Learning Stochastic Logic Programs

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Learning Stochastic Logic Programs

• written by Stephen Muggleton
• Taeyoung Jeong

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Contents

• Stochastic
• Logic Programming
• Stochastic Logic Programming
• Reconsideration of Bayes Theorem
• Results

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Stochastic

• Stochastic, from the Greek "stochos" or "aim,
  guess", means of, relating to, or characterized
  by conjecture and randomness. A stochastic
  process is one whose behavior is
  non-deterministic in that a state does not fully
  determine its next state. (wikipedia.org,
  stochastic)
• Fuzzy? Monte-Carlo? Statistic?

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Logic Program

• Given theory P, set of clauses
• Find a Model M, that satisfies a clause G

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Logic Program Example

• 1) ancestor(x,y)?parent(x,y)2)
  ancestor(x,y)?parent(x,z)?anc(z,y)3) parent(A,B)
  4) parent(B,C)
• Find u,v that satisfies ancestor(u,v)

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Proof in LP Resolution

• Assume that for all (u,v), ancestor(u,v) is
  false. It is the same to the sentence, false
  ? ancestor(u,v)
• If we prove this sentence is false for all (u,v),
  it means that ancestor(u,v) is true for all
  (u,v).

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Proof in LP Resolution

• Refutation Completeness If a (set of) sentence
  is not satisfiable, the resolution will always be
  able to derive a contradiction.(To prove
  Refutation Completeness, we use Herbrands
  Theorem, Ground Resolution Theorem, Lifting
  Lemma, etc)
• If we can derive the contradiction within the
  restriction D, that is the prove for it is
  satisfiable for D.

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Logic Program Example again.

• 1) ancestor(x,y)?parent(x,y)2)
  ancestor(x,y)?parent(x,z)?anc(z,y)3) parent(A,B)
  4) parent(B,C)
• Find u,v that satisfies ancestor(u,v)

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Sample continues

• false?ancestor(u,v)
• substitute u/x1,v/y1 with 2)false?parent(x1,z1)
  ?ancestor(z1,y1)
• with 3) x1/A,z1/B, false?ancestor(B,y1)
• 1) x2/B,y2/y1, false?parent(B,y1)
• 4) y1/c makes, false?true Contradiction!
• linear resolution ltG, C2, C3, C1, C4gt applied.
• u/x1,v/y1 x1/A,z1/B x2/B,y2/y1 y1/c gt
  uA, vC

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Stochastic Logic Program

• Given SLP S, a set of labeled clauses pC, where
  C is clause and p is probability. For each
  predicate symbol p, sum of probability pp must be
  pp 1.
• example (complete pp 1)0.5 coin(head)
  ?0.5 coin(tail) ?
• example (incomplete pp lt 1)0.3 likes(X,Y) ?
  pet(Y,X), cat(Y)

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Proof of SLP

• refutation sequence ltG, C1, C2.... gt becomes
  lt1G, p1C1, p2C2, .... gt
• Every step, pG and qCk makes pqR.
• We can find incomplete probability Q(aS)
  ?DS,(?a)Q(DS,(?a)), for all derivations DS,(?a)
  available.

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Model of SLP

• We want to find a distributional L-model M, which
  is Q(aS)M(a) for each ground term a.
  (unreasonable definition?)
• Because we dont have the complete information,
  we cannot bound Q(aS).

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Stochastic LP Model Example 1

• 0.5 coin(head) ?0.5 coin(tail) ?
• Q(coin(head)S) 0.5Q(coin(tail)S) 0.5
• 0.5coin(head), 0.5coin(tail) is a model
  of S.
• 0.4coin(head), 0.6coin(tail) is not a
  model of S.

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Stochastic LP Model Example 2

• Suppose we have in language L,two predicate
  symbols King and Humantwo constant John
  and Arthur0.5 Human(X) ? King(X)0.5
  King(John)
• 0.5 Human(John) ? King(John)0.5
  King(John)--------------------------0.25
  Human(John)

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Example 2 Truth table(?)

• Q(Human(John)S) 0.25Q(King(John)S) 0.5
• ?(Human(John)?King(John)) ? ?Human(John)?King(Jo
  hn)

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Example 2 goes on

• We cannot say about the incomplete area, because
  we dont have full information.
• 1Human(John), 0Human(Arthur)1King(John),
  0King(Arthur) is a model of S.
• 0.1Human(John), 0.9Human(Arthur)0.5King(John),
  0.5King(Arthur) is not a model of S.

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With Bayes theorem

• Bayes Theorem
• Our object Maximize p(SE)
• We will take the random examples, thus p(ES)
  ?p(eiS) ?Q(eiS)

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What is good learning?

• Find sentences that covers all the example
• Sentence should be small enough
• Not losing the generality Low derivation cost
  (leads to the error!)
• Size of sentence and the generality of the
  hypothesis is in the relation of
  tradeoff.(Learning from Positive Data,
  S.Muggleton. 2000)

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Bayes Theorm cont.

• With logarithm, it is converted into log2p(S)
  means number of bits needed to represent S, while
  ?log2Q(eiS) is to represent derivations.
• We can calculate ?Q(eiS) in a short time, with
  LP resolution!

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So the author

• Designed an SLP search strategy, like LP
  construction or Parameter estimation, by
  redefining the compression function with the
  User-defined evaluation function in Progol4.5

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Works like this
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Conclusion

• Works efficient, finds meaningful solutions
• Cannot find optimal solutionsa) LP construction
  is approximate since it involves greedy
  clause-by-clause construction.b) Parameter
  Estimation is only optimal in the case that each
  positive example has a unique derivation.

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Reference

• Learning Stochastic Logic Programs. S.Muggleton.
• Learning from Positive Data. S.Muggleton.
• Semantics and derivation from Stochastic Logic
  Programs. S.Muggleton.
• ltArtificial Intelligence, a Modern Approach. 2nd
  editiongt. S.Russell P.Norvig.