A Preference Semantics. for Ground Nonmonotonic Modal Logics. logics, a family of nonmonotonic modal logics obtained by means of a

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1 A Preference Semantics for Ground Nonmonotonic Modal Logics Daniele Nardi and Riccardo Rosati Dipartimento di Informatica e Sistemistica, Universita di Roma \La Sapienza", Via Salaria 113, I Roma, Italy fnardi,rosatig@dis.uniroma1.it Abstract. In this paper we provide a semantic characterization of ground logics, a family of nonmonotonic modal logics obtained by means of a variant of the well known Mc Dermott and Doyle xed point equation. The term ground logics carries the idea of restricting the negative introspection capabilities of the reasoning agent to the objective (i.e. non modal) part of the theory. This intuition was nicely formalized for modal logic S5 by a semantic denition based on a preference relation on Kripke models, which was obtained as the semantic counterpart to the notion of minimal knowledge initially dened by Halpern and Moses. We have then found a preference relation on Kripke models that both generalizes this notion of minimal knowledge and provides a semantic characterization for a signicant subset of ground logics. 1 Introduction The interplay of nonmonotonic reasoning and reasoning about knowledge and belief has been recognized from the beginning of the research on commonsense reasoning. Since then modal epistemic logics have been studied with the aim of characterizing the reasoning of an agent who is capable to perform introspective reasoning by making assumptions on its own knowledge. The rst formalizations of nonmonotonic reasoning based on the use of a modal operator have been proposed in [13, 15]. The knowledge of an agent is characterized in terms of a xed-point equation that expresses its introspective capabilities. A dierent approach for dening nonmonotonic modal logics was taken in [3, 22, 7], where the knowledge of the agent is characterized on a semantic ground, following the idea of selecting those models in which knowledge is minimal. Recently, there have been a number of attempts to reconcile the xed-point and semantic characterizations of modal nonmonotonic logics. In order to introduce the issue, let us recall McDermott and Doyle equation which applies to the consequence operator of a monotonic modal logic. The equation is a general scheme for dening expansions, namely possible sets of sentences representing the knowledge of an agent reasoning introspectively from an initial body of knowledge. We use L to denote a xed propositional language, L K to denote its modal extension with the only modality K; the set of formulae I L K stands for the initial knowledge of the agent.

2 Given a modal logic S, a consistent set of formulae T is an S MDD -expansion for a set of initial knowledge I L K if T = Cn S (I [ f:k' j ' 62 Tg); (1) where Cn S is the consequence operator in (classical) modal logic S. The resulting consequence operator is dened as the intersection of all S MDD - expansions for I. Such operator is in general nonmonotonic: thus for every modal logic S, the (nonmonotonic) modal logic S MDD is obtained by means of equation (1). The McDermott and Doyle's family of nonmonotonic modal logics has been extensively studied [15, 11, 10, 12]: McDermott [13] analyzed the case of S = S5 and found out that the resulting logic S5 MDD is monotonic, in the sense that the intersection of all S5 MDD -expansions of a theory I is exactly the set of consequences of I in monotonic S5. Schwarz [17] proved the equivalence of Moore's autoepistemic logic [15] with logic KD45 MDD. He also dened in [18] a preference semantics for McDermott and Doyle's family, thus giving a true possible-world semantic characterization of this class of modal nonmonotonic formalisms. This result allows us to study properties of logics S MDD by reasoning on Kripke structures, which is often easier than analyzing sets of modal formulae. From this semantic viewpoint it is easy to show that S5 MDD is not the only degenerate (i.e. monotonic) case for logics S MDD. More specically, it can be shown that for every normal modal logic S, characterized by a class of Kripke models whose accessibility relation is symmetric, its nonmonotonic variant S MDD is equivalent to monotonic logic S5. Therefore there is a whole class of logics in the McDermott and Doyle's family degenerating to the case of monotonic S5, precisely the logics characterized by a class of Kripke models whose accessibility relation is symmetric, i.e. all logics in which every instance of the modal axiom schema B, that is :K:K' ', is valid. Let us now turn the attention to the nonmonotonic modal logics that have been proposed on the basis of semantic considerations and, more precisely, on the intuition that in the models the knowledge attributed to the agent should be minimal. This principle was introduced in [3] and it was enforced by minimizing the set of objective facts known by the agent. In its most general version, this notion of minimal knowledge can be stated as follows: Denition1. A model M is a model of minimal knowledge for I L K in the logic S, if M is a model for I in S and for every model M 0 of I in S, T h(m 0 ) \ L 6 T h(m) \ L, where T h(m) = f' 2 L K j M j= 'g. We say that a logic S is a logic of minimal knowledge if for every theory I L K, every model for I in S is a model of minimal knowledge for I in S. In [3] the notion of minimal knowledge is applied to modal logic S5 and in [7] a possible-world semantics for this logic is given as follows. A model M is constituted by a set of interpretations: a sentence K' is true in a world w belonging to M if ' is true in all worlds w 0 belonging to M. This amounts to considering

3 the so-called universal S5 models, i.e. connected structures whose accessibility relation is reexive, symmetric and transitive. However, not every such structure that satises the initial assumptions I of the agent is taken into consideration: the interesting models are the minimal ones, namely those which do not have any proper superset satisfying I. In other words, an S5-model is minimal if it satises I and cannot be extended by adding a new possible world. Therefore, minimization of knowledge is obtained by maximizing the set of possible worlds, sometimes explained as maximizing ignorance. The nonmonotonic character of this construction becomes evident when looking at the case in which I is the empty set. In this case one can conclude :Kp for every atomic sentence p. This idea has been further developed in [7] 1 and in [9] where it is embedded in a bimodal logic that combines minimization of knowledge with justied assumptions. In [21] the minimization of knowledge is formulated in terms of a preference criterion on Kripke models which diers from the one shown in [7]: in particular, a dierent class of models is taken into consideration, that is the class of models characterizing modal logic S4F. However, this way of minimizing knowledge does not correspond to the minimization of objective facts, so this logic is not a logic of minimal knowledge in the sense of the denition above. The idea of minimal knowledge can be easily captured by a xed point equation in the McDermott and Doyle style, by bounding introspection in the righthand side of Equation (1) to modal-free formulae only. Given a normal modal logic S, a consistent set of formulae T is a ground S-expansion for a set I L K if T = Cn S (I [ f:k' j ' 2 L n Tg): (2) This equation denes a family of logics S G called ground nonmonotonic modal logics (see [19, 23, 4]). These logics are logics of minimal knowledge in the sense of Denition (1), as stated by the following proposition, which directly follows from a property of minimality of ground S G -expansions proved in [12]. Proposition2. If a theory T L is a ground S-expansion for I L K, then T is a model of minimal knowledge for I. Hence every ground logic S G is a logic of minimal knowledge. The goal of our work is to study the generalization of the idea of minimal knowledge intended as the minimization of objective facts, or, in other terms, to study the family of ground logics, for which we nd an appropriate semantic characterization, that has been advocated in [21]. In particular, the characterization we present is an instance of the preference semantics introduced by Shoham [22], where the preference criterion is given by a partial ordering over possible-world models. We show the correspondence between such semantic characterization 1 A more recent version of this work [8] contains a technical dierence which makes the resulting logic a logic of \minimal belief", that is no longer captured by our denition.

4 and the xpoint denition of ground logics for a subclass of normal modal logics, called cluster-decomposable logics, which includes the most studied cases in nonmonotonic modal logics. In the next section we present some preliminary denitions. We then introduce the semantic characterization of the family of ground logics. We nally briey discuss some interesting properties of this family of logics. 2 Preliminaries In this section we recall some basic denitions and theorems which will be referred to in the rest of the paper (see [12] for further details). A Kripke model M is dened as usual by a triple hw; R; V i, where W is a set (whose elements are called worlds), R is a binary relation on W (called the accessibility relation on M), and V is a function assigning a propositional valuation to each world w 2 M. The following is the denition of concatenation between Kripke models. Denition3. Given Kripke models M = hw; R; V i, M 1 = hw 1 ; R 1 ; V 1 i, M 2 = hw 2 ; R 2 ; V 2 i, such that W 1 \ W 2 = ;, M = M 1 M 2 if W = W 1 [ W 2, V = V 1 [ V 2 and R = R 1 [ (W 1 W 2 ) [ R 2. Now we recall the denitions of stable theory and of canonical model for a stable theory, which will be used in the proof of the semantical characterization of ground nonmonotonic modal logics. Denition4. A theory T L K is stable if 1. T is closed under propositional consequence; 2. for every ' 2 L K, if ' 2 T then K' 2 T ; 3. for every ' 2 L K, if ' 62 T then :K' 2 T. Denition5. The canonical model for a stable theory T is a Kripke model M T = hw; R; V i such that W consists of all propositional valuations in which all formulas from T \ L are true, R is the universal relation on W (hence M T is an S5-model) and V (w) = w for every w 2 W. ST (S) will be used to indicate the (unique) stable theory T such that T \L = Cn(S), where Cn is the propositional consequence operator. The following denition is derived from [12], Denition Denition6. A class C of Kripke models is cluster-decomposable if every model in C is of the form M 1 M 2, where M 2 is a universal Kripke model, and for every such model M 1 M 2 and every universal model M 0 2 whose set of worlds is disjoint from that of M 1, the model M 1 M 0 2 is in C. We remark that most of the modal logics studied in the nonmonotonic setting, in particular the logics S5; KD45; S4f and Sw5, are all characterized by a clusterdecomposable class of Kripke models.

5 The following denition characterizes a class of modal logics for which Theorem 8 states an important characterization of ground expansions in terms of minimal McDermott and Doyle's expansions. Denition7. A logic S characterized by a class C of models satises the terminal cluster property if for every M = hw; R; V i 2 C and for every world w 2 W there is a terminal cluster for w, i.e. a maximal subset Y of W such that Y Y R and: 1. for every w 0 2 Y; (w; w 0 ) 2 R; 2. for every w 0 2 Y and every w 00 2 W n Y; (w 0 ; w 00 ) 62 R. Theorem 8. Given a modal logic S such that K S S5, if S satises the terminal cluster property then a theory T is a ground S-expansion for a theory I L K i T is a minimal S MDD -expansion for I, i.e. for every stable theory T 0 containing I, T \ L T 0 \ L. Notice that modal logics S4F, KD45 and SW 5 satisfy the terminal cluster property, hence for such logics the notions of ground S-expansion and of minimal S MDD -expansion coincide. 3 Minimal model semantics for ground nonmonotonic modal logics We now present the semantic characterization for a relevant subset of ground nonmonotonic modal logics. The path we follow is similar to the one used in [18, 12], to provide a semantic characterization for the McDermott and Doyle's family of logics. Although the steps of the construction are the same, there are signicant dierences in the proofs, that will be pointed out in the presentation. The structure is as follows. We rst introduce a notion of ground-intended model, that constitutes a rst step to the semantic characterization of theories formulated through a xed point equation. We then focus on the preference relation, starting from the intuition that the syntactic notion of minimization of objective sentences can be formulated in terms of a partial ordering relation on Kripke models. This is achieved by enforcing the preference relation dened in [18], which in turn is obtained by weakening the preconditions for the comparison of Kripke models. The preference relation is then used to characterize groundminimal models. The nal step shows the correspondence between the notions of ground-intended and ground-minimal models. We start by dening the notion of ground-intended model, which follows from the properties a reasoning agent should satisfy. Such properties correspond to those stated in [12], par. 9.1, but for the formulae it is possible for the agent to assume: in particular, we restrict the introspection capability of the agent to modal-free formulae only. Denition9. Given a normal modal logic S S5 characterized by the class C of Kripke models and a theory I L K, a model M 2 C is ground C-intended for I i:

6 1. M j= I; 2. for every model N 2 C, if N j= I [f:k' j ' 2 LnTh(M)g, then T h(m) = T h(n ). Then, we show that the notion of ground C-intended model exactly corresponds to that of ground S-expansion, if C is the class of Kripke models characterizing modal logic S. Theorem 10. Given a modal logic S such that K S S5, let C be the class of Kripke models characterizing S. An S5-model M is a ground C-intended model for I L K i T h(m) is a ground S-expansion for I. Proof. First, assume M is ground C-intended for I, and dene T = T h(m). Since M is an S5-model, T is stable, therefore T f:k' j ' 2 L n T h(m)g. Besides, I T and S S5, consequently T Cn S (I [ f:k' j ' 2 L n T h(m)g) Since M is ground C-intended, every model of I [ f:k' j ' 2 L n T h(m)g in C is a model of T, therefore T Cn S (I [ f:k' j ' 2 L n T h(m)g) Hence, T is a ground S-expansion for I. Conversely, assume that T is a ground S-expansion for I. Then, T = T h(m) = Cn S (I [ f:k' j ' 2 L n T h(m)g). Consider a model N 2 C such that N j= I [ f:k' j ' 2 L n T h(m)g; it follows that T T h(n ). And if ' 2 L n T, then N j= :K', which implies that N 6j= '. Therefore LnT f' 2 L j N 6j= 'g, from which we obtain T h(n ) \ L T \ L. But T = T h(m) T h(n ), consequently T h(m) \ L = T h(n ) \ L, and since T is stable, we can conclude T h(m) = T h(n ) (see [12], Theorem 8.16). Therefore M is a ground C-intended model for I. ut Now we turn our attention to Kripke models and dene a notion of minimality based on a partial ordering relation on models. First, we need to dene a relation between Kripke models that dier only with respect to the accessibility relation. Denition11. Given the two Kripke models M 1 = hw 1 ; R 1 ; V 1 i and M 2 = hw 2 ; R 2 ; V 2 i, M 2 G M 1 if W 1 = W 2, V 1 = V 2 and R 2 R 1. Using the G relation, we are now able to dene a partial ordering relation on Kripke models. Denition12. Given two Kripke models M 1, M 2, M 2 < G M 1 if there exists a Kripke model M such that: 1. M 2 G M M 1 ; 2. there exists a world a 2 W 2 n W 1 such that for each world b 2 W 1, V 2 (b) 6= V 2 (a).

7 The above notion of partial ordering among Kripke models can informally be explained as follows: M 2 is \less than" M 1 if M 2 is built on top of M 1, by adding at least one world whose corresponding interpretation is dierent from those contained in M 1, in such a way that each new world must be connected to all the worlds belonging to M 1. Moreover, connections between worlds belonging to M 1 and the new worlds are allowed. Finally, minimal models are characterized using the < G ordering, as a special case of Shoham's preference semantics [22]. Denition 13. Given a normal modal logic S characterized by the class of Kripke models C, a model M 2 C is a ground C-minimal model for I if M j= I and for every model M 0 2 C such that M 0 j= I, M 0 6< G M. The preference criterion obtained through the denition of the relation < G can be seen as a stronger version of the minimality criterion found by Schwarz for Mc Dermott and Doyle's logics [18]. We try to give an intuitive argument to this claim. Informally, the dierence between Schwarz's ordering relation and < G is the following: Schwarz compares the S5-model M with all S MDD -models N such that N = M 0 M, therefore N is such that 1. every world of M 0 is connected to every world in M; 2. no world in M is connected to any world in M 0. In the ground case the second condition does not hold, therefore connections between worlds in M 0 and worlds in M are allowed (this is the intuitive meaning of N G M 0 M). Therefore, when checking for the minimality of a model M, the ground criterion allows more monotonic models of the theory to be compared with M. Hence, every ground model is also a McDermott-Doyle's model for I, while the converse in general does not hold. In the rest of this section we show the correspondence between the notion of ground expansion and the semantic notion of ground minimal model, which in turn establishes a preference semantics for ground nonmonotonic modal logics. To prove such a correspondence, we need the following lemmata. Lemma 14. Let N be a Kripke model and M be an S5-model such that: 1. N j= f:k' j ' 2 L n T h(m)g; 2. T h(n ) \ L = T h(m) \ L. Then, T h(n ) = T h(m). Proof. The proof easily follows from the fact that T h(m) is a stable theory and from [12], Theorem ut Lemma 15. Let M 0, M 00 be S5-models. If T h(m 0 ) \ L = T h(m 00 ) \ L then T h(n M 0 ) = T h(n M 00 ) for any Kripke model N.

8 Proof. Since M 0 and M 00 are S5-models, it follows that T h(m 0 ) and T h(m 00 ) are stable theories, hence T h(m 0 ) \ L = T h(m 00 ) \ L implies T h(m 0 ) = T h(m 00 ), which in turn implies T h(n M 0 ) = T h(n M 00 ) (see [12], Lemma 9.19). ut Finally, we state the equivalence between the syntactic and semantic characterizations above dened. This equivalence is proved under the restriction of cluster-decomposable classes of Kripke models. Theorem 16. Given a normal modal logic S, characterized by a cluster-decomposable class of Kripke models C, a theory I L K and a stable theory T, let M be the canonical model for T. Then T is a ground S-expansion for I if and only if M is a ground C-minimal model for I. Proof. The proof makes use of the intermediate semantic notion of intended model above dened. From Theorem 10 it follows that theory T is a ground S-expansion for I if and only if M is a ground C-intended model for I, hence we only have to show that the model M is ground C-intended for I if and only if M is a ground C-minimal model for I. First, assume M is ground C-intended. Then M j= I and for any M 0 2 C, if M 0 j= I [ f:k' j ' 2 L n T h(m)g, then T h(m) = T h(m 0 ). Now, suppose M is not ground C-minimal for I. Then, there exists a Kripke model N 2 C such that N j= I and N < G M. Since M is ground C-intended, it cannot be the case that N j= f:k' j ' 2 L n T h(m)g, therefore there exists a formula ' 2 L such that M j= :K' and N 6j= :K'. This implies that there exists a world b in M such that V M (b) 6= V N (a), for every world a 2 N. But this contradicts the hypothesis N G M, which concludes the proof of the rst part of the theorem. Next, assume M = hw; R; V i is ground C-minimal for I, i.e. M j= I and for any M 0 2 C, if M 0 j= I, then M 0 6 G M. Then, suppose M is not ground C-intended for I, i.e. there exists a model N = hw N ; R N ; V N i 2 C such that N j= I [ f:k' j ' 2 L n T h(m)g and T h(m) 6= T h(n ). Since N j= f:k' j ' 2 L n T h(m)g, it follows that if ' 2 T h(n ) \ L, then ' 2 T h(m) \ L (otherwise N j= :K'), hence T h(n ) \ L T h(m) \ L. Now, it cannot be the case that T h(n ) \ L = T h(m) \ L, otherwise by lemma 14 we would conclude T h(n ) = T h(m), which contradicts the hypothesis. Therefore, T h(n ) \ L T h(m) \ L, thus there exists a world a 2 W N such that V N (a) 6= V (b), for each world b 2 W. Now, the class C is cluster-decomposable, therefore there exists a model N 0 and an S5-model M 0 such that N = N 0 M 0. Since N j= f:k' j ' 2 L n T h(m)g, it follows that M 0 = hw 0 ; R 0 ; V 0 i is such that for each world wbelonging to W there exists a world w 0 belonging to W 0 such that V (w) = V 0 (w 0 ). In other words, M 0 must contain all the interpretations belonging to M. Thus, we only have two possible cases: 1. there exists a world a 2 W 0 such that V 0 (a) 6= V (b), for each world b 2 W. In this case we have M 0 < G M, and since N j= I implies M 0 j= I, it follows that M is not a ground C-minimal model for I, thus contradicting the hypothesis.

9 2. the interpretations belonging to M 0 are exactly the same belonging to M, i.e. T h(m)\l = T h(m 0 )\L. So, consider the Kripke model M 00 = N 0 M: since C is cluster-decomposable, it follows that M 00 2 C. Moreover, we have M 00 < G M. Finally, by Lemma 15 we have that M 00 j= I. Therefore, M is not a ground C-minimal model for I, which again contradicts the hypothesis. Consequently, M is a ground C-intended model for I. ut The last theorem provides a semantic characterization for a subset of the family of ground non-monotonic logics S G, since it relates the solutions of equation 2 to ground-minimal models. In particular, the correspondence is shown to hold for the ground logics built from modal logics S5; KD45; S4f; Sw5. The semantic characterization above presented can be used to prove some properties of ground logics, that we briey outline in the following. { none of the ground logics collapses into a monotonic logic (in particular for S = S5 the logic dened in [3] is obtained, as stated in [19]); { all cluster-decomposable ground logics S G are insensitive to the axiom schema D (i.e. schema K' :K:'), in the sense that, for each ground logic S G, every istance of schema D is a tautology in S G ; { S5 G shows monotonicity with respect to objective formulae, in the sense that for each I; ' 2 L K and for each 2 L, if I j= S5G then I [ f'g j= S5G. Therefore, when adding new information in S5 G, only modal formulae only can be lost, no longer being derivable in the resulting theory. { the above property implies that defaults [16] are not expressible in S5 G, in the sense that there exists no faithful modular translation (i.e. a translation which translates each default into a modal formula, independently from other defaults and from the theory) from default logic to S5 G. Since for no other ground logic the property of monotonicity with respect to objective formulae holds, this behaviour seems to be restricted to the logic S5 G only: for example in the logic S4F G (and in any normal modal logic contained in S4F G ) defaults are expressible with the same translation used in the corresponding Mc Dermott and Doyle's logic (see [2] for a detailed analysis on the embedding of defaults into ground logics). { given any two cluster-decomposable modal logics S, S 0, each containing the axiom schema D, S is equivalent to S 0 if and only if their ground nonmonotonic extensions S G, SG 0 are equivalent. Namely, with the exception of schema D, every cluster-decomposable modal logic S produces a dierent nonmonotonic logic S G. 4 Conclusions In this paper we have presented a semantic characterization of ground logics [19, 23, 4], a family of nonmonotonic modal logics obtained by a modication of the well known Mc Dermott and Doyle xpoint equation [14]. Ground logics owe

10 their name to the idea that the negative introspection capability of the reasoning agent is bounded to his objective knowledge only. This intuition was nicely formalized for modal logic S5 by a semantic denition based on a preference relation on Kripke models, which was obtained as the semantic counterpart to the notion of minimal knowledge initially dened by Halpern and Moses [3, 22, 7]. Therefore our minimal model characterization can be viewed as a generalization of this semantic account of minimal knowledge to a broader class of modal logics. Moreover, the preference relation can be viewed as an enforcing of the one used by Schwarz [18] to provide a minimal model semantics for Mc Dermott and Doyle's logics. We are currently working at a generalization of the semantic characterization above presented, with the aim of identifying the semantic counterpart to the xpoint denition for every ground modal logic. Moreover, we are addressing the use of ground logics in knowledge representation: our rst results in this direction are reported in [1, 2]. Acknowledgements We would like to thank Francesco Maria Donini for many discussions on the subject of the paper. References 1. F. M. Donini, D. Nardi and R. Rosati. Ground Nonmonotonic Modal Logics for Knowledge Representation. To appear in Proceedings of WOCFAI F. M. Donini, D. Nardi and R. Rosati. Non-rst-order features in concept languages. To appear in Proceedings of AI*IA-95 - Fourth congress of the Italian Association for Articial Intelligence. 3. J. Halpern and Y. Moses. Towards a theory of knowledge and ignorance: preliminary report. In K. Apt editor, Logics and models of concurrent systems, pages 459{476, Springer-Verlag, M. Kaminski. Embedding a default system into nonmonotonic logic. Fundamenta Informaticae, 14:345{354, K. Konolige. On the relationship between default and autoepistemic logic. Articial Intelligence Journal, 35:343{382, H. J. Levesque. All I know: a study in autoepistemic logic. Articial Intelligence, 42:263{310, V. Lifschitz. Nonmonotonic databases and epistemic queries. In Proc. of the 12th Int. Joint Conf. on Articial Intelligence IJCAI-91, Sydney, V. Lifschitz. Minimal belief and negation as failure. Articial Intelligence Journal, 70:53{72, F. Lin and Y. Shoham. Epistemic semantics for xed-point non-monotonic logics. Articial Intelligence Journal, 57:271{289, W. Marek, G.F. Shvarts and M. Truszczynski. Modal nonmonotonic logics: ranges, characterization, computation. In Proceedings of the 2nd international conference on principles of knowledge representation and reasoning (KR-91), pages 395{404, Morgan Kaufmann, 1992.

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