Derivation reduction of metarules in meta-interpretive learning - - PowerPoint PPT Presentation

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Derivation reduction of metarules in meta-interpretive learning - - PowerPoint PPT Presentation

Aug 04, 2023 321 likes •843 views

Derivation reduction of metarules in meta-interpretive learning Andrew Cropper & Sophie Tourret Input Output Examples Background knowledge Logic program Bias Biases - Mode declarations (Progol, ILASP, Aleph, XHAIL, ) - Metarules

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Derivation reduction of metarules in meta-interpretive learning

Andrew Cropper & Sophie Tourret

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Input Output Examples Background knowledge Bias Logic program

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  • Mode declarations (Progol, ILASP, Aleph, XHAIL, …)
  • Metarules (Metagol, MIL-Hex, ∂ILP, ProPPR, Clint, MOBAL …)

Biases

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∃PQ∀AB P(A,B) ← Q(A,B) ∃PQR∀AB P(A,B) ← Q(A),R(A,B) ∃PQR∀ABC P(A,B) ← Q(A,C),R(C,B)

Metarules

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P(A,B) ← Q(A,B) P(A,B) ← Q(A),R(A,B) P(A,B) ← Q(A,C),R(C,B)

P,Q,R are existentially quantified second-order variables A,B,C are universally quantified first-order variables

Metarules

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Input Output

% background parent(ann,amy)← parent(amy,amelia)← % example grandparent(ann,amelia)← % metarule P(A,B)←Q(A,C),R(C,B)

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Input Output

% background parent(ann,amy)← parent(amy,amelia)← % example grandparent(ann,amelia)← % metarule P(A,B)←Q(A,C),R(C,B) grandparent(A,B)← parent(A,C), parent(C,B) { P\granparent, Q\parent, R\parent }

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Completeness cannot learn grandparent/2 with only P(X)←Q(X) Efficiency more metarules = larger hypothesis space Usability Users do not want to provide metarules

Why?

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The Horn clause C is entailment redundant in the Horn theory T ∪ {C} when T ⊨ C

Remove redundant metarules [ILP14]

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Entailment redundancy

C1 = h(A,B)←s(A,B) C2 = h(A,B)←s(A,B),u(B) C3 = h(A,B)←s(A,B),u(A,B) C4 = h(A,B)←s(A,B),u(A,B),v(A,B)

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Entailment redundancy

{C1} ⊨ {C2,C3,C4} C1 = h(A,B)←s(A,B) C2 = h(A,B)←s(A,B),u(B) C3 = h(A,B)←s(A,B),u(A,B) C4 = h(A,B)←s(A,B),u(A,B),v(A,B)

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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)

? Entailment reduction of metarules [ILP14]

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P(A,B)←Q(B,A) P(A,B)←Q(A,C),R(C,B)

Entailment reduction of metarules [ILP14]

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 redundancy

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)

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{C1} ⊨ {C2,C3,C4}

Entailment redundancy

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)

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father(A,B)←parent(A,B),male(A) ✖ {C1} ⊨ {C2,C3,C4}

Entailment redundancy

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)

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The Horn clause C is derivationally redundant in the Horn theory T ∪ {C} when T ⊢ C

Derivation redundancy SLD-resolution

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Derivation redundancy

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)

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Derivation redundancy

{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) ✔

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While there is a clause in T such that T - {C} ⊢k C: Set T = T - {C}

Derivation redundancy

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P(A)←Q(A) ✔ P(A,B)←Q(A,C) ✔ P(A,B)←Q(A,B),R(B,D),S(D,B) ✔ P(A)←Q(B) ✖ P(A)←Q(A),R(B,C) ✖ P(A,B)←Q(A,B),S(C) ✖

Connected clauses

body literals are connected to the head literal

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P(A,B)←Q(A,B) ✔ P(A)←Q(A,B),R(B) ✔ P(A,B,C)←Q(A,B,C) ✖ P(A)←Q(A,B,C),R(B,C) ✖

H2m

restriction on literal arity

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P(A,B)←Q(A,B) ✔ P(A,B)←Q(A,C),R(C,B) ✔ P(A)←Q(A) ✖ P(A,B)←Q(A,B),R(B) ✖

H2=m

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P(A,B)←Q(A,B) ✔ P(A)←Q(A,B,C),R(B,C) ✔ P(A)←Q(A),R(A),S(A) ✖ P(A,B)←Q(A),R(B),S(A,B) ✖

Ha2

restriction on number of body literals

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P(A)←Q(A),R(A) ✔ P(A,B)←Q(A,B),R(A,B) ✔ P(A)←Q(A) ✖ P(A,B)←Q(A,B),R(B) ✖

Ha2=

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P(A)←Q(A) ✔ P(A,B)←Q(A,B) ✔ P(A,B)←Q(A,C),R(C,B) ✔ P(A,B)←Q(A) ✖ P(A)←Q(A,B) ✖ P(A)←Q(A),R(A) ✖

each first-order variable appears exactly twice

Exactly-two connected

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Idea

  • 1. Run derivation reduction with a SLD-resolution depth

bound of 10 on sub-fragments of an infinite fragment.

  • 2. Study the results.
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E2=5

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E-reduction D-reduction

P(A,B)←Q(B,A) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(B,A) P(A,B)←Q(A,C),R(C,B)

E2=5

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E-reduction D-reduction

P(A,B)←Q(B,A) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(B,A) P(A,B)←Q(A,C),R(C,B)

E2=5

E2=2 ⊢ E2=∞ ✔

Same as ILP14 paper

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E25

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E-reduction D-reduction

P(A)←Q(A) P(A)←Q(A,B),R(B) P(A,B)←Q(B,A) P(A,B)←Q(A),R(B) P(A,B)←Q(A,C),R(C,B) P(A)←Q(A) P(A)←Q(A,B),R(B) P(A,B)←Q(B,A) P(A,B)←Q(A),R(B) P(A,B)←Q(A,C),R(C,B)

E25

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E-reduction D-reduction

P(A)←Q(A) P(A)←Q(A,B),R(B) P(A,B)←Q(B,A) P(A,B)←Q(A),R(B) P(A,B)←Q(A,C),R(C,B) P(A)←Q(A) P(A)←Q(A,B),R(B) P(A,B)←Q(B,A) P(A,B)←Q(A),R(B) P(A,B)←Q(A,C),R(C,B)

E25

E22 ⊢ E2∞ ✔

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P(A)←Q(A) ✔ P(A)←Q(A),R(A) ✔ P(A,B)←Q(A,B),R(B) ✔ P(A,B)←Q(A,C),R(C,B) ✔ P(A,B)←Q(A) ✖ P(A)←Q(A,B) ✖ P(A)←Q(A),R(A,B) ✖

each first-order variable appears at least twice (i.e. prevents singleton variables) Two connected

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K2=5 two connected

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E-reduction D-reduction

P(A,B)←Q(B,A) P(A,B)←Q(B,A) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(A,B),R(A,B) P(A,B)←Q(A,B),R(A,C),S(C,D),T(C,D) P(A,B)←Q(A,C),R(A,C),S(B,D),T(B,D) P(A,B)←Q(A,C),R(A,D),S(B,C),T(B,D),U(C,D)

K2=5 two connected

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E-reduction D-reduction

P(A,B)←Q(B,A) P(A,B)←Q(B,A) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(A,B),R(A,B) P(A,B)←Q(A,B),R(A,C),S(C,D),T(C,D) P(A,B)←Q(A,C),R(A,C),S(B,D),T(B,D) P(A,B)←Q(A,C),R(A,D),S(B,C),T(B,D),U(C,D)

K2=5 two connected

K2=2 ⊬ K2=∞ ✖

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K25 two connected

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E-reduction D-reduction

P(A)←Q(A) P(A)←Q(A) P(A)←R(A,B),Q(A,B) P(A)←R(A,B),Q(A,B) P(A)←Q(A),R(A) P(A)←Q(B),R(A,B) P(A,B)←Q(B,A) P(A,B)←Q(B,A) P(A,B)←Q(A),R(B) P(A,B)←Q(A),R(B) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(A,B),R(A,B) P(A,B)←Q(A),R(A,B) P(A,B)←Q(A,C),R(A,D),S(C,B),T(B,D),U(C,D)

K25 two connected

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E-reduction D-reduction

P(A)←Q(A) P(A)←Q(A) P(A)←R(A,B),Q(A,B) P(A)←R(A,B),Q(A,B) P(A)←Q(A),R(A) P(A)←Q(B),R(A,B) P(A,B)←Q(B,A) P(A,B)←Q(B,A) P(A,B)←Q(A),R(B) P(A,B)←Q(A),R(B) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(A,B),R(A,B) P(A,B)←Q(A),R(A,B) P(A,B)←Q(A,C),R(A,D),S(C,B),T(B,D),U(C,D)

K25 two connected

K2=2 ⊬ K2=5 ✖

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E-reduction D-reduction

P(A)←Q(A) P(A)←Q(A) P(A)←R(A,B),Q(A,B) P(A)←R(A,B),Q(A,B) P(A)←Q(A),R(A) P(A)←Q(B),R(A,B) P(A,B)←Q(B,A) P(A,B)←Q(B,A) P(A,B)←Q(A),R(B) P(A,B)←Q(A),R(B) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(A,C),R(C,B) P(A,B)←Q(A,B),R(A,B) P(A,B)←Q(A),R(A,B) P(A,B)←Q(A,C),R(A,D),S(C,B),T(B,D),U(C,D)

K25 two connected

K2=∞ cannot be reduced ✖

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Why not?

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Does it matter?

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Accuracies

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Learning times

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% target program f(X):-has_car(X,C1), long(C1), two_wheels(C1), has_car(X,C2), long(C2), three_wheels(C2).

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% E-reduction f(A):-has_car(A,B),f1(A,B). f1(A,B):-has_car(A,C),f2(C,B). f2(A,B):-long(A),three_wheels(B).

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% D-reduction f(A):-f1(A),f2(A). f1(A):-has_car(A,B),three_wheels(B). f2(A):-has_car(A,B),roof_open(B).

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% D*-reduction f(A):-f1(A),f2(A). f1(A):-has_car(A,B),three_wheels(B). f2(A):-has_car(A,B),f3(B). f3(A):-long(A),two_wheels(A).

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% D*-reduction f(A):- has_car(A,B), three_wheels(B), has_car(A,C), long(C), two_wheels(C). % target program f(X):- has_car(X,C1), long(C1), two_wheels(C1), has_car(X,C2), long(C2), three_wheels(C2).

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  • Study derivation reduction problem
  • Other fragments
  • Triadics
  • Connected
  • Unconstrained resolution

Todo