Representing Defaults in the Framework of ... - Semantic Scholar

Representing Defaults in the Framework of Possibility Theory  Churn-Jung Liau Institute of Information Science Academia Sinica, Taipei, Taiwan Abstract

In this paper, we present two results on combining approximate and default reasoning. First, we investigate the relationship between default logic and the formalism introduced by Yager to represent default knowledge in the framework of possibility theory. Second, we describe a natural representation of default priorities in Yager's formalism. INDEX TERMS : Default logic, possibility theory, fuzzy set, prioritized default logic.

1 Introduction Commonsense reasoning is undoubtedly one of the most important aspects of human cognitive process. In face of complicated environment, human usually need to make their decisions with incomplete and uncertain information. To act with incomplete information, human usually jump to conclusions according to some empirical rules even though the conclusions are not deductively derived from the available knowledge. Unlike the ordinary mathematical theorems, the conclusions so drawn are just plausible and defeasible. They may be revised or retracted later when more information is acquired. The rules used to draw the plausible conclusions are not universally true but with exceptions. They are also called defaults. The process to draw conclusions from available information by using defaults is called default reasoning. Among the formalisms to model human default reasoning, the default logic proposed by Reiter[1980] has received much attention. Another important feature of human reasoning is the ability to deal with vague information. Vagueness refers to the uncertainty in the meaning of what is actually stated. It is widely recognized that many linguistic terms used in everyday life do not have precise meaning. However, human can still use these terms in natural languages and communicate with each other without too much diculty. Since classical logical systems provide no methods for representing vague information, such systems are inadequate for dealing with commonsense knowledge. To cope with the problem, Zadeh[1978a] proposes possibility theory to serve as a framework for the representation and treatment of vague information.  Preliminary results

of the paper have appeared in [Liau, 1994a, b].

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Although Reiter's default logic and its variants[Brewka,1991a, Delgrande and Jackson,1991, Froidevaux and Mengin,1992, Lukaszewicz,1988] provide an elegant way to treat default knowledge, they are all based on the classical logic language. In these systems, all known facts are represented as classical logic formulas and defaults make it possible to jump to conclusions which are also represented in the same language. Thus a conclusion drawn from the theory must not involve any vague information although the conclusion may be defeasible1 . That is, default theories based on classical logics provide no way to reasoning with vague information. However, many natural language examples show that defaults with vague prerequisites, justi cations, and/or consequent are often encountered in everyday life. just as in the following examples provided by Zadeh[1984]. Icy roads are slippery. Tall men are not very agile. Overeating cause obesity. What is rare is expensive. .. . etc.

In these examples, \tall", \slippery", \agile", \obesity", \rare", and \expensive" are some typical vague terms. Therefore, to handle default and vague information simultaneously, it is desirable to have an integrated formalism. In Zadeh's approach[1984], default knowledge is viewed as a collection of dispositions, i.e., propositions with fuzzy quanti ers, so a default can be translated into a corresponding sentence in fuzzy logic and then the inference rules of fuzzy logic[Bellman and Zadeh,1977] may be employed to do default reasoning. Consequently, in this framework, it is unnecessary to distinguish default and vague information because defaults are just statements with vague quanti ers. On the other hand, Yager suggests to combine default logic and possibility theory in a more modular way[Yager,1987a, b,1988]. That is, all known vague facts are represented as possibility distributions while defaults are translated into if-then rules whose antecedent and consequent are also possibility distributions with possibilistic quali cation. Then these rules are applied to known vague facts to derive default conclusions. These conclusions are still 1 Some variants mentioned above associate labels with classical logical formulas. however the information carried by such labeled formulas is still crisp.

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possibility distributions, so they may be both vague and tentative. Thus Yager's proposal preserves the spirit of Reiter's default logic while having the capability to represent vague information. Although from a representational viewpoint, Yager's formalism can be considered as a generalization of Reiter's default logic, the results produced by the former is substantially dierent from those by the latter even our knowledge base and defaults do not involve any vague knowledge. In this paper, we discuss how the problem can be circumvented. Furthermore, we show that priorities of defaults can be naturally represented in Yager's formalism, so a prioritized default theory[Brewka,1991a, b] can be easily translated into it. The structure of the paper is as follows. First, Reiter's default logic and the approximate reasoning method based on possibility theory are brie y described in Sec. 2 and 3 respectively. Then, we introduce Yager's proposal to represent default knowledge in the framework of possibility theory. The formalism is called possibilistic default logic. Its relationship with Reiter's default logic is discussed. In Sec. 5, we present a translation of prioritized default logic into the possibilistic one. Finally, a brief summary and some possible further works are given as conclusions.

2 Basic Default Logic We restrict our attention to propositional default logic. Let L denote the set of all well-formed formulas (ws) in a propositional language. For any S  L, de ne ThL(S) = f j 2 L; is closed, and S ` g, where S ` means that is classically provable from S. A default is expressed in the following form,  : 1 ; 2 ; : : :; n ; (1)

where , i 's and are ws of L.  is called the prerequisite of the default, i 's are its justi cations and

is its consequent. A default theory(DT) is a pair hW; Di where W  L and D is a set of defaults. The intended meaning of a default is as follows. if  is known, and none of 1 ; : : :; n are known to be false. then can be assumed by default. The intuitive meaning motivates the following de nition:

De nition 1 ([Reiter,1980]) Let
= hW; Di be a DT based on a propositional language L. (1) For any S  L let ?(S) be the smallest set satisfying the following three properties: D1. W  ?(S) D2. ThL (?(S)) = ?(S)

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D3. If : 1 ; 2;:::; 2 D and  2 ?(S), and : 1 ; : : :; : n 62 S then 2 ?(S). n

(2) A set E  L is an extension of
i ?(E) = E . (3) The generating defaults of E is de ned as

GD(E;
) = f  : 1 ; : : :; n j 2 E and : 1 ; : : :; : n 62 E g (4) If D is any set of defaults then

CON(D) = f j  : 1 ; : : :; n 2 Dg and

PRE(D) = fj  : 1 ; : : :; n 2 Dg

An important result is that if E is an extension of
then E = ThL (W [ CON(GD(E;
))):

(2)

3 Approximate Reasoning Let U be a universe of discourse and X be a variable (that may be a given attribute of a given object) taking its values on U. A piece of vague or fuzzy information about X can usually be considered as some elastic constraints on X[Zadeh,1978b]. A basic statement to express these constraints is of the form: X is A where A is a fuzzy subset of U. The intent of this statement is to indicate that the possible values of X are constrained by the set A. Two extreme cases are that A is a crisp singleton set when we have precise information about X and A = U when we are completely ignorant of the value of X. In general, the statement induces a possibility distribution X : U ! [0; 1] so that: X (u) = A(u); where A(u) is the degree of membership of u in A. Compound statements are composed of basic ones via ordinary logical connectives, :(not), ^(and), and _(or). The basic mechanism for approximate reasoning is the so-called combination/projection procedure[Dubois and Prade,1991]. The rst step of the procedure is to induce the joint possibility distribution of a compound statement. Usually, dierent variables that take their values on dierent universes may occur in a compound statement. To represent the joint possibility distributions induced by such compound 4

statements, the notion of cylindric extension is used[Klir and Folger, 1988]. Consider any compound statement S. Let X1 ; X2 ; : : :; Xn be all variables occurring in S with respective universes U1 ; U2; : : :; Un. Suppose I  f1; 2; : : :; ng = In . De ne X = (Xi )i2I , XI = (Xi )i2I , U = i2I Ui , and UI = i2I Ui . If u 2 U and v 2 UI , then v is said to be a subsequence of u, denoted by u  v, i ui = vi for all i 2 I. Suppose A is a fuzzy subset of UI . Then the cylindric extension of A (to U), denoted by Ae , is a fuzzy subset of U such that Ae (u) = A(v) for all u  v. While \XI is A" induces an elastic constraint on XI , the statement \X is Ae " induces the same constraint on X. Thus, in particular, any statement \Xi is Ai " occurring in S can be replaced by \X is Aei " without distorting the original information contents. On the other hand, if A is a fuzzy set of U, then the projection of A on I, denoted by A # I, is a fuzzy subset of UI such that A # I(v) = supuv A(u). To induce a joint possibility distribution from S, we rst translate S into a corresponding set-theoretic expression. Let T denote the translation mapping. Then 8 e Ai ; if S = Xi is Ai ; > > > > > if S = :S1 ; < T(S1 ); T(S) = > > T(S1 ) [ T(S2 ); if S = S1 _ S2 ; > > > : T(S1 ) \ T(S2 ); if S = S1 ^ S2 : The expression T(S) is then evaluated to a fuzzy subset of U. Let the resultant fuzzy set be A, then the joint possibility distribution corresponding to S is X : U ! [0; 1] such that X (u) = A(u) for all u 2 U. We note that T(S) (or A) is in fact a semantic representation of S, so to simplify the presentation, we will use fuzzy subsets as a standard representation formalism of vague information from now on. Thus, a knowledge base is just a class W = fA1; A2; : : :; Ak g where Ai 's are all fuzzy subsets of a common T universe U. The joint possibility distribution of W is equivalent to the membership function of ki=1 Ai . Finally, we can project X on I for any I  In to obtain the information about XI . Throughout the remaining of the paper , the symbol U will be reserved to denoting the joint universe, i.e., the Cartesian product of all the domains on which our concerned variables take their values. The symbols A; B; C (possibly with indices) will denote any fuzzy subsets of U. Since a knowledge base can be represented as a single fuzzy set, the symbol F will be used to denote a knowledge base of this form. Besides, two indices that measure the degree of consistency and containment between two fuzzy sets are used extensively in approximate reasoning. These two indices, called possibility and certainty, are de ned as Poss[B jA] = sup [min(A(u); B(u))]; n

u2U

Cert[B jA] = 1 ? Poss[B jA]: 5

n

The former measures the degree of A intersecting with B, and the latter measures the degree of A being a subset of B. When A and B are crisp sets, Poss[B jA] = 1 i A \ B 6= ; and Cert[B jA] = 1 i A  B. Also note Poss[B jA] = Poss[AjB]. Based on the possibility index, a level 2 fuzzy set [Klir and Folger,1988] can be constructed from an ordinary one. Let F (U) denote the class of all fuzzy subsets of U. A level 2 fuzzy set is a fuzzy subset of F (U). We will use A~ to denote a general level 2 fuzzy set. For each A, a special level 2 fuzzy set A+ can be de ned with the membership function A+ (G) = Poss[AjG] for all G 2 F (U).

4 Possibilistic Default Logic To integrate approximate and default reasoning, Yager[1987b] suggests to represent a possibilistic default in the following form, A : B1 ; B2; : : :; Bn ; (3) C

where A; B1 ; : : :; Bn, and C are all fuzzy subsets of a joint universe U. Thus a possibilistic default theory (DT) is a pair
 = hF; D i, where F is a fuzzy subset of U and D is a set of possibilistic defaults. If F and A, Bi 's and C mentioned above are all crisp,
 is called a crisp DT. As indicated in the preceding section, a knowledge base induces elastic constraints on our interested tuple of variables, X. These constraints are represented as a fuzzy subset F in a DT. Then the possibilistic defaults will impose further constraints on X. However, it remains unclear how these constraints should be imposed by using the possibilistic defaults. What has been pointed out by Yager[1987b] is just that the intended meaning of a default in (3) is \if X is A and for 1  i  n, X is Bi is possible, then X is C". To ful ll the intuition behind possibilistic defaults, there may be some dierent methods to constrain X by defaults in a DT. A direct method is to generalize the de nition of extensions in Reiter's default logic. Though Reiter's extension lacks some important properties such as cumulativity, speci city, and guaranteed existence etc., it shows the essential feature of xed-point de nition, so we will focus on its generalization in this paper. Other more sophisticated de nitions can be obtained by modifying the following de nition appropriately.

De nition 2 Let
 = hF; D i be a DT, then for each V 2 F (U), de ne ? (V ) as the largest (least speci c) fuzzy subset of F such that for each default in D of form (3) and u 2 U , ? (V )(u)  max(Poss[Aj? (V )]; 1 ? Poss[B1 jV ]; : : :; 1 ? Poss[Bn jV ]; C(u)): A fuzzy set V is called a possibility restriction(-restriction) of
 i ? (V ) = V .

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For each crisp DT, we can also de ne ?c (V ) as the largest crisp subset of F such that If A:B1 ;BC2 ;:::;B 2 D and ?c (V )  A, and for 1  i  n; Bi \ V 6= ; then ?c (V )  C. n

Then, we have the following lemma.

Lemma 1 If
 is a crisp DT, and ? and ?c are the respective mappings associated with
 , then for each crisp subset V of U , ?c (V ) = V i ? (V ) = V .

Proof: (if) Easy.

(only if) If ?c (V ) = V but ? (V ) 6= V , then V  ? (V ) and ? (V ) is noncrisp. Let V 0 be the smallest crisp subset containing ? (V ), then ? (V )  V 0 and since F is crisp, then by the de nition of ? (V ), V 0 is also a subset of F. Furthermore, the inequality in De nition 2 is satis ed if we replace ? (V ) by V 0 . This contradicts with the de nition of ? (V ). 2 Viewing propositional symbols as binary variables, we can easily translate a DT into a DT. First, let U be the set of all truth assignments on the language L. De ne Mod( ) = fu 2 U j u j= g as the set T of all models of for each 2 L and let Mod(W) = 2W Mod( ) for each W  L. Then given a DT
= hW; Di, its translation into DT is Tr(
) = hMod(W); Mod(D)i, where 1 ); : : :; Mod( n ) j  : 1 ; : : :; n 2 Dg: Mod(D) = f Mod() : Mod( Mod( ) Obviously, Tr(
) is a crisp DT.

Lemma 2 If ?c is the mapping associated with Tr(
) for some nite default theory
and V is a crisp subset of U , then ?c (V ) = V i there exists an extension E of
such that V = Mod(E).

Proof: This follows from the fact that ?c is just a reformulation of ? in model-theoretic terms.2

It will be interesting to explore the relationship between the extensions of a DT and the -restrictions of Tr(
). Let us consider a very simple DT
= h;; f >p:p ; >:::pp gi. This toy example, though rather arti cial, may occur as a fragment of a big DT. Obviously,
has two extensions, E0 = ThL (f:pg) and E1 = ThL(fpg). As for Tr(
), the joint universe U has two elements u0 and u1 that correspond to the two interpretations assigning 0 and 1 to p respectively. Let us denote any fuzzy subset of U by a pair (a; b) where a (resp. b) is the membership degree of u0 (resp. u1) in the fuzzy subset. Then it can be veri ed that for all a; b 2 [0; 1], (a; b) is a -restriction of Tr(
) i a + b = 1. The two crisp subsets (1; 0) and (0; 1) are equal to Mod(E0) and Mod(E1) respectively. However, for other a; b such that 0 < a; b < 1 and a + b = 1, (a; b) does not correspond to any extensions of the original DT. This shows the substantial dierence between a DT and its translation into DT. Theoretically, this can be explained as follows. Because there are two con ict defaults in the example and we do not know 7

which one is the more reliable, we should be ignorant of the truth of p. In Reiter's default logic, we have only two choices, p is true or p is false, since our base language is classical logical language. Thus, there are two extensions that can be considered as candidates of acceptable belief sets. Nevertheless, in a DT, there are other possible choices. Because we are interested in the possible values of p, in addition to the possibility that p is completely true or completely false, other intermediary possibility should also be considered. In general, the number of dierent -restrictions for a DT is in nite. Thus, from a pragmatic viewpoint, it will be helpful to shrink the number by some additional criteria. A reasonable criterion is to require that the -restrictions are sequentially generated from the defaults in some or other ways. This can be achieved by the sequential application scheme proposed by Yager[1988] in the case of nite DT. From now on, let us assume
 = hF; fdig1imi is a DT with nite defaults where each default (4) di = Ai : Bi1; BCi2 ; : : :; Bin : i According to Yager[1987b], we can de ne i

+ ; Cig; i = f(Ai )+ ; Bi+1; : : :; Bin i

for each di . Let V  U and  = 1 [ 2 be a class of level 1 or 2 fuzzy sets so that all elements in 1 are level 1 and those in 2 are level 2, we can de ne [V ] so that for each u 2 U ~ )jA~ 2 1g [ fB(u)jB 2 2 g): [V ](u) = max(fA(V We can further abbreviate V \ [V ] as V \ . Then for each permutation  on Im = f1; 2; : : :; mg, we can de ne V0 = F and de ne Vi = Vi?1 \ (i) for 1  i  m. Finally, let V  = Vm .

De nition 3 Let V be a fuzzy subset of U , then V is said to be sequentially generated from
 i there

exists a permutation  on the index set of the defaults such that V = V  .

If
is a nite DT, then there is an exact correspondence between its extensions and the sequentially generated -restrictions of Tr(
).

Lemma 3 Let
be a nite DT. If E is an extension of
, then Mod(E) is sequentially generated from

Tr(
).

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Proof: Let K = fijdi 2 GD(E;
)g and m0 = jK j. Then by Eq. (2), we have Mod(E) = F \ Ti2K Ci.

We will nd a permutation  so that V  is equal to the right-hand side of the equation. The set K can be strati ed naturally by the groundness of GD(E;
)[Froidevaux and Mengin,1992]. S Namely, GD(E;
) = n2N Dn , where the sequence (Dn )n2N is de ned by:

 D0 = ;,  Dn+1 = fd 2 GD(E;
) j W [ CON(Dn ) ` PRE(fdg)g for all n 2 N. De ne the level function L : f1; 2; : : :; mg ! N as follows  if i 2 K; i 2 Dn ) L(i) = n(d 1 + maxi2K L(i) if i 62 K; where n(di 2 Dn ) is the least n such that di 2 Dn 2. Let  be a permutation so that for 1  i  m ? 1, L((i))  L((i + 1)), then it can be veri ed that  satis es our requirement. 1. By properties of  and induction, for 1  i  m0 , (i) 2 K, Vi?1  A(i) , Mod(E)  Vi?1 , Vi?1 \ B(i)j 6= ; for 1  j  n(i) (by (a) and (c)), and Vi = Vi?1 \ C(i) (by (b) and (d)). T 2. Vm0 = F \ i2K Ci . (a) (b) (c) (d) (e)

3. By induction, for m0 < i  m, (a) either Vi?1 6 A(i) or there exists 1  j  n(i) so that Vi?1 \ B(i)j = ;, (b) and Vi = Vi?1 . 4. Vm = Vm0 . Thus, V  = Vm = Mod(E). 2 Combining Lemma 1{3, we have the following theorem.

Theorem 1 Let
be a nite DT and V be a fuzzy subset of the universe (i.e. the set of all truth

assignments). Then V is a -restriction sequentially generated from Tr(
) i there is an extension E of
such that V = Mod(E). 2

The notation is from recursion theory[Lewis and Papadimitrious,1981].

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Proof: ()): Because V is sequentially generated from Tr(
), V is crisp, then by Lemma 1 and 2, the

result follows. ((): By Lemma 3, if V = Mod(E), then V is sequentially generated from Tr(
). Furthermore, because Mod(E) is crisp, V is a -restriction of Tr(
) by Lemma 1 and 2. 2 For a general nite DT, we can enumerate all fuzzy subsets V that are sequentially generated from it and test whether ? (V ) = V . In this way, we can nd all sequentially generated -restrictions of a nite DT and deduce plausible results from them.

5 Representation of Priorities Prioritized default logic is proposed by Brewka[1991b] to represent priorities among defaults. The de nition is as follows.

De nition 4 ([Brewka,1991b]) 1. Let W be a set of ws in L and Di be a set of classical defaults for 1  i  k, then
= hW; D1; : : :; Dk i is called a prioritized default theory(PDT). 2. E is a PDT-extension of
i there exist sets of ws E1; : : :; Ek such that Ei is an extension of hEi?1; Di i for 1  i  k and E = Ek , where for convenience, E0 = W .

We will show that a PDT can be translated into a DT naturally, and the results produced by the translating DT re ect the certainty of the default conclusions in an explicit way. To represent the translation, we need to use the notion of fuzzy default value de ned by Dubois and Prade[1988]. According to the notion, the default knowledge \Usually, X is A" is translated into the statement \X is A ", where  2 [0; 1] and the membership function of A is de ned as A(u) = max(A(u); 1 ? ): This means that though in general X takes its value inside A, it may take its value outside A with the possibility 1 ? . That is,  estimates the certainty that the value of X is not an exception. When  = 1, A is reduced to A, i.e., the knowledge is certain without exceptions. , de ne the translation of d with certainty  as For each default d = : 1 ;:::;

t(d; ) = A : B1C; :: :; Bn ; where A = Mod(), Bi = Mod( i ) for 1  i  n and C = Mod( ). Then given a PDT
= hW; D1 ; : : :; Dk i and an arbitrary strictly decreasing function f : Ik ! ( 12 ; 1], we can de ne a DT n

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Trf (
) = hF; D i, where F = Mod(W) and D = ft(d; f(i)) j d 2 Di ; 1  i  kg: The function f is called certainty assignment function. Though dierent certainty assignment functions may result in dierent translations for a PDT, the dierence may be ignored when only the ordering is ). relevant. Thus, we will assume a xed f from now on(e.g. f(i) = 2(ii+2 +1) To investigate the relationship between prioritized default logic and the possibilistic one, let us consider a given nite PDT
= hW; D1; : : :; Dk i such that D1 [ : : : [ Dk = fdij1  i  mg, where each di = i : i1 ; i2 ; : : :; in : i Let us assume that the index set is partitioned so that di 2 Dj i i 2 I j , where I j = fijmj ?1 +1  i  mj g for 1  j  k and 0 = m0 < m1 < : : : < mk = m. For each di , de ne i

+ i = f(Ai )+ ; Bi+1; : : :; Bin ; Cig; i

and

+ ; Ci g; i = f(Ai )+ ; Bi+1; : : :; Bin i

i

i

where Ai = Mod(i), Bij = Mod( ij )(1  j  ni) , Ci = Mod( i ), and i = f(j) i i 2 I j , for 1  i  m. For convenience, we also assume 0 = 1. Let us rst consider the fuzzy sets sequentially generated from Trf (
). A permutation on Im is said to be compatible with Trf (
) i (i)  (i+1) for 1  i  m ? 1. We will only consider those fuzzy sets sequentially generated from Trf (
) by a permutation compatible with it. Without loss of generality, we may assume  is the identity permutation and drop the superscript . Then we have the following de nitions. S0 = W;  Si?1 ` i and 81  j  ni ; Si?1 6` : ij Si = SSi?1 [ f i g ifotherwise; i?1 V0 = F = Mod(W); and

Vi = Vi?1 \ i ; i

for 1  i  m. If V is a fuzzy subset of U, we de ne the binary approximation of V , denoted by Vb as the crisp subset such that u 2 Vb i V (u) > 12 . Then we can prove the following lemmas. 11

Lemma 4 For 0  i  m, the following results hold (1) Vbi = Mod(Si ), (2) for each u 2 U , Vi (u) 2 i =df fj ; 1 ? j j0  j  ig, and (3) if Vi?1 (u) < 21 , then Vi (u) = Vi?1(u) (1  i  m).

Proof: We prove all the three results by simultaneous induction on i. It is easily seen that the results hold for i = 0. Assume i > 0 and the results hold for all j  i ? 1, then there are two cases to be considered.

Case 1: If Si?1 ` i and 81  j  ni ; Si?1 6` : ij , then Vbi?1  Ai and Vbi?1 \ Bij 6= ; for all 1  j  ni. Thus, each of Poss[Ai jVi?1] and 1 ? Poss[Bij jVi?1 ] is equal to some 1 ?  where   i by (2) and the fact that i  i+1 for all i. Then i [Vi?1] = Ci and  i if u 2 Mod(Si?1 [ f: i g) Vi (u) = 1V ? (u) (5) otherwise: i?1 i

i

Then if u 62 Mod(Si?1 ), then Vi (u) = Vi?1(u) < 21 by (1). If u 2 Mod(Si?1 [ f: i g), then Vi (u) = 1 ? i < 21 . If u 2 Mod(Si?1 [ f i g) = Mod(Si ), then Vi (u) = Vi?1 (u) > 12 by (1). Thus Vi (u) > 21 i u 2 Mod(Si ) and this proves (1). Second, (2) follows from the induction hypothesis and Eq. 5 directly. Finally, we note that if Vi?1(u) < 12 , then u 62 Mod(Si?1 ) by (1), so Vi (u) = Vi?1 (u). Case 2: If Si?1 6` i or there exists j such that 1  j  ni and Si?1 ` : ij , then at least one of Poss[Ai jVi?1] and 1 ? Poss[Bij jVi?1] is equal to some   i by induction hypothesis for (1) and (2). Thus i [Vi?1] = Ci1? for some . This results in  u 2 Mod(Si?1 [ f: i g) Vi (u) = Vmin(V(u)i?1 (u); ) ifotherwise: (6) i?1 i

Thus Vbi = Vbi?1 and (3) holds. Since in this case Si = Si?1 , (1) also follows from it. (2) follows from the observation that  2 i?1.

2

De ne the threshold normalization of Vm as a fuzzy subset V so that  Vm (u)  12 V (u) = V1 m (u) ifotherwise:

Lemma 5 Let ' 2 L and V be the threshold normalization of Vm . De ne N(') = 1 ? supu62Mod ' V (u). ( )

Then

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1. if ' 2 ThL (Sm ) and i is the smallest index so that Si ` ', then N(') = i , 2. if ' 62 ThL (Sm ), then N(') = 0.

Proof: 1. If i = 0, then S0 = W ` ', so sup V0 (u) = sup V (u) = 0

u62Mod(')

u62Mod(')

by (3) of the preceding lemma. If i > 0, then Si?1 6` ' and Si ` '. Thus Case 1. of the preceding proof holds. Then by Eq. (5), supu62Mod(') Vi (u) = 1 ? i since for all u 62 Mod('), Vi (u) < 12 by Lemma 4(1) and there exists u 62 Mod(') such that u 2 Mod(Si?1 [f: g). Then the result follows from Lemma 4(3). 2. If ' 62 ThL (Sm ), there exists u 62 Mod(') such that u 2 Vbm . Thus supu62Mod(') V (u) = 1 since V is the threshold normalization of Vm . Then N(') = 0 by de nition.

2

Note that because we adopt threshold normalization, V is truly normalized only when height(Vm ) > 12 in the lemma. Thus, if height(Vm )  12 , then N(') > 0 for all ' 2 L, i.e., Sm is inconsistent. Let us de ne Ei = ThL(Sm ) for 0  i  k and call a w ' 2 Ek i-rooted i i is the smallest index so that ' 2 Ei. We assume 1 < 1 in the following theorem. i

Theorem 2 If ' and ' are i-rooted and j-rooted respectively, and i > j, then N(' ) < N(' ). 1

2

1

2

Proof: This follows from the preceding lemma and the fact that m < m when i > j since the i

j

certainty assignment function is required to be strictly decreasing. 2 This theorem shows that the fuzzy sets sequentially generated by Trf (
) provides useful information about the derivation process of default conclusions for
when Ek is a PDT-extension of
. However, it can also easily seen that not all compatible permutation produce PDT-extensions. This motivates the following de nition.

De nition 5 A fuzzy subset Vm sequentially generated from Trf (
) by a permutation compatible with it is called a prioritized -restriction of Trf (
) i for all 1  j  k, Vbm is a -restriction of hVbm ?1 ; Mod(Dj )i. j

j

Theorem 3 V is a prioritized -extension of Trf (
) i there is a PDT-extension E of
so that Vb = Mod(E).

13

Proof: This follows from de nitions, Theorem 1, and Lemma 4(1) directly.

2

The results of this section show the representational adequacy of possibilistic default logic for prioritized defaults, though its computational eciency remains to be explored. Finally, let us use an example to illustrate the results presented above.

Example 1 Consider the (extended) high school dropout example. The example has the following three

defaults

d1 =Typically high school dropouts are adults, d2 =Typically adults are employed, d3 =Typically high school dropout are not employed. Assume further that d1 and d3 has higher priority over d2. Thus D1 = fd1; d3g and D2 = fd2g. Let h; a, and e denote \Jack is a high school dropout", \Jack is an adult", and \Jack is employed" respectively. The eight truth assignments of the three propositional symbols are de ned in the following table u0 u1 u2 u3 u4 u5 u6 u7 h 0 0 0 0 1 1 1 1 a 0 0 1 1 0 0 1 1 e 0 1 0 1 0 1 0 1 For I  f0; 1; 2; : : :; 7g, let AI = fuiji 2 I g. Then 1 = fA+0123; A+2367; A2367g; 0 g; 20 = fA+0145; A+1357; A1357 3 = fA+0123; A+0246; A0246g; and assume 1 >  > 0 > 0:5 according to the priorities. Moreover, assume the knowledge base F = A4567, i.e., the only known fact is that Jack is a high school dropout. Then E0 = ThL (fhg) and E1 = E2 = ThL (fh; a; :eg). The fuzzy subset sequentially generated by the permutation (1) = 1; (2) = 3; (3) = 2, is V = (0; 0; 0; 0; 1 ? ; 1 ? ; ; 1 ? ) where the ith element of the tuple denotes V (ui ). By normalization, we get V = (0; 0; 0; 0; 1 ? ; 1 ? ; 1; 1 ? ). Thus N(h) = 1; N(a) = N(:e) = , and N(e) = 0. This means that h is known certainly, a and :e are drawn as default conclusions and e is not derivable. On the other hand, if F is equal to A2367 instead of A4567, i.e., only the fact that Jack is an adult is known, then the same  will produce V = (0; 0; 1 ? 0 ; 1; 0; 0; 1 ? 0 ; 1) that has been normalized. Thus N(a) = 1; N(h) = N(:h) = 0, and N(e) = 0 . This corresponds to the fact that E0 = E1 = ThL (fag) and E2 = ThL (fa; eg).

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6 Conclusions The results of the preceding sections can be summarized as follows.

 The notion of -restrictions is de ned for possibilistic default logic. The -restrictions sequentially

generated from a given DT are especially important. If the given DT is a translation of a DT, the -restrictions sequentially generated from it correspond to the models of the extensions of the original DT. On the other hand, if the DT is not a translation of any DTs, then it will shrink the number of -restrictions to consider the sequentially generated ones.

 Prioritized default theories are shown to have natural translations in DTs. The priorities can be encoded into the consequents of defaults and the strength of a conclusion is re ected in its necessity measure according to the fuzzy sets sequentially generated from the DT.

There are of course some limitations and possible generalizations about the work reported here.

 While we translate a DT into a DT by taking propositional symbols as binary variables here, there

may be more semantically-directed ways to do this. In other words, we can consider the case in which the universe U is not the set of all truth assignments though the sets F, Ai 's, Bij 's, and Ci 's are still crisp subsets of U. For example, the inheritance hierarchies explored in [Etherington,1987] may have a natural representation in DT by taking attributes as variables.

 Although the general framework DT is designed to represent both default and vague knowledge

in an integrated way, we devote most part of the paper to the discussion of the crisp case. When vague information is represented by fuzzy sets, the -restrictions may be fuzzy sets that do not have any explicit relation with the input fuzzy sets. Then, how can we nd linguistic terms that correspond to the -restrictions. This, we think, is still an important but dicult problem common to many fuzzy reasoning systems.

 We consider only propositional case in this paper and when we de ne sequential generatedness of

a -restriction, we restrict to nite default theories. Because in traditional default logics, an open default is considered as a (possibly in nite) set of closed default, if we can lift the second constraint (i.e. niteness), the rst one will become more irrelevant. Essentially, our de nitions can be easily adapted to the in nite case. However, the permutations on a nite index set will be replaced by trans nite sequences and the resultant de nition will not be constructive any more.

 Finally, the inference problem of possibilistic default logic should be considered. In the present

stage, possibilistic default logic only provides a representational formalism and to be useful, an 15

inference method is needed. It seems that the graded default logic[Froidevaux, et. al., 1991] can provide the inference mechanism for prioritized -restrictions. The general mechanism remains to be found.

Acknowledgement I would like to thank two anonymous referees for valuable comments. The research is partially supported by National Science Council of ROC under the grant number NSC 82-0113-E-001-015-T. I am grateful to T. F. Fan for her extensive discussion with me in the initial stage of the research.

References [1] Bellman, R. E. and Zadeh L. A.[1977] \Local and fuzzy logics ". In G. Epstein, editor, Modern Uses of Multiple-Valued Logic, pp. 103{165. D. Reidel Publishing Company. [2] Brewka, G.[1991a], \Cumulative default logic: In defense of nonmonotonic inference rules". Arti cial Intelligence, 50:183{205. [3] Brewka, G.[1991b], Nonmonotonic Reasoning : Logical Foundation of Commonsense. Number 12 in Cambridge Tracts in Theoretical Computer Science. Cambridge University Press. [4] Delgrande, J. P. and Jackson W. K.[1991], \Default logic revisited". In J. Allen, R. Fikes, and E. Sandewall, editors, Proccedings of the 2nd International Conference on Principles of Knowledge Representation and Reasoning, pp. 118{127. Morgan Kaufmann. [5] Dubois, D. and Prade H.[1988], \Default reasoning and possibility theory". Arti cial Intelligence, 35:243{257. [6] Dubois, D. and Prade H.[1991], \Fuzzy sets in approximate reasoning. Part 1 : Inferences with possibility distributions". Fuzzy Sets and Systems, 40:143{202. [7] Etherington, D. W.[1987], \Formalizing nonmonotonic reasoning systems". Arti cial Intelligence, 31:41{85. [8] Froidevaux, Ch., Chatalic P., and Mengin J.[1991], \Graded default logics". In R. Kruse and P. Siegel, editors, Symbolic and Quantitative Approaches to Uncertainty, LNCS 548, pp. 70{75. Springer-Verlag. [9] Froidevaux, C. and Mengin J.[1992], \A framework for default logics ". In D. Pearce and G. Wagner, editors, Proceedings of Logics in AI, LNAI 633, pp. 154{173. Springer-Verlag. 16

[10] Klir, G. J. and Folger T. A.[1988], Fuzzy Sets, Uncertainty, and Information. Prentice-Hall International Inc.. [11] Lewis, H. R. and Papadimitriou C. H.[1981], Elements of the Theory of Computation. Prentice-Hall. [12] Liau, C. J.[1994a], \Application schemes of possibility theory-based defaults". In R. Trappl, editor, Proceedings of the 12th European meeting on cybernetics and Systems Research, pp. 343{350. WorldScienti c. [13] Liau, C. J.[1994b], \Representibg priorities of defaults in possibility theory". In Proceedings of the 3rd IEEE International Conference on Fuzzy Systems, pp. 1280{1284. [14] Lukaszewicz, W.[1988], \Considerations on default logic { an alternative approach". Computational Intelligence, 4:1{16. [15] Reiter, R.[1980], \A Logic for default reasoning". Arti cial Intelligence, 13:81{132. [16] Yager, R. R.[1987a], \Possibilistic quali cation and default rules". In B. Bouchon and R. R. Yager, editors, Uncertainty in Knowledge-Based Systems, LNCS 286, pages 41{57. Springer-Verlag. [17] Yager, R. R.[1987b], \Using approximate reasoning to represent default knowledge ". Arti cial Intelligence, 31:99{112. [18] Yager, R. R.[1988], \A generalized view of nonmonotonic knowledge: A set-theoretic perspective". International Journal of General Systems, 14:251{265. [19] Zadeh, L. A.[1978a], \Fuzzy sets as a basis for a theory of possibility ". Fuzzy Sets and Systems, 1:3{28. [20] Zadeh, L. A.[1978b], \PRUF - a meaning representation language for natural languages". Int. J. Man-Machine Studies, 10:395{460. [21] Zadeh, L. A.[1984], \A theory of commonsense knowledge". In H. J. Skala, S. Termini, and E. Trillas, editors, Aspects of Vagueness, volume 39 of Theory and Decision Library, pp. 257{295. D. Reidel Publishing Company.

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