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Fast Strong Planning for FOND
Problems with Multi-Root DAGs
Jicheng Fu, Andres Calderon Jaramillo - University of Central Oklahoma
Vincent Ng, Farokh B. Bastani, and I-Ling Yen - University of Texas at Dallas
ABSTRACT
We present a planner for addressing a difficult, yet
under-investigated class of planning problems:
Fully Observable Non-Deterministic planning
problems with strong solutions. Our strong planner
employs a new data structure, MRDAG (multi-root
directed acyclic graph), to define how the solution
space should be expanded. We further equip a
MRDAG with heuristics to ensure planning towards
the relevant search direction. We performed
extensive experiments to evaluate MRDAG and the
heuristics. Results show that our strong algorithm
achieves impressive performance on a variety of
benchmark problems: on average it runs more
than three orders of magnitude faster than the
state-of-the-art planners, MBP and Gamer, and
demonstrates significantly better scalability.
Goal
Goal
Initial
State
Initial
State
Figure 1(a). A weak plan.
Figure 1(b). A strong cyclic plan.
There is at least one successful
path to the goal.
Plan may use actions that can cause cycles
but will likely succeed eventually.
Goal
Initial
State
Figure 1(c). A strong plan.
Goal is achieved from any state without using actions
that cause cycles.
BACKGROUND
In its broadest terms, artificial intelligence
planning deals with the problem of designing
algorithms to find a plan in order to achieve a goal
under certain constraints. In this context, a
domain is a structure that describes the possible
actions that can be used in finding a plan. A
planning problem for a given domain specifies
the initial state of a system and a set of goals to
achieve. A planner is an algorithm that solves a
planning problem by finding a suitable set of
actions in the domain to take the system from the
initial state to at least one goal state.
FOND problems assume that each state in a
system can be fully observed and that some
actions in the domain may have more than one
possible outcome (non-determinism). Solutions can
be classified in three categories [Cimatti et al.,
2003]: weak plans, strong cyclic plans, and strong
plans. See Figure 1 and Figure 2.
B
C
Initial State
The procedure continues until the only nonexpanded states are goal states, in which case a
strong plan is returned. If dead-ends are
encountered, the algorithm backtracks to a
previous stage. If the algorithm has to backtrack
from the initial state, a strong plan can not exist.
At each expansion, the planner checks that no
cycle is produced.
Each stage produces a multi-root directed acyclic
graph (MRDAG), where the roots of the graph are
the states with more than one applicable action.
See Figure 3.
We use two heuristics to inform our planner:
•
•
Most Constrained State (MCS): expands
states with fewer applicable actions first.
Least Heuristic Distance (LHD): uses
applicable actions with the least estimated
distance to the goal first.
A
Goal
B
C
A
pick-up(B, A)
C
B
A
Figure 2. Example of a simple strong plan.
The action pick-up(x, y) is non-deterministic as it can succeed or fail (block x may fall on the table).
The action put-down(x) is deterministic.
MRDAG2
MRDAG3
MRDAG4
…
MRDAG1
…
OUR PLANNER
Our planner finds a strong plan if one exists. At
each stage, states with a single applicable action
are expanded until states with more than one
applicable action are encountered. A set of actions
is then selected to be applied to the latter set.
put-down(B)
…
Initial
State
Goal
Figure 3. Expansion of the solution space.
This graph illustrates how MRDAGs are structured and expanded. Dark green nodes are roots of a
MRDAG. Light green nodes are states with exactly one applicable action.
EVALUATION
Among the planners that are capable of solving strong FOND problems, the
two that are most well-known are arguably MBP [Cimatti et al., 2003] and
Gamer [Kissmann and Edelkamp, 2009]. We used domains derived from
the FOND track of the 2008 International Planning Competition [Bryce and
Buffet, 2008]. Gamer outperformed MBP in all domains. Nevertheless, our
planner could perform 2 to 4 orders of magnitude faster than Gamer with
comparable plan sizes in most cases.
REFERENCES
[Bryce and Buffet, 2008] Daniel Bryce and Olivier Buffet. International Planning Competition
Uncertainty Part: Benchmarks and Results, In Proceedings of International Planning Competition,
2008.
[Cimatti et al., 2003] Alessandro Cimatti, Marco Pistore, Marco Roveri, and Paolo Traverso. Weak,
strong, and strong cyclic planning via symbolic model checking, Artificial Intelligence, 147(1-2):35–
84, 2003.
[Kissmann and Edelkamp, 2009] Peter Kissmann and Stefan Edelkamp. Solving Fully-Observable
Non-Deterministic Planning Problems via Translation into a General Game, In Proceedings of the
32nd Annual German Conference on Advances in Artificial Intelligence (KI'09), pages 1–8, Berlin,
Heidelberg: Springer-Verlag.