330 SU Kaile Vol.16 Truszczyński [12] constructed an example of denumerable family of pairwise inconsistent theories which cannot be represented by no

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1 Vol.16 No.4 J. Comput. Sci. & Technol. July 2001 Constraints on Extensions of a Default Theory SU Kaile (±ΞΠ) Department of Computer Science, Zhongshan University, Guangzhou , P.R. China State Key Laboratory for Novel Software Technology, Nanjing University Nanjing , P.R. China isskls@zsu.edu.cn Received February 17, 2000; revised December 13, Abstract In this paper, the representability of a family of theories as the set of extensions of a default theory is studied. First, a new necessary condition is given for the representability by means of general default theories; then a suicient one is presented. The families of theories represented by default theories are also fully characterized. Finally, the paper gives an example of denumerable families of mutually inconsistent theories that are represented by a default theory but not by normal ones. Keywords default logic, extension, representability 1 Introduction Reiter's default logic [1] is one of the most prominent formalizations of nonmonotonic reasoning, and is extensively investigated by the community of logical foundations of Artificial Intelligence (see [2 10] for example). A default theory describes a collection of subsets of the set of all formulas, namely the family of all extensions of, which represents a family of belief sets of a reasoning agent. An important issue is, then, to characterize those families of sets that can be represented as the set of extensions of a certain default theory, and to identify those collections of sets that can be represented as the set of extensions of a default theory in a specific class. This issue is called the theory of representability for default logic, and was first studied in a deep-going way in [11, 12]. By ext( ) we denote the family of all extensions of a default theory. It is well-known that the family ext( ) must be non-including [1]. In [11], Marek and Truszczyński show that this constraint is the only one on finite families that can be represented by default theories. Unfortunately, this is not true for infinite families as Marek, Treur and Truszczyński actually constructed a denumerable family of non-including theories that is not representable [12]. In this paper, we present a stronger constraint on families that can be represented by default theories. In addition, we present suicient conditions for representability for infinite families of theories, which are much weaker than those in [12]. We further obtain a necessary and suicient condition for representability for infinite families of theories. Not only is the family of extensions of a normal default theory non-including, but all its members are pairwise inconsistent [1]. Again, this stronger constraint could not fully characterize those families of theories represented by normal default theories. Marek and Truszczyński [11] raised an interesting question: Whether the family ext( ) of all extensions of a default theory is represented by a normal default theory under the conditions that ext( ) is not empty and all its members are pairwise inconsistent? Marek, Treur and This research was supported in part by the National Natural Science Foundation of China under grant Nos and , in part by the Guangdong Provincial Natural Science Foundation under grant No , in part by the Foundation for University Key Teachers by The Ministry of Education, China and in part by the National `863' High-Tech Programme of China (No ZT ). Part of Section 5 in this paper has appeared in Proceedings of the Sixth Asian Logic Conference.

2 330 SU Kaile Vol.16 Truszczyński [12] constructed an example of denumerable family of pairwise inconsistent theories which cannot be represented by normal default theories. However, this example is not represented by any default theory and thus does not answer the question above. In this paper, we negatively answer this question by constructing another denumerable family of pairwise inconsistent theories, which is represented by a default theory but not by any normal default theory. 2 Preliminaries In this paper, by L we denote a language of propositional logic with a denumerable set of atoms At. We denote propositional provability by ` and the corresponding consequence operator by Cn [11]. By a theory we always mean a subset of L closed under propositional provability. Let B be a set of standard monotonic inference rules, and we obtain a formal proof system, denoted by PC+B, in which derivations are built by means of propositional provability and rules in B. The corresponding provability operator is analogously denoted by `B and the consequence operator by Cn B ( ). A default is an expression d of the form : fi, where and fi are formulas from L and is a finite list of formulas from L. The formula is called the prerequisite of d (p(d), in symbols) and fi is called the consequent of d (c(d), in symbols). The set of formulas is called the justification set of d and is denoted by j(d). This terminology is naturally extended to a set of defaults D. Namely, the prerequisite, consequent and justification sets of D, in symbols p(d), c(d) [ and j(d), are defined [ by: [ p(d) = fp(d)g; c(d) = fc(d)g; j(d) = j(d): d2d d2d If p(d) is a tautology, d is called prerequisite-free. In such case, p(d) is usually omitted from the notation of d. If j(d) = fc(d)g, d is called normal. If j(d) = fc(d) ^ flg for some formula fl, d is called seminormal. By a default theory we mean a pair = (D; W), where D is a set of defaults and W a set of formulas. The set W is called the objective part of (D; W). We say that (D; W) is prerequisite-free, if all defaults in D are prerequisite-free; normal, if they are normal; and seminormal, if they are seminormal. Following [11], we give an alternative definition of extension, which is proved equivalent to the original one by Reiter. First, we need the following concepts. For a default d and a set of defaults D define: Mon(d) = p(d) c(d) d2d and Mon(D) = ρ p(d0 ) c(d 0 ) fid0 2 D Given a set of formulas S, those defaults d such that S 6` :fl for every fl 2 j(d) are called S-applicable. A set D of defaults is S-applicable if all its members are S-applicable. We define: D S = Mon(fd 2 D : S 6` :fl for every fl 2 j(d)g): A theory S is an extension of a default theory (D; W), if and only if S = Cn DS (W ): The family of all extensions of (D; W) is denoted by ext(d; W). We now define some key concepts concerning the representability issue for default logic. Definition 1. Two default theories and 0 are said to be equivalent to each other (denoted by ß 0) if they have the same extensions, i.e., ext( ) = ext( 0). A default :

3 No.4 Constraints on Extensions of a Default Theory 331 theory is said to be representable in a class of default theories if the default theory is equivalent to some default theory in the class. Definition 2. Let T be a family of theories contained in L. The family is represented by a default theory if ext( ) = T. A family (of theories) is called representable if it can be represented by a default theory. 3 Representability of General Default Theories We get a constraint on extensions of a default theory, which is stronger than the wellknown constraint obtained first by Reiter [1] that extensions should be pairwise non-including. We then present some suicient conditions. To prove these results, we need a lemma that allows us to replace any default theory with an equivalent default theory in which all defaults are prerequisite-free. This result was obtained independently by Schaub [13], Bonatti and Eiter [14], and Marek et al. [12]. However, our argument shows that the equivalent theory can be simpler, in the inference-free form defined as follows. Definition 3. A default theory (D; W) is called inference-free if (1) the default theory is prerequisite-free, (2) the objective part W is empty, (3) for prerequisite-free defaults d; d 0, if d 2 D and d 0 has the same justification set as d and c(d 0 ) is logically inferred from c(d), then d 0 2 D also, (4) for defaults d; d 0 2 D, the default :j(d)[j(d0 ) c(d)^c(d is also in D, 0 ) Lemma 4. For every default theory there is an inference-free default theory 0 equivalent to. Proof. Given a default theory = (D; W), we define another default theory 0 = (D 0 ; ;) as follows: ρ : j(d D 0 00 ) = jd 00 is a finite subset of D and W 00 `Mon(D ) ' ' It is trivial to check that the default theory 0 is inference-free. We now prove that 0 is equivalent to, i.e., they have exactly the same extensions. Recalling the definition of an extension, suice it to show that, for all sets of formulas S, Cn DS (W ) = Cn D0 S(;) or that for every formula ', W `DS ' i `D0 S ': Assume first that W `DS '. Then, there is a finite subset B of D S such that W `B '. There must be, by the definition of D S, a set D 00 of defaults such that Mon(D 00 ) = B and D 00 is S-applicable. Hence, the default :j(d00 ) ' is in D 0 and is S-applicable. Since Mon( :j(d00 ) ' ) = > ', where > is a tautology, we have that the rule > ' is in D00 S, which leads to `D0 S ': To prove the converse implication, assume that `D0 S '. Then, there must be a finite subset D Λ of D 0 such that all defaults in D Λ are S-applicable and `Mon(DΛ ) '. By the definition of D 0, we can assume: ρ : j(d D Λ 00 i = ) j i < n ; ' i By ; we denote the empty set. :

4 332 SU Kaile Vol.16 where n is a natural number. Observe that, for each i, W `Mon(D 00 i ) ' i : Moreover, since :j(d00 i ) ' i is S-applicable, those defaults in Di 00 that `Mon(DΛ ) ' is equivalent to f' i j i < ng ` '. Hence, are also S-applicable. Observe W `Mon( [ Di 00) ': i<n It follows that W `DS ': 2 We now introduce three notions of a non-including family of theories. Definition 5. Let F be a family of theories. Then we say that F is (1) non-including if F is an antichain (that is, for T;T 0 2 F, if T T 0 then T = T 0 ); (2) compactly non-including if for all theories E 2 F and formulas 2 E, there is a finite list of families F 0 ;:::;F k such that (a) [ i»k F i = ft 2 F : =2 T g, (b) F i E is a non-empty set, for each F i ; (3) supercompactly non-including if for all theories E 2 F, there is a finite list of families F 0 ;:::;F k such that (a) [ i»k F i = F feg, (b) F i E is a non-empty set, for each F i. Note that, in the last two items, the list ff 0 ;:::;F k g can be empty. Clearly, if a family of theories F is supercompactly non-including, then it is compactly non-including, but not vice versa (see the example in the proof of Proposition [16]). In addition, we have that the compact non-including of a family implies the non-including of the family also. Proposition 6. If F is compactly non-including, then it is non-including. Proof. Suppose that a family F is compactly non-including. We must show it is nonincluding also. Assume conversely that F 1 F 2 held for some distinct theories F 1 ;F 2 2 F, and hence there was a formula 2 F 2 F 1. By the compact non-including of F, there is a finite list of families F 0 ;:::;F k such that (a) [ i»k F i = ft 2 F : =2 T g, (b) F i F 2 is a non-empty set, for each F i. As =2 F 1, there was a j» k such that F 1 2 F j and hence F j F 1. So, F j F 2 was an empty set, contradicting the latter assertion about F i s. 2 It is easy to present an example of families that are non-including but not compactly nonincluding. Let fp 0 ;p 1 ;:::g be a set of propositional atoms. Define T i = fp i g, i = 0; 1;:::, and T = ft i : i = 0; 1;:::g. This family T is desired. To gain more intuitive impression on these three notions, we introduce the concept of an hitting set, which was used by Kleer [15] and Reiter [16]. We say that a set H is a hitting set for a family F of theories if H F 6= ; holds for every F 2 F. Proposition 7. Let F be a family of theories. (1) F is non-including i, for each F 2 F, there is a hitting set for F ff g which is not a hitting set for F. (2) F is compactly non-including i, for each F 2 F and 2 F, there is a finite hitting set for ft 2 F : =2 T g which is not a hitting set for ft 2 F : =2 T g[ffg. (3) F is supercompactly non-including i, for each F 2 F, there is a finite hitting set for F ffg which is not a hitting set for F. Proof. The result of (1) follows immediately by the definition. It is easy to prove (2). To prove the `if' part, we can associate each formula ' j in the hitting set being considered with the family of those theories that contain the formula. For

5 No.4 Constraints on Extensions of a Default Theory 333 the `only if' part, given F 2 F and 2 F, we then get the families F j as in the definition of compact non-including. The result follows if we select a formula j in F j F for each j and consider the set of those j s. The proof of (3) is similar to that of (2). 2 The next proposition says that the three notions above are equivalent to each other for finite families. Proposition 8. Let F be a finite family of theories. Then the following three propositions are equivalent to each other. (1) F is non-including. (2) F is compactly non-including. (3) F is supercompactly non-including. Proof. As we have demonstrated already that (1) implies (2) and (2) implies (3), suice it to prove that (3) implies (1). But this is trivial by Proposition 7 and the finiteness of F. 2 What follows is a new constraint on those families represented by default theories. Theorem 9. If F is represented by a default theory, then it is compactly non-including. Proof. Suppose that F is represented by a default theory, E a theory in F and a formula in E. By Lemma 4, we can assume that is inference-free. Let = (D; ;). As the extensions of the default theory (D; ;) are exactly those theories in F, E is an extension of (D; ;). By the definition of extension, we then have that E = Cn DE (;), and hence 2 Cn DE (;). It follows that there is a finite subset D 0 of D such that each default in D 0 is E-applicable and `Mon(D0 ) holds, i.e., c(d 0 ) ` since the default theory (D; ;) is inference-free. We now have that the default d = : j(d0 ) 2 D: Suppose that j(d 0 ) consists of 0 ;:::; k. For each j, let F j = ft 2 F j : j 2 T and 62 T g: Then F j ft 2 F j =2 T g. On the other hand, for T 2 F such that =2 T, we have that =2 Cn DT (;), which yields that d is not T -applicable. Thus there is an j such that : j 2 T, and we have that T 2 F j. So we have proved that [ i»k F i = ft 2 F j =2 T g: Now we need only to show that, for each j» k, F j E is non-empty. But it is very easy since : j 2 F j and : j =2 E. 2 Since a family being compactly non-including implies the family being non-including, we get the following well-known result [1;11] as a corollary. Corollary 10. If F is represented by a default theory, then it is non-including. The next result completely characterizes subfamilies of representable families. Theorem 11. A family F is a subset of a representable family if and only if it is compactly non-including. Proof. The `only if' part follows immediately from Theorem 9 and the fact that if a family is compactly non-including, so are its subfamilies. As for the `if' part, assume that a family of theories F is compactly non-including. Given E 2 F and 2 E, we obtain, by the assumption above, a finite list of families F 0 ;:::;F k such that [ j»k F j = ft 2 F : =2 T g

6 334 SU Kaile Vol.16 and for each j» k, there is an j such that j 2 ( F j E): By d E;, we denote the default :f:0;:::;: kg. Now consider the default theory (D; ;), where D = fd T;' j T 2 F and ' 2 T g: We want to prove that T ext(d; ;), i.e., for each T 2 F, T = Cn DT (;): On one hand, for each ' 2 T, d T;' is T -applicable, and ' 2 Cn DT (;). On the other hand, for all T -applicable defaults d T ;0, we must prove that T. By the construction of d T ;0, 0 Swe can assume that j(d T ;0) = 0 f0 0 ;:::;0 k g, and there are families F 0 0 ;:::;F k 0 such that j»k F j 0 = fs 2 F : 0 =2 Sg and, for all js, :j 0 2 F j 0. Suppose that 0 2 T did not hold. Then we would have that T 2 Fj 0 for some j. Thus, :0 j 2 T, contradicting the assumption that the default d T 0 ;0 is T -applicable. 2 We now propose a suicient condition for the representability of a family. Theorem 12. If a family F is supercompactly non-including, then it is represented by a default theory. Proof. Assume that a family of theories F is supercompactly non-including. Since the empty family is representable, we assume further that F is a non-empty family. Given an arbitrary E 2 F, we obtain, by the assumption above, a finite list of families F 0 ;:::;F k such that [ F j = F feg j»k and for each j» k, there is an j such that j 2 ( F j E): For each E 2 F, we fix one such finite set of : j s, denoted by J E. As in the proof of Theorem 11, by d E;, we denote the default :JE. Now consider the default theory (D; ;), where D = fd T;' j T 2 F and ' 2 T g: We shall prove that F is represented by the default theory (D; ;). First, we show that F ext(d; ;). Given T 2 F, we must prove T = Cn DT (;). On one hand, for each ' 2 T, d T;' is T -applicable, and hence ' 2 Cn DT (;). On the other hand, for all T - applicable defaults d T 0 ;0, we prove 0 2 T as follows. By the construction of d T 0 ;0, we can S assume that j(d T ;0) = 0 f:0 0 ;:::;:0 k g, and there are families F 0 0 ;:::;F k 0 such that j»k F j 0 = F ft 0 g and, for all js, j 0 2 F j 0. Suppose that 0 2 T did not hold. Then we would have that T 6= T 0 for 0 2 T 0, and hence T 2 Fj 0 for some j. Thus, j 0 2 T, contradicting the assumption that the default d T 0 is T -applicable. ;0 Now, we show that F ext(d; ;). Given an E 2 ext(d; ;), we must prove E 2 F. Without loss of generality, we assume that E 6= Cn(;); otherwise, all extensions of (D; ;) are Cn(;), and the result follows since F ext(d; ;) and F 6= ;. As E = Cn DE (;) 6= Cn(;), there is an E-applicable default d T in D, where T F. Hence, for all ' 0 2 T 0, d T ;0 0 ;'0 is also E-applicable. This leads to ' 0 2 Cn DE = E. Thus, T 0 E. As a result, T 0 = E since we have proved that T 0 is an extension of (D; ;). This completes the proof. 2 We say that a family F of theories has a strong system of distinct representatives (SSDR, for short) if for every T 2 F there is a formula ' T 2 T which does not belong to any other theory in F (see [12]).

7 No.4 Constraints on Extensions of a Default Theory 335 Proposition 13. If a family F of theories is compactly non-including and has an SSDR, then it is supercompactly non-including. Proof. Suppose that F is a compactly non-including family that has an SSDR. By the definition, for each E and 2 E, there is a finite list of families F 0 ;:::;F k such that and for each j» k, [ j»k F j = ft 2 F : =2 T g ( F j E) 6= ;: Consider the case where is the formula ' T 2 T, which does not belong to any other theory in F. It follows immediately that F is supercompactly non-including, since we have that ft 2 F : ' T =2 T g = F feg: 2 As a corollary of Theorem 12 and the proposition above, we obtain another suicient condition for the representability. Corollary 14. If a family F is represented by a default theory, then every family G F that has an SSDR is represented by a default theory. Corollary 14 improves a closely related result in [12]. Corollary 15 (Theorem 4.2 in [12]). If a family F is represented by a default theory and has an SSDR, then every family G F is represented by a default theory. Proof. The result follows immediately from the fact that if a family F has an SSDR, so does every subfamily of F. 2 The result of Theorem 4.2 in [12] essentially says that a family with an SSDR is representable under the condition that it is a subfamily of a representable family with an SSDR, while Corollary 14 weakens the condition by that it is a subfamily of a representable family. 4 Complete Characterization We start by demonstrating that the compact non-including of a family is not a suicient condition for the representability of this family, and the supercompact non-including is not a necessary one. We then attempt to address the open problem of finding a complete characterization of families of theories that are represented by general default theories [12]. However, we do not think the presented result is satisfying enough for us to construct a denumerable and compactly non-including family which is not representable. Proposition 16. (1) There is a family which is compactly non-including but not representable. (2) There is a denumerable family which is representable but not supercompactly non-including. Proof. The result of (1) follows easily from a cardinality argument. There are continuummany default theories in the given (denumerable) language, while there are more than continuum-many families that are compactly non-including since there is a default theory with continuum-many extensions and hence all subsets of ext( ) are compactly nonincluding. As for (2), let us construct an example of denumerable and representable families that are not supercompactly non-including. Let P = fp i : i 2!g and Q = fq i : i 2!g be two disjoint sets of propositional atoms. Define: S j = fp i : i < jg[fq i : i jg for all j 2! By!, we denote the set of all natural numbers.

8 336 SU Kaile Vol.16 and F = fcn(s j ) : j 2!g[fCn(P )g: We prove that F is not supercompactly non-including as follows. Assume H is a finite hitting set for fcn(s j ) : j 2!g, we will show that it is a hitting set for F also. By the finiteness of H, there is a formula ' such that ' 2 Cn(S j ) for infinitely many S j s. Let k be a natural number such that those atoms q i, where i k, do not occur in '. Let k 0 k be such that ' 2 Cn(S k 0), i.e., S k 0 ` '. Replacing q i s, i k 0, in S k 0 ` ' by a tautological formula >, we obtain that fp i : i < k 0 g ` ' (see the following remark for the justification of this technique). Hence ' 2 Cn(P ) and H is a hitting set for F. To complete the proof, we must show that F is representable. Denoting the set f:q i : i < jg[f: g by j, we define: D = ρ :qi p i : i 2! [ρ i : i 2! and 2 S i We now consider the default theory (D; ;). It is easy to check that Cn(P ) and Cn(S i )s are extensions of this default theory, that is, F ext(d; ;). On the other hand, we shall prove that F ext(d; ;), whence F is represented by (D; ;). Suppose S 2 ext(d; ;), and we want to show S 2 F. There are two cases: Case 1. If q i =2 S for all is, then all defaults :qi a i in D are S-applicable, hence p i 2 Cn DS (;) = S. Accordingly, Cn(P ) S, which leads to Cn(P ) = S since extensions of a default theory are non-including. Case 2. If q i 2 S for some is, then let n be the least such is. Since :q n 2 i for all i > n, we have that all S-applicable defaults in D fall in the following set: D 0 = ρ :qi p i : i 6= n [ρ i : i» n and 2 S i Thus S = Cn DS (;) Cn(c(D 0 )). Clearly c(d 0 ) 6` p n and hence S 6` p n. Thus the defaults n, where 2 S n, are S-applicable, which leads to that S = Cn DS (;) S n. It follows that S = S n. In both cases, S is in F. So we have proved the result of (2). 2 Remark 17. In the proof above, there is a maneuver of replacing by > (true) and then inferring something. And similar technique also applies to the substitutions of > and? (false) in Lemma 26 and the paragraph after the statement of Fact 2 in Lemma 25. The justification for this technique is as follows: Given an arbitrary set S of propositional formulas, propositional formulas ' and, and primitive formula p, let '( p ) denote the formula obtained from ' when an occurrence of is substituted for each occurrence of p in ', and S( p ) = fi( p ) j i 2 Sg. Then, by the replacement theorem for propositional calculus, j= ' implies j= '( p ). And S j= ' i there are 1;:::; n 2 S such that j= ( 1 ^ :::^ n )! ', which by the same reason implies j= (( 1 ^ :::^ n )! ')( p ), that is, j= ( 1 ( p ) ^ :::^ n( p ))! '( p ), implying S( p ) j= '( p ). Definition 18. Let F be a family of theories. A set of prerequisite-free defaults D is acceptable by F if D satisfies the following two properties: Soundness: Given an arbitrary d 2 D, for E 2 F such that c(d) 62 E, d is not E- applicable. Completeness: And conversely, for each E 0 2 F and each 2 E 0, there is a prerequisitefree default d 2 D such that c(d) = and d is E 0 -applicable. Intuitively, the acceptability of D by F means that given an arbitrary belief set E 2 F, we obtain exactly those formulas in E by applying all E-applicable defaults in D. : :

9 No.4 Constraints on Extensions of a Default Theory 337 Theorem 19. A family of theories F is represented by a default theory if and only if there is a set of prerequisite-free defaults D such that F is the largest family, by which D is acceptable. Proof. Assume F is represented by a default theory. Then, by Lemma 4, F is represented by an inference-free default theories (D; ;). That is, F = ext(d; ;). It follows immediately that D is acceptable by F. To prove that F is the largest family, by which D is acceptable, we must show that F 0 F for those families F 0 such that D is acceptable by F 0. Suice it to show that E = Cn DE (;) for each E 2 F 0. On one hand, for every 2 E there is an E-applicable default d 2 D with c(d) = by the completeness in Definition 18. That is, E Cn DE (;). On the other hand, by the soundness in Definition 18, for every E-applicable default d 2 D, we have that c(d) 2 E. Thus, E Cn DE (;). This completes the `if' part of the theorem. As for the `only if' part, assume that F is the largest family, by which D is acceptable. Then, by the same argument as above, we have that F ext(d; ;). Clearly, the default set D is applicable by ext(d; ;) also. Thus, ext(d; ;) F. We therefore have proved that F is represented by the default theory F. 2 5 Normal Default Theories In this section we present a new essential feature distinguishing normal default theories from unrestricted ones. Using this feature we construct an example of default theory which has exactly infinitely and countably many pairwise inconsistent extensions, but is not representable in the class of normal default theories Λ. This gives a negative answer to the question of Marek and Truszczyński whether the following proposition holds without the assumption that the extensions are finitely generated [11;p:130]. Proposition 20 (Corollary 5.10 in [11]). If a default theory (D; W) has at least one extension and all the extensions for (D; W) are finitely generated and pairwise inconsistent, then the default theory is representable in the class of normal default theories. Moreover, the above example refutes an assertion of Marek and Truszczyński [11;p:129] that if a default theory (D; W) has at least one and at most countable pairwise inconsistent extensions, then the conclusion of the above proposition holds even without the assumption that extensions of (D; W) are finitely generated. What follows is a known result [12], which is trivial for finite default theories since we are essentially encoding the extension's generating defaults into single defaults. Lemma 21 (Theorem 3.1 in [12]). Every normal default theory is representable in the class of prerequisite-free normal default theories. It is well known that there exists an extension and two dierent extensions are inconsistent for every normal default theory (i.e., the existence and orthogonality of extensions, see [1, Theorems 3.1 and 3.3]). Marek and Truszczyński showed that the two properties mentioned above, in several cases, are essentially the only two features distinguishing normal default theories from unrestricted ones. The following theorem, which gives a new essential feature of normal default theories, is closely related to Reiter's corresponding result [1; Theorem 3:3], i.e., the inconsistency of any two extensions of normal default theory, which says that there is a sentence in one extension but the negation of the sentence is in the other. Somewhat stronger than Reiter's theorem, our theorem says that this sentence can be such one as if : were a default of the default theory being considered, in other words, for every extension E Λ, the consistency of E Λ [fg implies 2 E Λ. Λ This example was previously presented in [7].

10 338 SU Kaile Vol.16 Theorem 22. Let (D; W) be a normal default theory, then for all extensions E, E 0 of (D; W) such that E 6= E 0 there is a sentence such that 2 E and E 0 [fg is inconsistent, and for every extension E Λ of (D; W), either 2 E Λ or E Λ [fg is inconsistent. Proof. By Lemma 21 suice it to prove the theorem only in the case that (D; W) is prerequisite-free normal default theory. For extensions E 1 and E 2 there exist D 1 and D 2 D such that E i = Cn(W [ c(d i ) (i = 1; 2) (by [3, p.47]). Suppose that E 1 6= E 2. Then E 1 6 E 2, and there is a default i = :fi fi 2 D 1 such that fi 62 E 2. Hence :fi 2 E 2, otherwise fi 2 Cn DE 2 (W ) = E2. Moreover fi 2 E 1 (by :fi fi 2 D 1), and for every extension E of (D; W), fi 2 E if :fi 62 E, i.e., we have that fi 2 E, or :fi 62 E. 2 According to the presented feature of normal default theories, we now construct an example of default theory, which has exactly countably many pairwise inconsistent extensions, but is not representable in the class of normal default theories. Example 23. The default theory (D 0 ;W 0 ): Let W 0 = fq i! : ji 6= jg D 0 = f : q i p i j i 2!g[f : a ij a ij j i; j 2!g where q i s, s, and a ij s (i» j) are dierent atoms, and a ji = :a ij for i < j. Theorem 24. There is a default theory (D; W) which has exactly countably many extensions and these extensions are pairwise inconsistent, however, it is not representable in the class of normal default theories. The above theorem can be proved directly by the following two lemmas and Theorem 22. Lemma 25. Let E j = Cn(W 0 [f g[fa ij ji 2!g) for all j 2!. Then E j s are exactly all extensions of (D 0 ;W 0 ) and E i [ E j is not consistent if i 6= j. Proof. It is easy to prove that E j s are extensions of (D 0 ;W 0 ) and E i [E j is not consistent if i 6= j. We only show that for an arbitrary extension E of (D 0 ;W 0 ) there is a j such that E = E j. First, we have (by [3, Proposition 6.2.9]) E = Cn(W 0 [ c(gd(e))), where GD(E) is the set of all generating defaults for E, i.e., GD(E) = f : q j jj 2!; : 62 Eg [f : a ij j 2 E;:a ij 62 E; i; j 2!g: Second, it is easy to see that there exists a j such that 2 E, otherwise pj:aij a ij 62 GD(E) for all i; j, and c( :qj ) 62 Cn(W 0 [c(gd(e))), hence :qj 62 GD(E) for all j. Thus GD(E) = fg, we have E = Cn(W 0 ), which contradicts that E is an extension. Finally, suppose 0 2 E, we shall show that E = E j0 by proving the following facts: Fact 1. :qj 62 GD(E) for j 6= j 0. If j 6= j 0 then q j! :0 2 W 0 E. Since Cn(E) = E, we have :q j 2 E. Thus :q j 62 GD(E) is proved. Fact E and hence pj:aij a ij 62 GD(E) for all j 6= j 0. By Fact 1, E = Cn(W 0 [f0 g[f ), where F fa ij ji; j 2!g. If 2 E for some j 6= j 0, then W 0 [f0 g[f `. If we replace by? (false), then a ij W 0 0 [fq i! :? ji 6= jg [f0 g[f `? where W 0 0 W 0 (See the remark after the proof for justifications). Since q i! :? is valid, we have W 0 0 [f0 g[f `?, which contradicts that E is consistent (since W 0 is).

11 No.4 Constraints on Extensions of a Default Theory 339 Fact 3. : q j 0 2 GD(E), 0 : a ij0 2 GD(E) for all i. 0 a ij0 By Fact 1 and Fact 2, E Cn(W 0 [f0 g [ fa ij0 ji 2!g). Since W 0 [f0 g [fa ij0 ji 2!g 6` :q j0 and W 0 [f0 g[fa ij0 ji 2!g 6` :a ij0 for all i, we have :q j0 62 E and a ij0 62 E for all i. Thus :qj 0 2 GD(E) and pj 0 :aij 0 0 a ij 2 GD(E) for all i. 0 From these facts, it is easy to see that GD(E) = f :qj 0 g [ f :pj 0 :aij 0 0 a ij ji 2!g. Hence 0 E = Cn(W 0 [f0 g[fa ij0 ji 2!g) = E j0. 2 Lemma 26. Let E j s be the same as in Lemma 25, fl an arbitrary sentence such that fl 2 E 0 and :fl 2 E 1. Then there is an E i such that fl 62 E i and :fl 62 E i. Proof. Suppose that fl is a sentence such that fl 2 E 0 and :fl 2 E 1. There exists a number k such that fl contains no atoms in f jk» jg [ fa ij ji» j, k» jg. Let j be a number greater than k. We shall prove W 0 [f g[fa ij ji 2!g 6` fl. Otherwise, assume that W 0 [f g[fa ij ji 2!g ` fl, i.e., W 0 [f g[fa ij ji» jg[f:a ji jj < ig ` fl. If we substitute > for a ij (i» j), and by? for a ji (j < i), we would have W 0 [ f g ` fl. If we substitute > for, we would obtain that W 0 0 [fq i! >ji 6= jg [ f>g ` fl; where W 0 0 W 0. Then W 0 ` fl. This contradicts that :fl 2 E 1. Thus we have that fl 62 E j. By the same reason, :fl 62 E j for all j k. 2 6 Conclusion and Open Questions The representability theory is of key importance for default logic. Representability by certain classes of default theories provides insights into the expressive power of the corresponding branches of default logic. This kind of study will be helpful for the users to find simpler representation for their default theories. The usefulness of Reiter's default logic for specifying multiple belief sets of an agent was investigated by Marek, Treur and Truszczyński [12]. They, however, have not found a complete characterization of families of theories that are represented by general default theories. In this paper, besides giving a new necessary condition for the representability by means of general default theories and presenting several suicient ones, we have fully characterized the families of theories represented by general default theories. This paper has interestingly improved some related results of [12] also. For example, while Marek et al. showed that every subfamily of a representable family that has an SSDR is representable, we have obtained that every subfamily of a representable family is representable whenever the subfamily has an SSDR. It was shown in [11] that if a family of belief sets is non-empty and the belief sets in it are pairwise inconsistent, then this family is represented by a normal default theory under the assumption that those belief sets are all finitely generated. However, it is not known whether the assumption is removable under the condition that this family is represented by a default theory. In this paper, we have given a new feature of extensions of normal default theories, by which we construct a default theory that is not representable in the class of normal default theories but has exactly countably many extensions and those extensions are pairwise inconsistent. It gives a negative solution to the problem mentioned above, and refutes the assertion that if a default theory (D; W) has at least one and at most countable pairwise inconsistent extensions, then (D; W) is equivalent to a normal default theory [11;p:129]. However, we have not found some essential and distinguishing features for other important classes of default theories such as that of semi-normal default theories [2], and that of unitary default theories, i.e., those default theories where each default has exactly one justification. The following problems seem more diicult and remain open.

12 340 SU Kaile Vol.16 Problem 1. Is each default theory equivalent to a semi-normal default theory? Problem 2. Is each default theory equivalent to a unitary default theory? Problem 3. Is each unitary default theory equivalent to a semi-normal default theory? We note that if Problem 1 or Problem 2 has an airmative answer, then, from a representation viewpoint, the user is allowed to characterize families of belief sets for his/her agents by using only semi-normal default theories or unitary default theories instead of general ones. References [1] Reiter R. A logic for default reasoning. Artif. Intell., 1980, 12: [2] Etherington D W. Reasoning with Incomplete Information. London: Pitman, [3] Besnard P. An Introduction to Default Logic. Berlin: Springer-Verlag, [4] Brewka G. Nonmonotonic Reasoning: Logical Foundation of Commonsense. Cambridge University Press, Cambridge, UK, [5] Su K, Ding D. Default logic about assertions. Science in China (Series A), 1994, 37(11): [6] Cadoli M. Tractable Reasoning in Artificial Intelligence. Berlin: Springer-Verlag, [7] Su K, Chen H. A solution to a problem of Marek and Truszczyński. In Proceedings of the 6th Asian Logic Conference, Singapore: World Scientific Publishing Co. Ltd., 1998, pp [8] Zhang M. A new research into default logic. Information and Computation, 1996, 129(2): [9] Antoniou G. Nonmonotonic Reasoning. London MA: The MIT Press, [10] Su K, Li W. Computation of extensions of seminormal default theories. Fundamenta Informaticae, 1999, 40(1): [11] Marek W, Truszczyński M. Nonmonotonic Logic: Context-dependent Reasoning. Heiderberg: Spring- Verlag, [12] Marek W, Treur J, Truszczyński M. Representation theory for default logic. Annals of Mathematics and Artificial Intelligence, 1997, 21(2-4): [13] Schaub T. On constrained default theories. In Proceedings of the 11th European Conference onartificial Intelligence, ECAI'92, John Wiley and Sons, 1992, pp [14] Bonatti P A, Eiter T. Querying disjunctive databases through nonmonotonic logics. Theoretical Computer Science, 160(1-2): [15] de Kleer J. Local methods for localizing in electronic circuits. In MIT AI Memo 394, Cambridge, MA, [16] Reiter R. A theory of diagnosis from first principles. Artif. Intell., 1987, 32: SU Kaile was born in He received his Ph.D. degree from Nanjing University in Then, he worked as a postdoctoral researcher at the National Key Lab. of Parallel and Distributed Computing in National University of Defense Technology. Since 1999, he has been working in Department of Computer Science, Zhongshan University, where he is now a professor. Professor Su has been engaged in theoretic research of artificial intelligence for ten years. He has published more than 30 reviewed journal papers, including two papers in Science in China and several papers in famous international journals such as Information and Computation, Journal of Logic and Computation, and Fundamenta Informaticae.