2007/9/15AIAI '07 (Aix-en-Provence, France)1 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

2007/9/15AIAI '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

2007/9/15AIAI '07 (Aix-en-Provence, France)3 Induction Explanatory Induction  Definition [Muggleton 95] Given: B and E Find: H such that B ∧ H E Descriptive Induction  Definition [Helft 89] Given: B and E Find: H such that comp (B ∧ E ) H the same logical form as abduction a formalization of nonmonotonic reasoning

2007/9/15AIAI '07 (Aix-en-Provence, France)4 Example  Explanatory induction  H = Bird (x) ⊃ Flies (x)  Inductive leap: B ∧ H Flies (b)  Descriptive induction  H = Flies (x) ⊃ Bird (x)  Incompleteness: B ∧ H Flies(a) Background knowledge: B = Bird(a) ∧ Bird(b) Observations: E = Flies(a) inductive leap: deduction of new facts not stated in given observations incompleteness: inability to explain observations

2007/9/15AIAI '07 (Aix-en-Provence, France)5 hypothesis by explanatory induction hypotheses by descriptive induction Inductive Leaps and Incompleteness facts in E facts not in E inductive leaps uncovered

2007/9/15AIAI '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.

2007/9/15AIAI '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

2007/9/15AIAI '07 (Aix-en-Provence, France)8 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 ∧ H E descriptive induction explanatory induction

2007/9/15AIAI '07 (Aix-en-Provence, France)9 Example  Explanatory induction  H = Bird (x) ⊃ Flies (x)  Inductive leap: Flies (b)  Descriptive induction  H = Flies (x) ⊃ Bird (x)  incomplete: B ∧ H Flies(a)  Circumscriptive induction  H = Bird (x) ∧ (x ≠b) ⊃ Flies (x)  conservative and complete for a new fact Bird (c ), B ∧ H Flies (c ) Background knowledge: B = Bird(a) ∧ Bird(b) Observations: E = Flies(a)

2007/9/15AIAI '07 (Aix-en-Provence, France)10 Inductive Leaps and Conservativeness For a clausal theory S, a predicate p, a test set of induction leap TS (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.

2007/9/15AIAI '07 (Aix-en-Provence, France)11 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 ∧ H E Advantage of Circumscriptive Induction 1

2007/9/15AIAI '07 (Aix-en-Provence, France)12 Solitary in Z  A formula A(Z ) is solitary in predicates Z if A (Z ) can be written in the form of N (Z ) ∧ (K ≦ Z ), where  N(Z ) is a formula not containing any predicates P positively  K is a predicates not containing any predicate in Z

2007/9/15AIAI '07 (Aix-en-Provence, France)13 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.

2007/9/15AIAI '07 (Aix-en-Provence, France)14 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

2007/9/15AIAI '07 (Aix-en-Provence, France)15 General Inductive Leap For a clausal theory S, a predicate set P, a general test set of induction leap GTS (S, P ) is GTS (S, P ) = {A | S A, A is a formula involving no positive atom whose predicate is in P }. GTS allows a formula to be disjunctive. Example: P 1 (s) ∨ P 1 (t), P 1 (s) ∨ P 2 (s) …

2007/9/15AIAI '07 (Aix-en-Provence, France)16 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.

2007/9/15AIAI '07 (Aix-en-Provence, France)17 If Circ[B ∧ E ; P ; Z ] |= H, then H is strongly conservative ， i.e., GTS (B ∧ H, P ) ⊂ＧＴＳ (B ∧ E, P ). Sufficient Condition for Strong Conservativeness Strong Conservativeness of Correct Answers If H is a correct solution to 〈 B, E, P, Z 〉, then H is strongly conservative and complete

2007/9/15AIAI '07 (Aix-en-Provence, France)18 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.

2007/9/15AIAI '07 (Aix-en-Provence, France)19 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 ).

2007/9/15AIAI '07 (Aix-en-Provence, France)20 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 disjunctive formula A. Pointwise circumscription PWC[A ;P ]: A (P ) ∧ ∀ x (P (X ) ⊃￢ A [P/λu (P(u) ∧ u≠x )] )

2007/9/15AIAI '07 (Aix-en-Provence, France)21 Pointwise Circumscription for Circumscriptive Induction 1. Pointwise circumscription is a new computation method i.e., a first-oder approximation method which just uses first-order concepts/tools. 2. Pointwise circumscription often generates interesting correct answers for circumscriptive induction.

2007/9/15AIAI '07 (Aix-en-Provence, France)22 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 ∧ H E Pointwise Circumscriptive Induction descriptive induction explanatory induction

2007/9/15AIAI '07 (Aix-en-Provence, France)23 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 〉,

2007/9/15AIAI '07 (Aix-en-Provence, France)24 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.

2007/9/15AIAI '07 (Aix-en-Provence, France)25 How to derive a correct answer from pointwise circumscription?  Minimal extension formulas and the ordinary resolution can often interesting correct answers.

2007/9/15AIAI '07 (Aix-en-Provence, France)26 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 ] Pointwise Formula  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)

2007/9/15AIAI '07 (Aix-en-Provence, France)27 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; 1. If P (t ) occurs in a definite clause, then P (t ) is replaced by t ≠x 2. 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)

2007/9/15AIAI '07 (Aix-en-Provence, France)28 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 ] )

2007/9/15AIAI '07 (Aix-en-Provence, France)29 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 not depends on individual pairs of B and E. Min[A ;P ;X] and PWC[B ∧ E;P] often generates interesting correct answers for circumscriptive induction.

2007/9/15AIAI '07 (Aix-en-Provence, France)30 Example1: Definite Case  Bird (x) ≡ Min[B ∧ E ; Bird ; x]; 1. x = a ⊃ Bird (x) 2. x = b ⊃ Bird (x) 3. Bird (x) ⊃ x =a ∨ x =b  Flies (x) ≡ Min[B ∧ E; Flies ; x]; 4. x=a ⊃ Flies (x) 5. Flies (x) ⊃ x=a  We can obtain H just by resolution to the clauses (3) and (4), H ： Bird (x) ∧ (x ≠b ) ⊃ Flies (x) Background knowledge: B = Bird(a) ∧ Bird(b) Observations: E = Flies(a)

2007/9/15AIAI '07 (Aix-en-Provence, France)31 Example 2: Disjunctive Case  Bird (x) ≡ Min[B ∧ E ; Bird ; x]; 1. (Bird (a) ⊃ x =a) ∧ (Bird (b) ⊃ x =b ) ⊃ Bird (x) 2. Bird (x) ∧ Bird (a) ⊃ x =a 3. Bird (x) ∧ Bird (b) ⊃ x =b  Flies (x) ≡ Min[B ∧ E ; Flies ; x]; 4. x=a ⊃ Flies (x) 5. x=c ⊃ Flies (x) 6. Flies (x) ⊃ [x=a ∨ x=c ]  By resolving the clauses (2) and (4), we can obtain H: H: Bird (x) ∧ Bird (a) ⊃ Flies (x) Background knowledge: B = Bird(a) ∨ Bird(b) Observations: E = Flies(a) ∧ Flies(c) Conditional hypothesis for treating the disjunctive situtaion

2007/9/15AIAI '07 (Aix-en-Provence, France)32 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.

2007/9/15AIAI '07 (Aix-en-Provence, France)33 Thank you for your attention !!