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Reconsideration of Circumscriptive Induction with Pointwise Circumscription

Reconsideration of Circumscriptive Induction with Pointwise Circumscription. Koji Iwanuma 1 Katsumi Inoue 2 Hidetomo Nabeshima 1. 1 University of Yamanashi 2 National Institute of Informatics. Contents. Background Explanatory Induction and Descriptive Induction

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Reconsideration of Circumscriptive Induction with Pointwise Circumscription

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  1. Reconsideration of Circumscriptive Induction with Pointwise Circumscription Koji Iwanuma1 Katsumi Inoue2 Hidetomo Nabeshima1 1 University of Yamanashi 2 National Institute of Informatics AIAI '07 (Aix-en-Provence, France)

  2. Contents • Background • Explanatory Induction and Descriptive Induction • Circumscriptive Induction for unifying both induction • Reconsideration of Circumscriptive Induction • General Inductive Leap and Strong Conservativeness • Pointwise Circumscription, i.e., a first-order approximation of circumscription, as • Yet Another Induction Framework • Conclusions and Future Works AIAI '07 (Aix-en-Provence, France)

  3. the same logical form as abduction a formalization of nonmonotonic reasoning Induction Explanatory Induction • Definition [Muggleton 95] Given: B and E Find: H such that B∧HE Descriptive Induction • Definition [Helft 89] Given: B and E Find: H such that comp (B∧E ) H AIAI '07 (Aix-en-Provence, France)

  4. Example Background knowledge: B = Bird(a) ∧ Bird(b) Observations: E = Flies(a) • Explanatory induction • H = Bird (x) ⊃ Flies (x) • Inductive leap: B∧HFlies (b) • Descriptive induction • H = Flies (x) ⊃ Bird (x) • Incompleteness: B ∧HFlies(a) inductive leap: deduction of new facts not stated in given observations incompleteness: inability to explain observations AIAI '07 (Aix-en-Provence, France)

  5. hypotheses by descriptive induction hypothesis by explanatory induction uncovered inductive leaps Inductive Leaps and Incompleteness facts in E facts not in E AIAI '07 (Aix-en-Provence, France)

  6. Difficult to combine • Explanatory induction • complete • non-conservative (i.e., inductive leaps) • Descriptive induction • incomplete • conservative (i.e., no inductive leap) • Circumscriptive induction [Inoue and Saito 04] • unify both induction for keeping each merit. AIAI '07 (Aix-en-Provence, France)

  7. Circumscription • Definition [McCarthy 80, Lifschitz 85] CIRC[A;P;Z ]≡A(P,Z)∧∀ｐｚ (p＜P ⊃￢A(p,z)) • Policy • Minimized predicates P predicates whose extensions are minimized • Variable predicates Z predicates whose extensions are allowed to vary in minimizing predicates of P • Fixed predicates Q the rest of predicates whose extensions are fixed AIAI '07 (Aix-en-Provence, France)

  8. descriptive induction explanatory induction Circumscriptive Induction [Inoue and Saito 04] Circumscriptive Induction Problem Given clausal theories B and E, disjoint predicates P and Z, 〈B, E, P, Z 〉 is a circumscriptive induction problem Circumscriptive Induction H is a correct solution to the 〈B, E, P, Z 〉 if • CIRC[B∧E ;P ;Z ] H • B ∧HE AIAI '07 (Aix-en-Provence, France)

  9. for a new fact Bird (c ), B∧HFlies (c ) Example Background knowledge: B = Bird(a) ∧ Bird(b) Observations: E = Flies(a) • Explanatory induction • H = Bird (x) ⊃ Flies (x) • Inductive leap: Flies (b) • Descriptive induction • H = Flies (x) ⊃ Bird (x) • incomplete: B ∧HFlies(a) • Circumscriptive induction • H = Bird (x) ∧(x ≠b) ⊃ Flies (x) • conservative and complete AIAI '07 (Aix-en-Provence, France)

  10. Inductive Leaps and Conservativeness For a clausal theory S , a predicate p , a test set of induction leapTS (S, p) is TS (S, p ) ={A | S A , A is a ground atom whose predicate is p } For clausal theories B , E , and H , H realizes an induction leap if there is p in B ∧E s.t. TS (B ∧ H, p ) － TS (B ∧ E, p ) ≠Φ Otherwise, H is said to be conservative. AIAI '07 (Aix-en-Provence, France)

  11. Advantage of Circumscriptive Induction 1 Consistency If B is consistent and H is conservative, then B ∧H is consistent. Completeness If H is a correct solution to 〈B, E, P, Z 〉, then H explain all observations E : B∧HE AIAI '07 (Aix-en-Provence, France)

  12. Advantage of Circumscriptive Induction 2 Conservativeness If B ∧E is solitary in Z , then H is conservative. Corollary: If Z appears only in heads of B ∧E and H is a correct solution to 〈B, E, P, Z 〉,then H is complete and conservative. AIAI '07 (Aix-en-Provence, France)

  13. Our Goals • Reconsideration of circumscriptive induction: • to generalize the concept of induction leap, and • to strengthen the conservativeness. • Study pointwise circumscription, a first-order approximation of circumscription, as • Yet Another Induction Framework AIAI '07 (Aix-en-Provence, France)

  14. General Inductive Leap For a clausal theory S , a predicate set P , a general test set of induction leapGTS (S, P )is GTS (S, P ) ={A | S A , A is a formula involving no positive atom whose predicate isin P }. GTS allows a formula to be disjunctive. Example: P1(s) ∨P1(t) , P1(s) ∨P2(s)… AIAI '07 (Aix-en-Provence, France)

  15. Strong Conservativeness For clausal theories B , E , and H ,a predicate set P , H realizes an general inductive leap if GTS (B ∧ H, P ) － GTS (B ∧ E, P ) ≠Φ. Otherwise, H is strongly conservative. AIAI '07 (Aix-en-Provence, France)

  16. Sufficient Condition for Strong Conservativeness If Circ[B∧E ; P ; Z ] |= H , then H is strongly conservative，i.e., GTS (B ∧ H, P ) ⊂ＧＴＳ (B ∧ E, P ). Strong Conservativeness of Correct Answers If H is a correct solution to 〈B, E, P, Z 〉, then H is strongly conservative and complete AIAI '07 (Aix-en-Provence, France)

  17. Problems of Circumscriptive Induction • It is unclear what kinds of formulas can be correct answers? • Second-order formulation makes it difficult to effectively compute. Pointwise circumscription could be a solution for the above problems, because it is a first-order approximation of circumscription. AIAI '07 (Aix-en-Provence, France)

  18. Pointwise Circumscription[Lifschitz 85] PWC[A ;P ]≡def A (P ) ∧∀x (P(X )⊃￢A [P/λu (P(u)∧u≠x )]) • where [P /λu (P (u)∧u≠x )] denotes the substitution of all occurrences of P by λu (P (u )∧u ≠x ). AIAI '07 (Aix-en-Provence, France)

  19. Pointwise circumscription PWC[A ;P ]: A (P ) ∧　　∀x (P (X ) ⊃￢A [P/λu (P(u)∧u≠x )] ) • PWC[A ;P ] semantically states that it is impossible to obtain a model of A by eliminating exactly one element from the extension of P . • PWC[A ;P ] is a first-order approximation of CIRC[A;P ], i.e., CIRC[A;P ] PWC[A ;P ]. • PWC[A ;P ] is an extension of predicate completion for disjunctiveformula A. AIAI '07 (Aix-en-Provence, France)

  20. Pointwise Circumscription for Circumscriptive Induction • Pointwise circumscription is a new computation method i.e., a first-oder approximation method which just uses first-order concepts/tools. • Pointwise circumscription often generates interestingcorrect answers for circumscriptive induction. AIAI '07 (Aix-en-Provence, France)

  21. descriptive induction explanatory induction Pointwise Circumscriptive Induction Pointwise Circumscriptive Induction Problem Given clausal theories B and E, disjoint predicates P, 〈B, E, P〉 is a pointwise induction problem Pointwise Circumscriptive Induction H is a correct solution to the 〈B, E, P 〉 if • PWC[B∧E ;P ;Z ] H • B ∧HE AIAI '07 (Aix-en-Provence, France)

  22. Strong Conservativeness If H is a correct solution to 〈B, E, P 〉, then H is strongly conservative and complete Soundness of Pointwise Circumscriptive Induction for Circumscriptive Induction If H is a correct solution to a poitwise circumscriptive induction 〈B, E, P 〉, then for any variable predicates Z, H is a correct answer for circumscriptive induction 〈B, E, P , Z〉, AIAI '07 (Aix-en-Provence, France)

  23. Soundness of Pointwise Circumscriptive Induction • If H is a correct solution to a pointwise circumscriptive induction 〈B, E, P 〉, then • for any variable predicates Z s.t. Z∩P=φ, • H is a correct answer for circumscriptive induction 〈B, E, P , Z〉. Pointwise circumscription can be used as an approximation computation framework for circumscriptive induction. AIAI '07 (Aix-en-Provence, France)

  24. How to derive a correct answer from pointwise circumscription? • Minimal extension formulas and the ordinary resolution can often interesting correct answers. AIAI '07 (Aix-en-Provence, France)

  25. Pointwise Formula PWC[A ;P ]≡A (P ) ∧　　∀x (P (x) ⊃￢A [P/λu (P(u)∧u≠x )] ) • We call the above subformula ￢A [P/λu (P (u)∧u ≠x )]pointwise formula, denoted as Pwf[A ;P ;x ] • Example Suppose B ; Bird(a) ∨Bird(b)and E ; Flies(a)∧ Flies(c) Pwf[B ; Bird ; x] = (Bird (a) ⊃x =a)∧(Bird (b) ⊃ x =b ) Pwf[E; Flies ; x] = (x=a )∨ (x=c) AIAI '07 (Aix-en-Provence, France)

  26. Minimal Extension Formula as Revised Pwf[A ;P ;X] • The minimal extension formula Min[A ;P ;X ] is ￢B where B is obtained from A by replacing every positive occurrence of P in A as follows; • If P (t ) occurs in a definite clause, then P (t ) is replaced by t ≠x • Otherwise P (t ) is replaced by P (t )∧t≠x • Example Suppose B ; Bird(a) ∨Bird(b)and E ; Flies(a)∧ Flies(c) Min[B ; Bird ; x] = (Bird (a) ⊃x =a)∧(Bird (b) ⊃ x =b ) Min[E; Flies ; x] = (x=a )∨ (x=c) AIAI '07 (Aix-en-Provence, France)

  27. Some Properties [Iwanuma et al. 90] • For any first-order formula A and any predicate P • PWC[A ;P ] ∀x (P (X ) ≡ Pwf[A; P ;X ] ) • PWC[A ;P ] ∀x (P (X ) ≡ Min[A ;P ;X ] ) • For any first-order formula A and any predicate P • A ∀x (Min[A ;P ;X ] ⊃P (X )) • 　 ∀x (Pwf[A; P ;X ] ⊃ Min[A ;P ;X ] ) AIAI '07 (Aix-en-Provence, France)

  28. Entailment Power of Min [A ;P ;X] and PWC[B∧E;P] CIRC[B∧E ;P ;Z ] ∀x (P (X )≡ Min[B∧E ;P ;X ]) is always guaranteed. However, whether B ∧ ∀x (P (X ) ≡ Min [B∧E ;P ;X ]) E or notdepends on individual pairs of B and E. Min[A ;P ;X] and PWC[B∧E;P]often generates interestingcorrect answers for circumscriptive induction. AIAI '07 (Aix-en-Provence, France)

  29. Example1: Definite Case Background knowledge: B = Bird(a) ∧ Bird(b) Observations: E = Flies(a) • Bird (x) ≡ Min[B∧E ; Bird ; x]; • x = a ⊃ Bird (x) • x = b ⊃ Bird (x) • Bird (x)⊃ x =a ∨ x =b • Flies (x) ≡ Min[B∧E; Flies ; x]; • x=a ⊃ Flies (x) • Flies (x) ⊃ x=a • We can obtain H just by resolution tothe clauses(3) and (4), H：Bird (x) ∧(x ≠b ) ⊃ Flies (x) AIAI '07 (Aix-en-Provence, France)

  30. Example 2: Disjunctive Case Background knowledge: B = Bird(a) ∨ Bird(b) Observations: E = Flies(a)∧ Flies(c) • Bird (x) ≡ Min[B∧E ; Bird ; x]; • (Bird (a) ⊃x =a)∧(Bird (b) ⊃ x =b ) ⊃ Bird (x) • Bird (x)∧Bird (a) ⊃ x =a • Bird (x)∧Bird (b) ⊃ x =b • Flies (x) ≡ Min[B∧E ; Flies ; x]; • x=a ⊃ Flies (x) • x=c ⊃ Flies (x) • Flies (x) ⊃ [x=a ∨x=c ] • By resolving the clauses (2) and (4), we can obtain H: H: Bird (x) ∧Bird (a) ⊃ Flies (x) Conditional hypothesis for treating the disjunctive situtaion AIAI '07 (Aix-en-Provence, France)

  31. Conclusion and Future Work • Conclusion • Reconsideration to circumscriptive induction: • General induction leap and strong conservativeness • Propose pointwise circumscription, as a new method for induction tools • Future work • Study Extended Pointwise Circumscription, which is a more accurate first-order approximation of circumscription, where minimal models are considered with k-elements difference relation. Notice that pointwise circumscription just consider one-element difference relation between its models. AIAI '07 (Aix-en-Provence, France)

  32. Thank you for your attention !! AIAI '07 (Aix-en-Provence, France)

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