x3 grid problem.

Initially the planner does not select the optimal action, since it leads to a state with a greater distance from the goals, according to the heuristic. In order to decide to move the robot towards the key, the planner should go through all the other valid plans, then backtrack and move the robot to worse states (this requires that the planner maintains a closed list of visited states and does not revisit them). In difficult problems, the number of states that the planner has to visit before following the optimal direction, is extremely large. This is the main reason why Grt, like many other heuristic planners, does not handle grid-like domains efficiently.

For the 3x3 grid problem of Figure 2, an ideal planner should detect that, in order to move the key from n0_2 to n2_2, it is necessary that the robot gets the key, so the fact (at R n0_2) should be achieved before the fact (at R n2_0). However, the planner does not know that the facts (at R n0_0), (at R n2_0) and (at R n0_2) are related in some way, because the domain definition does not provide this piece of information. Therefore, it is necessary to provide the planner with information about relations that hold between the facts of the problem.

 

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Next: Defining XOR-constraints Up: The GRT Planning System Previous: Using XOR-constraints to avoid Local Optimal States

Ioannis Refanidis

14-8-2001