Learning

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• Learning
• From your mistakes, or
• By thinking

St Andrews
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for constraint satisfaction problems
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Why learn?

• Improve performance
• stop making the same mistakes!
• Provide explanations
• why is there no solution?
• why cant I have a chocolate tea pot?
• React to change
• when constraints are retracted/relaxed

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Why bother?
all good sat solvers learn
constraint programmers dont learn
But some are starting to (palm, the Germans, )
cp might be ignoring a win
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Learning while searching
Brief/incomplete history of learning

• Dechters deep and shallow learning (late 80s
  ?)
• The reactive scheduling agent (IJCAI91)
• DAS (PhD thesis, 1990)
• Ginsbergs Dynamic Backtracking (DB or not DB?,
  JAIR 93)
• Conflict-directed backjumping (CI 9(3))
• MAC-CBJ (apes report)
• Relevance bounded learning (help!)
• Nogood recording in local search (Richards2 CP98
  (BMS external?))
• Quick Xplain (Ulrich Junkers, IJCAI 2001)
• Sat solvers (see paper by Innes)

Almost all are learning from mistakes
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Learning by thinking
Reformulate the problem and think about it for as
long as you want
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Reformulate a csp as a problem of finding a
maximal independent set of a hypergraph
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Whats that then?

• A hypergraph G (V,E)
• V is a set of vertices
• E is a set of hyperedges
• an edge with 2 or more vertices

• An independent set S
• assume vertices(e) is set of vertices in
  hyperedge e

• Maximal independent set S
• there is no independent set S that subsumes S

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Show Me!
A Hypergraph
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Show Me!
An Independent Set
You could add vertex 3 or vertex 8!
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Show Me!
A Maximal Independent Set
There are 11 maximal independent sets of size 6
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Show Me!
The Largest Independent Set
There is only one for this graph
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Show Me!
A Minimal Maximal Independent Set
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CP/Choco
But what about maximality?
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CP/Choco
Encoding Maximality
An example, vertex 2
That is, we state when a variable MUST be
selected and when it MUST NOT be selected
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CP/Choco
Example, vertices 1,2, and 3
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More Generally
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So?
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You can reformulate a csp as a problem of finding
a independent set of a hypergraph (this is not
new) The independent set has to be of size n It
is also maximal
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An Example
X Y Z 2 where X, Y and Z are in 0,1
Give me an independent set of size n
We have n.m vertices
A hyper edge for each nogood
An m-clique for each variables domain
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Another Example
Golomb Ruler

• A ruler with N ticks
• Distance between every pair of ticks is
  different
• Ruler is of length L

Independent Set encoding N.L vertices,
corresponding to the L possible values for each
of the N ticks N cliques of size L (binary
nogoods) Hyper edges of arity 3
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Maximality learning
Learning by thinking

• Use maximality as learning, at the top of
  search?
• For a csp with n variables, what is a maximal
  independent set of size less than n, say k?
• It is a nogood, but what kind of nogood?

• It is a set of instantiations of k variables such
  that
• no other variable can be instantiated with any
  value
• i.e. EVERYTHING is in conflict

We can find a maximal independent set in polytime
It is NPC to find a maximal independent set of a
specified size
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Conclusion

• We might learn by thinking
• reformulate the problem
• look for maximal independent sets
• post these as new constraints
• will this be useful?

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