The Relationship between KLM and MAK Models for Nonmonotonic Inference Operations J U R G E N DIX and DAVID M A K I N S O N Institute for Logic, University of Karlsruhe, P.O. Box 6980, 7500 Karlsruhe, Germany and Les Etangs B2, La Ronce, 92410 Ville d'Avray, France (Received 10 April, 1991; accepted 18 December, 1991)

Abstract. The purpose of this note is to make quite clear the relationship between two variants of the general notion of a preferential model for nonmonotonic inference: the models of Kraus, Lehmann and Magidor (KLM models) and those of Makinson (MAK models). On the one hand, we introduce the notion of the core of a KLM model, which suffices to fully determine the associated nonmonotonic inference relation. On the other hand, we slightly amplify MAK models with a monotonic consequence operation as additional ingredient. We give two equivalent characterizations of the cores of KLM models: they are precisely the amplified MAK models whose satisfaction relation: - may be expressed as the intersection of some non-empty family of satisfaction relations each of which is classically well-behaved; or - satisfies certain syntactic conditions. This gives corollary characterizations of certain particular classes of KLM models, notably those that are (in their terminology) cumulative and more specifically those they call preferential. Key words: nonmonotonic logic, knowledge representation 1. I N T R O D U C T I O N Preferential m o d e l s w e r e i n t r o d u c e d b y S h o h a m (1986, 1988) as a g e n e r a l i z a tion o f the c o n s t r u c t i o n s u s e d in circumscription. G i v e n a first-order l a n g u a g e and a partial o r d e r i n g ~ o f all the classical m o d e l s o f that l a n g u a g e , o n e m a y define a n i n f e r e n c e relation ~ b e t w e e n sets q~ o f f o r m u l a e and individual f o r m u l a e ff b y the r u l e : ~q5 iff ~ is satisfied in e v e r y classical m o d e l o f 9 that is m i n i m a l ( u n d e r -<) a m o n g the m o d e l s that satisfy q~.

Journal of Logic, Language, and Information 1: 131-140, 1992. (~) 1992 Kluwer Academic Publishers. Printed in the Netherlands.

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This notion was presented in rather more general fashion by Makinson (1989) and Kraus et al. (1990). Instead of restricting attention to classical models, they consider arbitrary objects (which, to avoid confusion, are usually called states). The ordering relation -< holds between such states and is taken as arbitrary, eventually satisfying certain conditions weaker than partial order. Whilst the formulations of Makinson (1989) and of Kraus et al. (1990) are evidently very close to each other, they are not quite the same. The most conspicuous difference is that the latter makes use of worlds as well as states, and a labelling function that associates a non-empty set of worlds with each state. Our main idea is to show that one can swap KLM's family of labelled sets of worlds for a background consequence operation and a single satisfaction relation on states, both satisfying appropriate constraints. We begin in Section 2 by recalling the definition of a Makinson model (MAK model), and adding a suitably constrained background consequence operation into its structure to form an amplified MAK model. We then recall in Section 3 the definition of a Kraus, Lehmann and Magidor model (KLM model), and the special subclasses of cumulative and preferential KLM models. We also define the core of any such model, which suffices to determine the induced nonmonotonic operation. With this background, we are then able to prove in Section 4 our principal results, which may be stated as follows: THEOREM Consider any language s closed under classical propositional connectives, and any structure S = (S, ~ , -<, Cn). Then the following three conditions are equivalent: (1) S is the core of a KLM model for ~; (2) S is an amplified M A K model whose satisfaction relation ~ may be expressed as the intersection of a non-empty family of satisfaction relations ~ i , each of which is classically well-behaved; (3) S is an amplified M A K model whose satisfaction relation ~ satisfies the following conditions for all s E S: (a) s ~ o~ and s ~ /3 implies s ~ ~ /~ /3, (b) There are formulae o~ and/3 with s ~ o~and not s ~ /3. COROLLARY Consider any language ~, closed under classical propositional connectives, and any structure S = ( S, ~ , -<, C n ). Then the following two conditions are equivalent: 1. S is the core of a preferential KLM model for ~;

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2. S is an amplified MAK model whose satisfaction relation ~ is classically well-behaved and whose preference relation is transitive and irreflexive. Our presentation is self-contained, but some familiarity with Makinson (1989) or Kraus et al. (1990) will help to motivate the reader. 2. MAK AND AMPLIFIED MAK MODELS MAK models are defined for any language 12regardless of whether it is closed under classical propositional connectives or not. We write F m l for the set of all formulae of the language 12. DEFINITION 2.1 (MAK MODEL). A MAK model S for the language 12 is a triple S = (S, ~ , -<) where: 1. S is an arbitrary set, called the set of states; 2. ~ is any subset of S x F m l , called the satisfaction relation; 3. -.< is any binary relation on $, called the preference relation. For any state s and any formula r E F m l , we say that s preferentially satisfies r and we write: s~_
all s in S with s~.
When there is no danger of confusion, we shall sometimes drop the subscript from Cs, writing simply ~ E C(ff). It is often convenient to express C as a relation ~ , writing itf

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We recall that a consequence operation (in the sense of Tarski) is an operation

Cn" 2 F~l

, 2 Fret with r C Cn(~) = Cn(Cn(~)) C Cn(cP')

for all ~ _C ~1 _C Fml. Each MAK model S also generates such a consequence operation Cns in a natural way, by putting

r E Cns(~)

iff s ~ r for every s E S with s~(~.

Noting that ~.< is a subrelation of ~ , we clearly have Cns(~) C C8(ff9) for all (I) C_ Fml, i.e. writing it more briefly, C n s < Cs. When the language s contains, and is closed under classical propositional connectives, so that the operation of classical consequence, denoted by Cno, may be applied to it, it is useful to add to the structure of a MAK model an additional ingredient: DEFINITION 2.3 (AMPLIFIED MAK MODEL)

An amplified MAK model for a language ~ closed under classical propositional connectives is a quadruple S = (S, ~ , -<, Cn ) where: 1. (S, ~ , -<) is a MAKmodel; 2. Cn is a consequence operation such that (c~) Cn is compact, i.e. whenever r E Cn(~b) then r E Cn(~o) for some finite (Po C_ ~, (/3) Cno < Cn < Cns, (~/) Cn satisfies "disjunction in the antecedent", i.e.for all ~U { o~,/3, "y} C_ Fml:

E Cn(

U { 4 ) n Cn(

u {/3}) then

E Cn(

U

V/3}).

Of course, in the particular case that Cn is chosen to be Cno, conditions (~) and (~,) are automatically satisfied. Clearly, a MAK model S = (S, ~ , -<) has an amplification (S, ~ , -<) iff Cno <_ Cns: amplifications of S need not in general be unique. The following well-known lemma will be useful for the proof of our theorem: LEMMA 2.4

Let Cn be any compact consequence operation with Cno <_ Cn. Then: (a) Every Cn-consistent set can be extended to one which is maximally Cnconsistent; (b) The following two conditions are equivalent: (i) Cn satisfies "disjunction in the antecedent",

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(ii) l f o~ f[ C n ( 4 ) then there is a Cn-consistent extension of 4 containing -~ol. We recall the easy proof. For (a) the compactness assumption on C n implies, by Zorn's Lemma, that any Cn-consistent set can be extended to a maximally Cn-consistent one. To prove (b), let C n satisfy "disjunction in the antecedent". If there is no Cn-consistent extension of 4 containing -~c~, then C n ( 4 U {~c~}) is inconsistent, thus a E C n ( 4 U { - ~ } ) (because Cno <_ Cn). Since obviously c~ E C n ( 4 U {c~}) (again because Cno <_ Cn) we get by disjunction in the antecedent

and w e are done.

For the converse, let 3, E C n ( 4 U {a}) rq C n ( 4 U {/3}). If-), ~ C n ( 4 U {c~ v/3}) then there is a Cn-consistent extension of 4 U {a V/3} containing -~9'. This extension can be extended (using (a) to a maximal Cn-consistent set M~ax. Since M~ax is maximally Cn-consistent with Cno <_ Cn, and a V/3 E M ~ , we have using (ii) that a E M r ~ or/3 E M m ~ . In both cases we have -y E Mmax which is a contradiction. 3. KLM MODELS AND THEIR CORES KLM models, due to Kraus et al. (1990), are defined only for languages 12 that are closed under classical propositional connectives: DEFINITION 3.1 (KLM MODEL) A KLM model is a quadruple S = (S, W, l, -4) where." 1. S is an arbitrary set, called the set of states. 2. W is a non-empty set of classical propositional models for 12, called the set of worlds. These worlds can equivalently be viewed as subsets of F m l , maximally consistent under Cno. It is not assumed that all the classical models for 12 are in W . However it is assumed that W is compact in the sense that there is a w E W with 4 c_ w whenever for every finite 4o C_ 4 there is a zoo E W with 4o C_ wo. 3. I 9 S , 2 w is a function associating a non-empty set of worlds with each state. It is called the labelling function. 4. -4 is any binary relation over S, called the preference relation. Note, that neither the set W of worlds nor the labelling function l occur in MAK models. We write ~ for the satisfaction relation determined by W , i.e. w ~ qJfor w(q5) = 1 or q5 E w according to the representation of a world that is used.

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From the satisfaction relation ~ on worlds, KLM define a derived satisfaction ~ on states by the rule s~

iff w ~ ohfor all w in l(s).

Given this ~, they define s~_
and there is no s' E S with s I -< s and s ' ~ .

The induced nonmonotonic inference operation is also defined as in the context of MAK models (definition 2.2), eg in the ~ - formulation: DEFINITION 3.2 (INDUCED NONMONOTONICINFERENCEOPERATION ~-~) For any KLM model S we define ~S by the rule:

~

iff s ~

for all s in S with s~.
Clearly then, once the derived satisfaction relation ~ is given, along with the set S of states and the preference relation -<, neither W nor l plays any further role in determining the induced nonmonotonic inference operation. W does however play a part in determining a natural consequence operation C n w , which is used by Kraus, Lehmann and Magidor to formulate certain properties of interaction with C. We therefore define: DEFINITION 3.3 (KLM CORE) Any KLM model S = (S, W, l, -<) induces a structure (S I, ~ , _<1 C n w ) , called its core, by putting 1. S I = S , 2. ~ is the derived satisfaction relation from S, 3..
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C n w ( ~ ) = Fml, may hold for a given K L M model but fail when W is enlarged. We are now in a position to prove our theorem identifying the cores of K L M models with certain amplified M A K models. But before we do so, some remarks on the literature may help avoid confusion. The most general models actually considered by Kraus, Lehrnann and Magidor are those K L M models, as defined above, in which the preference relation -< satisfies a certain condition of smoothness - also known as stoppering in the infinitary version of Makinson (1989). This is the condition that for every formula c~ and every s E S, if s~c~ then either s~.
(1) implies (2) Let (S, ~ , -<, C n w ) be the core of a K L M model S = (S, W, l, -<). First we need to show that it is an amplified MAK model, for which it suffices to check that C n w is a compact consequence operation with Cno <_ C n w <_ Cns, satisfying in addition "disjunction in the antecedent". That C n w is a consequence operation is clear from its definition. That it is compact follows immediately from the hypothesized compactness of W. That C n w satisfies disjunction in the antecedent is immediate from the assumption that the elements of W are classical models. That Cno <_ C n w is also immediate from the same assumption (the converse inclusion may fail when some classical models are missing from W). Finally, that C n w <_ C n s may easily be verified as follows. Suppose o~ E Cnw(~b) and suppose s~ff); we need to show that s~o~. Since s~ff),

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we have w ~ (I) for all w 9 l(s), so since a 9 Cnw('I') we have w ~ a for all w 9 l(s), so that s p a as desired. It remains to show that p may be expressed as the intersection of a non-empty set of satisfaction relations p i , each of which respects classical connectives. Let 27 be the set of all choice functions i : S ) W with i(s) 9 l(s). Recalling that l(s) is always non-empty, the axiom of choice implies that 7: is non-empty. For each i 9 2: define P i by putting s~ia

iff

i(s)~a.

Since the satisfaction relation ~ is classically well-behaved, so is each p i . Also we have ~ _ p i for each i 9 Z. For if s p a and w = i(s) 9 l(s) we have by the definition of the derived satisfaction relation p that w ~ a so

8PiC~. It remains to show that

N{P

i 9 z}

p

Suppose not spa. Then by the definition of p there is a w E l(s) with not w ~ a. By the axiom of choice, there is a choice function i : S ~ W with i(s) = w. Since not i(s) ~ a this gives us not s pia and we are done.

(2) implies (3) Let (S, p , -<, Cn) be an amplified M A K model such that p may be expressed as the intersection of a non- empty set of satisfaction relations each of which respects classical connectives. We need to check that p satisfies the two listed syntactic conditions. The verifications are straightforward, as follows: (a) Suppose s p a and sp/3. Then for all i E [: s p i a and spi/3, so since each P i is classically well-behaved, we have that for all i E I, s pia A fl so that s p a A/3 as desired. (b) Let s E S. Since each P i is classically well-behaved, we have s pi(,b v - - r so s P C v-1r Moreover, we know that I is non-empty; choose some i E I. Then p i is classically well-behaved, so not s P i e A -1r so not s p C A -1r

(3) implies (1) Let (S, p , -<, Cn) be an amplified M A K model satisfying the conditions (a) and (b). We want to show that it is the core of some K L M model (S, W, l, -<). We need to define W, l, and verify the required properties.

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First note that every amplified MAK model satisfies the following condition: (*) whenever s~(x and/3 E Cn({tx}) then s~fl. For suppose s ~ and fl E Cn({c~}). Since Cn < Cns, we have fl E Cns ({ix}), so, by the defmition of Cns, since s ~(~, we have s ~ f l as desired. Let W be the set of all maximal Cn-consistent sets of formulae in Fml. Using part (b) of our lemma 2.4, it is immediate that W consists of classical models. For each s E S put l(s) to be the set of all maximal Cn-consistent sets w of formulae suchthat {5" s~5} C w. Note that for all s E S

Cn({5" s~6}) ---- {(~" s~} # Fml. The equality follows easily from the compactness of Cn, Cno < Cn, the hypothesized property (a) and fact (*). The inequality is given directly by hypothesis (b). Using part (a) of lemma 2.4, this proves l(s), and thus also W, to be non- empty. The compactness of W is immediate from that of Cn. It remains to show that (S, ~, -% Cn) is indeed the core of (S, W, l, -.<). For Cn we have to show o~ E Cn(@) iff w ~ a for every maximal Cn-consistent w that includes q,. From left to right is immediate (Cn is a monotone consequence operation). The converse is exactly (the contraposition of) condition (b) (ii) of our lemma 2.4. For ~ , we need to show that

spa

iff w ~ a f o r a l l w E l ( s ) .

But since s~

i f f ~ E {6" s~5} = Cn({5" s ~ } )

then, applying the definition of l(s), this is merely a particular case of what we have just shown about Cn.

5. APPLICATIONS TO PARTICULAR CASES Note that the definition of smoothness (at the end of Section 3) concems only the ingredients S, -<, ~ (and not W, l). Hence the theorem continues to hold

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when KLM model is restricted to cumulative KLM model and MAK model to smooth MAK model. The definition of preferential KLM models (also at the end of Section 3) involves S, -4 and the labelling function l. To verify the corollary of our theorem (stated in Section 1) we need only run through the proof again, making two small additions. In the verification "(1) implies (2)" we note that if each l(s) is a singleton, then Z contains exactly one choice function, so that ~ is classically well-behaved. In the verification of "(3) implies (1)" note that since ~ is classically well-behaved, the set {6 : s ~ } = Cn({~ : s ~ } ) r Fml is maximally Cn0-consistent. Since Cno < Cn, this set is also maximally Cn-consistent, so that l(s) is a singleton. ACKNOWLEDGEMENT The authors would like to thank Daniel Lehmann for his helpful comments on a draft. REFERENCES Kraus, S., Lehmann, D., and Magidor, M., 1990, "Nonmonotonic reasoning, preferential models and cumulative logics," Artificial Intelligence 44(1), 167-207. Makinson, D., 1989, "General theory of cumulative inference," Non-Monotonic Reasoning. In Reinfrank, de Kleer, Ginsberg, and S andewall, eds., Lecture Notes in Artificial Intelligence 346, Springer-Verlag. Shoham, Y., 1986, Reasoning about Change: Time and Causation from the Standpoint of Artificial Intelligence. PhD thesis, Yale University. Shoham, Y., 1988, Reasoning about Change. Cambridge: MIT Press, USA.