An Abductive Framework for General Logic Programs and other ...
A n A b d u c t i v e F r a m e w o r k for G e n e r a l Logic P r o g r a m s a n d o t h e r N o n m o n o t o n i c Systems Gerhard Brewka G M D , Postfach 12 40 5205 Sankt Augustin, Germany
K u r t Konolige SRI International 333 Ravenswood Ave Menlo Park, CA 94025
Abstract We present an abductive semantics for general p r o p o s i t i o n a l logic programs which defines the m e a n i n g of a logic p r o g r a m in terms of its extensions. T h i s approach extends the stable m o d e l semantics for n o r m a l logic programs in a n a t u r a l way. T h e new semantics is equivalent to stable semantics for a logic p r o g r a m P whenever P is n o r m a l and has a stable m o d e l . T h e a b d u c t i v e semantics can also be applied to generalize default logic and autoepistemic logic in a like manner. O u r approach is based on an idea recently proposed by Konolige for causal reasoning. Instead of m a x i m i z i n g the set of hypotheses alone we m a x i m i z e the u n i o n of the hypotheses, along w i t h possible hypotheses t h a t are excused or refuted by the theory.
1
Background and M o t i v a t i o n
In this paper we investigate the relationship between a b d u c t i o n , logic p r o g r a m m i n g , and other n o n m o n o t o n i c f o r m a l i s m s . 1 T h i s investigation is interesting for several reasons. F i r s t l y , a b d u c t i o n as a f o r m of nonmonotonic reasoning has gained a lot of interest in recent years, and e x p l o r i n g the relationship between different forms of n o n m o n o t o n i c reasoning is of interest in itself. Secondly, as we w i l l show in this paper, it is possible to define a simple and elegant extension of Gelfond and Lifschitz's stable m o d e l semantics [9] based on a b d u c t i o n . T h i s new abductive semantics has the f o l l o w i n g properties: • T h e semantics is equivalent to stable model semantics for p r o g r a m s w h i c h possess at least one stable model. • A p r o g r a m P has a defined m e a n i n g unless P considered as a set of inference rules is inconsistent. I n p a r t i c u l a r , n o r m a l logic programs w i t h o u t stable models are not meaningless. • T h e a b d u c t i v e semantics i m i t a t e s the well-founded m o d e l in n o t assigning a t r u t h v a l u e to propositions *For simplicity we consider only finite propositional logic programs, that is programs w i t h finite Herbrand base, in this preliminary report. A l l definitions also apply to the genera] case.
whose assertion is self-contradictory. • T h e semantics is, w i t h o u t f u r t h e r m o d i f i c a t i o n , applicable to logic programs t h a t contain classical negation and so-called epistemic d i s j u n c t i o n [8]. • T h e semantics can be applied to other consistencybased n o n m o n o t o n i c f o r m a l i s m s , i n c l u d i n g default logic and autoepistemic logic. We consider all of these properties as h i g h l y desirable. Stable model semantics is currently the most widely accepted semantics for logic programs which have a stable m o d e l . We therefore believe t h a t an extension of stable model semantics should preserve the meaning of those programs. On the other h a n d , m a n y authors consider it a severe weakness of stable m o d e l semantics t h a t not all n o r m a l logic programs have stable models; by contrast, the well-founded semantics always exists. O u r semantics overcomes this weakness, in the same manner as the well-founded semantics, by a l l o w i n g a t r u t h v a l u e gap for self-contradictory propositions. At the same t i m e , it does not suffer f r o m the weakness of well-founded semantics, the " f l o a t i n g conclusions" p r o b l e m . 2 F i n a l l y , there has been a great a m o u n t of recent work t r y i n g to extend the expressiveness of n o r m a l logic programs by, among other things, a d d i n g classical negation and "epist e m i c " d i s j u n c t i o n . It turns o u t to be a n o n - t r i v i a l task to adapt existing semantics to more general logic p r o grams. It is therefore clearly an advantage if a simple semantics for n o r m a l programs can directly be applied to theses generalizations. A b d u c t i o n , i n f o r m a l l y , is the generation of explanations for a given fact p. G i v e n a background theory T and a set of possible hypotheses or abducibles H, an exp l a n a t i o n for p is a subset H' of H such t h a t is consistent and p is provable f r o m Usually, there is a further acceptability criterion t h a t distinguishes preferred explanations. In our approach we w i l l consider negated atoms as hypotheses, a logic p r o g r a m (viewed as a set of inference rules) as the background theory. Moreover, we introduce a simple c r i t e r i o n defining the acceptable explanations or, in our t e r m i n o l o g y , extension bases. We consider a p r o p o s i t i o n q derivable ' o m 2
Floating conclusions are conclusions that are intuitively justified by case analysis yet underivable in well-founded semantics. The standard example is Well-founded semantics does not conclude c.
Brewka and Konolige
9
10
Automated Reasoning
S 1 corresponds to the single stable m o d e l {a} of P 3 . 5 2 , a l t h o u g h m a x i m a l , does not correspond to a stable m o d e l . T h i s raises the question w h a t the " i n t e r e s t i n g " ones a m o n g the complete scenarios are. D u n g argues t h a t t w o views are reasonable: a skeptical view which considers the least complete scenario only, and a credulous view w h i c h considers all m a x i m a l complete scenarios. In examples like the one j u s t discussed there seems to be no reason to dispense w i t h stable semantics, u n less one adheres to the skeptical view. In a sense, Dung's credulous view seems to move too far away f r o m stable semantics, whereas the t w o abductive approaches m e n tioned earlier stick w i t h it t o o closely. T h e abductive f r a m e w o r k we present in this paper is d i s t i n c t f r o m this earlier work in the f o l l o w i n g respects: 1. We do n o t restrict the abducibles to atoms. T h i s has the advantage t h a t we can operate on the o r i g i n a l programs directly and do not have to use any k i n d of t r a n s f o r m a t i o n of the programs. Moreover, this makes the use of i n t e g r i t y constraints unnecessary. 2. T h e above approaches treat p r o g r a m rules as clauses and need some i m p l i c i t device to o b t a i n the directedness of rules. We consider the rules of a p r o g r a m as a set of inference rules, not as clauses. 3. We apply a new simple m a x i m a l i t y criterion t h a t guarantees t h a t the r i g h t sets of abducibles are chosen. T h i s criterion models the i n t u i t i o n t h a t undefinedness should be m i n i m i z e d . 4. O u r f r a m e w o r k is simpler t h a n the above approaches and can, unlike t h e m , be applied to logic programs w i t h classical negation and epistemic disj u n c t i o n , as well as other consistency-based nonm o n o t o n i c formalisms. T h e rest of the paper is organized as follows: in Section 2 we introduce our abductive f r a m e w o r k and show how it can be used to formalize n o r m a l logic programs. In Section 3 we treat general logic programs w i t h classical negation and epistemic d i s j u n c t i o n in the heads. Section 4 applies our a b d u c t i v e m e t h o d to default logic, Section 5 applies it to autoepistemic logic.
2
T h e abductive framework
In this section we introduce our abductive semantics for n o r m a l logic programs. We define the n o t i o n of an extension for a logic p r o g r a m . T h i s t e r m i n o l o g y reflects the s i m i l a r i t y t o other w o r k i n n o n m o n o t o n i c reasoning, in p a r t i c u l a r default logic.
Brewka and Konolige
11
• j u s t i f i a b i l i t y : every positive conclusion should be demonstratable using the directed rules of the program, •
m i n i m a l undefinedness: the number of undefined facts should be reduced as m u c h as possible.
As Sacca and Zaniolo show P-stable models do not guarantee m i n i m a l undefinedness: P-stable models can be proper subsets of other P-stable models. T h e authors therefore propose " t h a t the m i n i m a l undefinedness p r i n ciple should be enforced by r e s t r i c t i n g our a t t e n t i o n to the class of P-stable models t h a t are m a x i m a l . " U n f o r t u nately, m a x i m a l i t y is insufficient to guarantee m i n i m a l undefinedness as can be demonstrated by the p r o g r a m P 3 used earlier in the i n t r o d u c t i o n :
T h i s p r o g r a m has no stable m o d e l , yet it has an extension generated by the extension base { ~ c } Extensions thus have the well-known property of the well-founded semantics [20] in a l l o w i n g t r u t h v a l u e gaps, t h a t is, neither a nor ~a is in the extension of the above p r o g r a m . B u t like stable models, extensions do not suffer f r o m the p r o b l e m of " f l o a t i n g conclusions." In the f o l l o w i n g e x a m p l e , the well-founded semantics does not conclude p, w h i l e p is p a r t of every extension.
a
K u r t Konolige SRI International 333 Ravenswood Ave Menlo Park, CA 94025
Abstract We present an abductive semantics for general p r o p o s i t i o n a l logic programs which defines the m e a n i n g of a logic p r o g r a m in terms of its extensions. T h i s approach extends the stable m o d e l semantics for n o r m a l logic programs in a n a t u r a l way. T h e new semantics is equivalent to stable semantics for a logic p r o g r a m P whenever P is n o r m a l and has a stable m o d e l . T h e a b d u c t i v e semantics can also be applied to generalize default logic and autoepistemic logic in a like manner. O u r approach is based on an idea recently proposed by Konolige for causal reasoning. Instead of m a x i m i z i n g the set of hypotheses alone we m a x i m i z e the u n i o n of the hypotheses, along w i t h possible hypotheses t h a t are excused or refuted by the theory.
1
Background and M o t i v a t i o n
In this paper we investigate the relationship between a b d u c t i o n , logic p r o g r a m m i n g , and other n o n m o n o t o n i c f o r m a l i s m s . 1 T h i s investigation is interesting for several reasons. F i r s t l y , a b d u c t i o n as a f o r m of nonmonotonic reasoning has gained a lot of interest in recent years, and e x p l o r i n g the relationship between different forms of n o n m o n o t o n i c reasoning is of interest in itself. Secondly, as we w i l l show in this paper, it is possible to define a simple and elegant extension of Gelfond and Lifschitz's stable m o d e l semantics [9] based on a b d u c t i o n . T h i s new abductive semantics has the f o l l o w i n g properties: • T h e semantics is equivalent to stable model semantics for p r o g r a m s w h i c h possess at least one stable model. • A p r o g r a m P has a defined m e a n i n g unless P considered as a set of inference rules is inconsistent. I n p a r t i c u l a r , n o r m a l logic programs w i t h o u t stable models are not meaningless. • T h e a b d u c t i v e semantics i m i t a t e s the well-founded m o d e l in n o t assigning a t r u t h v a l u e to propositions *For simplicity we consider only finite propositional logic programs, that is programs w i t h finite Herbrand base, in this preliminary report. A l l definitions also apply to the genera] case.
whose assertion is self-contradictory. • T h e semantics is, w i t h o u t f u r t h e r m o d i f i c a t i o n , applicable to logic programs t h a t contain classical negation and so-called epistemic d i s j u n c t i o n [8]. • T h e semantics can be applied to other consistencybased n o n m o n o t o n i c f o r m a l i s m s , i n c l u d i n g default logic and autoepistemic logic. We consider all of these properties as h i g h l y desirable. Stable model semantics is currently the most widely accepted semantics for logic programs which have a stable m o d e l . We therefore believe t h a t an extension of stable model semantics should preserve the meaning of those programs. On the other h a n d , m a n y authors consider it a severe weakness of stable m o d e l semantics t h a t not all n o r m a l logic programs have stable models; by contrast, the well-founded semantics always exists. O u r semantics overcomes this weakness, in the same manner as the well-founded semantics, by a l l o w i n g a t r u t h v a l u e gap for self-contradictory propositions. At the same t i m e , it does not suffer f r o m the weakness of well-founded semantics, the " f l o a t i n g conclusions" p r o b l e m . 2 F i n a l l y , there has been a great a m o u n t of recent work t r y i n g to extend the expressiveness of n o r m a l logic programs by, among other things, a d d i n g classical negation and "epist e m i c " d i s j u n c t i o n . It turns o u t to be a n o n - t r i v i a l task to adapt existing semantics to more general logic p r o grams. It is therefore clearly an advantage if a simple semantics for n o r m a l programs can directly be applied to theses generalizations. A b d u c t i o n , i n f o r m a l l y , is the generation of explanations for a given fact p. G i v e n a background theory T and a set of possible hypotheses or abducibles H, an exp l a n a t i o n for p is a subset H' of H such t h a t is consistent and p is provable f r o m Usually, there is a further acceptability criterion t h a t distinguishes preferred explanations. In our approach we w i l l consider negated atoms as hypotheses, a logic p r o g r a m (viewed as a set of inference rules) as the background theory. Moreover, we introduce a simple c r i t e r i o n defining the acceptable explanations or, in our t e r m i n o l o g y , extension bases. We consider a p r o p o s i t i o n q derivable ' o m 2
Floating conclusions are conclusions that are intuitively justified by case analysis yet underivable in well-founded semantics. The standard example is Well-founded semantics does not conclude c.
Brewka and Konolige
9
10
Automated Reasoning
S 1 corresponds to the single stable m o d e l {a} of P 3 . 5 2 , a l t h o u g h m a x i m a l , does not correspond to a stable m o d e l . T h i s raises the question w h a t the " i n t e r e s t i n g " ones a m o n g the complete scenarios are. D u n g argues t h a t t w o views are reasonable: a skeptical view which considers the least complete scenario only, and a credulous view w h i c h considers all m a x i m a l complete scenarios. In examples like the one j u s t discussed there seems to be no reason to dispense w i t h stable semantics, u n less one adheres to the skeptical view. In a sense, Dung's credulous view seems to move too far away f r o m stable semantics, whereas the t w o abductive approaches m e n tioned earlier stick w i t h it t o o closely. T h e abductive f r a m e w o r k we present in this paper is d i s t i n c t f r o m this earlier work in the f o l l o w i n g respects: 1. We do n o t restrict the abducibles to atoms. T h i s has the advantage t h a t we can operate on the o r i g i n a l programs directly and do not have to use any k i n d of t r a n s f o r m a t i o n of the programs. Moreover, this makes the use of i n t e g r i t y constraints unnecessary. 2. T h e above approaches treat p r o g r a m rules as clauses and need some i m p l i c i t device to o b t a i n the directedness of rules. We consider the rules of a p r o g r a m as a set of inference rules, not as clauses. 3. We apply a new simple m a x i m a l i t y criterion t h a t guarantees t h a t the r i g h t sets of abducibles are chosen. T h i s criterion models the i n t u i t i o n t h a t undefinedness should be m i n i m i z e d . 4. O u r f r a m e w o r k is simpler t h a n the above approaches and can, unlike t h e m , be applied to logic programs w i t h classical negation and epistemic disj u n c t i o n , as well as other consistency-based nonm o n o t o n i c formalisms. T h e rest of the paper is organized as follows: in Section 2 we introduce our abductive f r a m e w o r k and show how it can be used to formalize n o r m a l logic programs. In Section 3 we treat general logic programs w i t h classical negation and epistemic d i s j u n c t i o n in the heads. Section 4 applies our a b d u c t i v e m e t h o d to default logic, Section 5 applies it to autoepistemic logic.
2
T h e abductive framework
In this section we introduce our abductive semantics for n o r m a l logic programs. We define the n o t i o n of an extension for a logic p r o g r a m . T h i s t e r m i n o l o g y reflects the s i m i l a r i t y t o other w o r k i n n o n m o n o t o n i c reasoning, in p a r t i c u l a r default logic.
Brewka and Konolige
11
• j u s t i f i a b i l i t y : every positive conclusion should be demonstratable using the directed rules of the program, •
m i n i m a l undefinedness: the number of undefined facts should be reduced as m u c h as possible.
As Sacca and Zaniolo show P-stable models do not guarantee m i n i m a l undefinedness: P-stable models can be proper subsets of other P-stable models. T h e authors therefore propose " t h a t the m i n i m a l undefinedness p r i n ciple should be enforced by r e s t r i c t i n g our a t t e n t i o n to the class of P-stable models t h a t are m a x i m a l . " U n f o r t u nately, m a x i m a l i t y is insufficient to guarantee m i n i m a l undefinedness as can be demonstrated by the p r o g r a m P 3 used earlier in the i n t r o d u c t i o n :
T h i s p r o g r a m has no stable m o d e l , yet it has an extension generated by the extension base { ~ c } Extensions thus have the well-known property of the well-founded semantics [20] in a l l o w i n g t r u t h v a l u e gaps, t h a t is, neither a nor ~a is in the extension of the above p r o g r a m . B u t like stable models, extensions do not suffer f r o m the p r o b l e m of " f l o a t i n g conclusions." In the f o l l o w i n g e x a m p l e , the well-founded semantics does not conclude p, w h i l e p is p a r t of every extension.
a
