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Logical reduction of metarules Andrew Cropper & Sophie Tourret - - PowerPoint PPT Presentation
Logical reduction of metarules Andrew Cropper & Sophie Tourret - - PowerPoint PPT Presentation
Logical reduction of metarules Andrew Cropper & Sophie Tourret ILP Examples Learner Computer program Background knowledge Mode declarations Progol ILASP Aleph XHAIL TILDE Metarules Metagol MIL-Hex ILP ProPPR Clint
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Mode declarations Progol ILASP Aleph XHAIL TILDE …
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Metagol MIL-Hex ∂ILP ProPPR Clint MOBAL … (almost all neural-ILP approaches) Metarules
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Metarules
P(A,B) ← Q(A,C),R(C,B)
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Metarules
P(A,B) ← Q(A),R(A,B)
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Metarules
P(A,B) ← Q(A),R(A,B)
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Input % background parent(alice,bob). parent(bob,charlie). % example grandparent(alice,charlie). % metarule P(A,B) ← Q(A,C),R(C,B)
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Output % subs Subs = {P\grandparent,Q\parent,R\parent} % program grandparent(A,B) ← parent(A,C), parent(C,B)
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Where do we get metarules from?
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Completeness Cannot learn grandparent/2 with only P(X)←Q(X) Efficiency More metarules = larger hypothesis space
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Idea: find logically minimal sets
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The clause C is entailment redundant in the clausal theory T ∪ {C} when T ⊨ C
Entailment redundant metarules [Cropper and Muggleton, ILP14]
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C1 = p(A,B)←q(A,B) C2 = p(A,B)←q(A,B),r(A) C3 = p(A,B)←q(A,B),r(A),s(B,C) Entailment redundancy
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{C1} ⊨ {C2,C3} C1 = p(A,B)←q(A,B) C2 = p(A,B)←q(A,B),r(A) C3 = p(A,B)←q(A,B),r(A),s(B,C) Entailment redundancy
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P(A,B)←Q(A,B) P(A,B)←Q(B,A) P(A,B)←Q(A,C),R(B,C) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(B,A),R(A,B) P(A,B)←Q(B,A),R(B,A) P(A,B)←Q(B,C),R(A,C) P(A,B)←Q(B,C),R(C,A) P(A,B)←Q(C,A),R(B,C) P(A,B)←Q(C,A),R(C,B) P(A,B)←Q(C,B),R(A,C) P(A,B)←Q(C,B),R(C,A)
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Entailment reduction
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P(A,B)←Q(B,A) P(A,B)←Q(A,C),R(C,B) Entailment reduction P(A,B)←Q(A,B) P(A,B)←Q(B,A) P(A,B)←Q(A,C),R(B,C) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(B,A),R(A,B) P(A,B)←Q(B,A),R(B,A) P(A,B)←Q(B,C),R(A,C) P(A,B)←Q(B,C),R(C,A) P(A,B)←Q(C,A),R(B,C) P(A,B)←Q(C,A),R(C,B) P(A,B)←Q(C,B),R(A,C) P(A,B)←Q(C,B),R(C,A)
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C1 = P(A,B)←Q(A,B) C2 = P(A,B)←Q(A,B),R(A) C3 = P(A,B)←Q(A,B),R(A,B) C4 = P(A,B)←Q(A,B),R(A,B),S(A,B)
- Entailment reduction problem
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{C1} ⊨ {C2,C3,C4} C1 = P(A,B)←Q(A,B) C2 = P(A,B)←Q(A,B),R(A) C3 = P(A,B)←Q(A,B),R(A,B) C4 = P(A,B)←Q(A,B),R(A,B),S(A,B)
- Entailment reduction problem
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father(A,B) ← parent(A,B), male(A) ✖
- Entailment reduction problem
C1 = P(A,B)←Q(A,B) C2 = P(A,B)←Q(A,B),R(A) C3 = P(A,B)←Q(A,B),R(A,B) C4 = P(A,B)←Q(A,B),R(A,B),S(A,B) {C1} ⊨ {C2,C3,C4}
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The clause C is derivationally redundant in the theory T ∪ {C} when T ⊢ C
Derivation redundancy [Cropper and Tourret, ILP18, JELIA19]
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The clause C is derivationally redundant in the theory T ∪ {C} when T ⊢ C
SLD-resolution in this work
Derivation redundancy
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C1 = P(A,B)←Q(A,B) C2 = P(A,B)←Q(A,B),R(A) C3 = P(A,B)←Q(A,B),R(A,B) C4 = P(A,B)←Q(A,B),R(A,B),S(A,B) Derivation redundancy
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{C1,C2,C3} ⊢ {C4} C1 = P(A,B)←Q(A,B) C2 = P(A,B)←Q(A,B),R(A) C3 = P(A,B)←Q(A,B),R(A,B) C4 = P(A,B)←Q(A,B),R(A,B),S(A,B) father(A,B) ← parent(A,B), male(A) ✔ Derivation redundancy
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MLJ Paper
We compare subsumption, entailment, and derivation reduction We theoretically show whether infinite fragments of metarules can be logically reduced to finite sets We run the reduction algorithms on finite sets of metarules to identify minimal sets We experimentally compare the learning performance
- f Metagol when supplied with reduced sets of
metarule
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- Q. Can we reduce M to a fragment with only two
body literals?
- Q. Can we reduce M to a fragment with finitely many
body literals?
- Q. If M has a finite reduction, what is that fragment?
Theoretical questions
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Idea
- 1. Generate big sets of metarules
- 2. Run the reduction algorithms on the sets
- 3. Study the results.
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Ham maximum arity a maximum body literals m
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Can the connected fragment C2∞ be reduced to two body literals?
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Can the connected fragment C2∞ be reduced to two body literals?
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C2∞ cannot be derivationally reduced to C22 Can the connected fragment C2∞ be reduced to two body literals?
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C2∞ cannot be derivationally reduced to a finite fragment! Can the connected fragment C2∞ be reduced to two body literals?
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Derivation reduction of connected fragment
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Why not?
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Can the Datalog D2∞ fragment be reduced to two body literals?
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Can the Datalog D2∞ fragment be reduced to two body literals? D2∞ cannot be derivationally reduced to a finite fragment
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Reduction summary
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Does it matter? Is there any difference in learning performance when using different reduced sets of metarules?
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Trains
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Accuracies
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Learning times
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String transformations
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String transformations
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Inducing game rules
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Inducing game rules
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Conclusions New form of logical reduction Negative theoretical results (especially for MIL) Little impact on practical performance
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