WHAT IS AN INCONSISTENT HEURISTIC? HOW DO YOU GET AN

Inconsistent Heuristics
What
How
is an
Inconsistent Heuristic?
Inconsistent Heuristic?
Consider this example:
A heuristic is inconsistent if |h(x) - h(y)| > dist(x, y) for
states x, y.
Consistent
Heuristic
4
2
3
2
1
2
0
Goal
PDB
3
1
1
2
x
2
2
4
5
3
4
5
x
x
5
6
7
8
6
7
8
x
x
8
3
1
|3-0| > 1
4
|3-4| ! 1
Goal State
0
Dual of Start
2
3
1
4
5
6
7
8
This research was sponsored by the Israeli Science foundation under grant number 828/06 to Ariel Felner.
are the benefits of using inconsistent heuristics?
Heuristic Value Distribution
Inconsistency Properties
Lookup Costs
1
1
Inconsistent
Heuristic
Heuristic Value
3
0
Regular PDB PDB Symmetry Dual Lookup Dual Symmetry
3
(Incorrect) convential wisdom about inconsistent heuritics:
“One has to work quite hard to concoct heuristics that are
admissible but not consistent” (Russell Norvig, 2005)
We show that it is easy to create inconsistent heuristics, and
that they can be quite useful for search.
Bidirectional Pathmax
For Inconsistent Heuristics
Previous research (Mero, ‘84; Felner, et. al. ‘05) proposed
using PathMax and Bidirectional PathMax (BPMX) for propagating heuristic information between parents and children
during search. BPMX can only be applied when heuristics
are inconsistent.
Parent must
have h-cost of 4
Heuristic difference is 3,
but edge cost is 1
5
4
2
1
1
1
1
5
2
6
4
7
3
1
5
8
2
6
4
7
1
5
8
2
1
6
4
7
3
5
8
2
1
6
4
7
5
3
Other child must
have h-cost of 3
The implementation of BPMX in IDA* is straightforward.
4
3
3
3
5
3
3
4
3
6
3
6
consistent
8
5
3
4
3
4
Expanded
5
Generated
3
Not Generated
inconsistent
In this example, the PDB provides 4 different heuristics.
•These lookups use the same pattern database.
•Unless we do the same lookups in every state,
the heuristic may be inconsistent.
•No single heuristic is always the best.
• Additional heuristic lookups give diminishing returns
Distribution of Heuristic Values (Rubik's Cube)
60
In general with K different consistent heuristics:
We can get a new consistent heuristic by:
• A priori choosing n out of the K heuristics
• At each state compute the n values and take the max
We get a new inconsistent heuristic by:
• Randomly consulting n of the K heuristics
-or • Using heuristics taken from the dual states
We show that inconsistent heuristics work well in
practice, even though previous work showed that
there are cases where inconsistency is bad.
Drawbacks?
Regular Lookup
50
Why would we create an inconsistent heuristic?
1
4
Dual Lookup
Random Lookup
40
Static Distribution
30
20
10
0
0
1
2
3
4
5
6
Heuristic Value
7
8
9
10
11
0
1
Each path through the
top structure results in
a partial expansion of
the bottom path.
28
27
The total cost is O(N2)
0
8
0
26
1
8
0
8
O(N) nodes
0
1
1
1
17
9
8
0
25
25
O(N) nodes
1
Start State
% Nodes
Edge cost
What
do you get an
As used in
Single-Agent
Search
Ariel Felner, Ben-Gurion University
Uzi Zahavi, Bar-Ilan University
Jonathan Schaeffer, University of Alberta
Nathan Sturtevant, University of Alberta