2007/9/15aiai '07 (aix-en-provence, france)1 reconsideration of circumscriptive induction with...

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

Reconsideration of Circumscriptive Induction with

Pointwise Circumscription Koji Iwanuma1

Katsumi Inoue2

Hidetomo Nabeshima1

1 University of Yamanashi2 National Institute of Informatics


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


Induction

Explanatory Induction Definition [Muggleton 95]

Given: B and EFind: H such that B∧H E

Descriptive Induction Definition [Helft 89]

Given: B and EFind: H such that comp (B∧E ) H

the same logical form as abduction

a formalization of nonmonotonic reasoning


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 explainobservations


hypothesis by explanatory induction

hypotheses by descriptive induction

Inductive Leaps and Incompleteness

facts in E

facts not in E inductive leaps uncovered


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.


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 Qthe rest of predicates whose extensions are fixed


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 InductionH is a correct solution to the 〈 B, E, P, Z 〉 if CIRC[B∧E ;P ;Z ] H B ∧H E

descriptive induction

explanatory induction


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 )




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 atomwhose 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.


ConsistencyIf B is consistent and H is conservative, then B ∧H is consistent.

CompletenessIf H is a correct solution to 〈 B, E, P, Z 〉 ,

then H explain all observations E : B∧H E

Advantage of Circumscriptive Induction 1


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.


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


General Inductive LeapFor 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: P1(s) ∨P1(t) , P1(s) ∨P2(s) 　…


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.


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


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.


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 ).


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 )] )


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.


Pointwise Circumscriptive Induction ProblemGiven clausal theories B and E, disjoint predicates P , 〈 B, E, P 〉 is a pointwise induction problem

Pointwise Circumscriptive InductionH 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


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 〉 ,


Soundness of Pointwise Circumscriptive Induction

If H is a correct solution to a pointwise circumscriptive induction 〈 B, E, P 〉 , thenfor 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.


How to derive a correct answer from pointwise circumscription?

Minimal extension formulas and the ordinary resolution can often interesting correct answers.


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)


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)


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 ] )


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.


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)




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 fortreating the disjunctive situtaion


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 m

ore 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.


Thank you for your attention !!

2007/9/15aiai '07 (aix-en-provence, france)1 reconsideration of circumscriptive induction with...

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e inductive leaps

bird x flies x inductive

e facts

z p p ap

flies x bird x incompleteness

clausal theories b

predicate p

disjoint predicates