a fuzzy version of default logic

A fuzzy version of default logicKumar S. Ray

Indian Statistical Institute, Kolkata, India, and

Arpan ChakrabortyNorth Carolina State University, Raleigh, North Carolina, USA

Abstract

Purpose – The importance of fuzzy logic (FL) in approximate reasoning, and that of default logic(DL) in reasoning with incomplete information, is well established. Also, the need for a commonsensereasoning framework that handles both these aspects has been widely anticipated. The purpose of thispaper is to show that fuzzyfied default logic (FDL) is an attempt at creating such a framework.

Design/methodology/approach – The basic syntax, semantics, unique characteristics andexamples of its complex reasoning abilities have been presented in this paper.

Findings – Interestingly, FDL turns out to be a generalization of traditional DL, with even bettersupport for non-monotonic reasoning.

Originality/value – The paper presents a generalized tool for commonsense reasoning which can beused for inference under incomplete information.

Keywords Artificial intelligence, Logic, Reasoning, Fuzzy logic

Paper type Research paper

1. IntroductionThe qualification problem is arguably one of the most important challenges in thedomain of knowledge representation and reasoning. Reiter’s (1980) default logic (DL) isa formalism that tries to answer this problem. It enables us to reason in the presence ofincomplete knowledge. Another important issue in artificial intelligence arises from thefact that most real-world knowledge is not discrete. Representation of vagueness thus,becomes a necessity for most systems that need to interact with the real world. Zadeh’s(1965) fuzzy logic (FL) is a popular approach that impressively handles vagueness.

Human intelligence seems to have both these characteristics (among severalother capabilities). We go about reasoning with commonsense and make implicitassumptions all the time (that we may not be consciously aware of either). Dealing withuncertainty is effortless with our ability to maintain varying degrees of belief.Certainly, these two are not the only factors that make human intelligence so powerful.Nonetheless, it is a worthy pursuit to try and unite these ideas in artificial systems.We suggest in this paper, a logic that incorporates the capabilities of both FL and DL,bringing us closer to a comprehensive framework for commonsense reasoning.

A number of authors have suggested different ways of combining these two (or similar)logic formalisms. For example, Yager (1987) has quite cogently demonstrated therepresentation of default knowledge in the form of possibility-qualified rules andBenferhat et al. (1999) have tried to unify both default reasoning and approximatereasoning under the common framework of possibility theory. Our formalism, fuzzyfieddefault logic (FDL), is simply about default reasoning in a fuzzy world.

The primary goal of FDL is to enable us to draw approximate conclusions underpartial presence or absence of knowledge; while in DL (a bi-valued logic), facts can

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A fuzzy versionof default logic

5

Received 3 September 2009Revised 2 August 2010

Accepted 21 August 2010

International Journal of IntelligentComputing and Cybernetics

Vol. 4 No. 1, 2011pp. 5-24

q Emerald Group Publishing Limited1756-378X

DOI 10.1108/17563781111115769

either be present or absent, but not midway. This is accomplished by fuzzyfying eachindividual fact or piece of knowledge, attaching a degree of truth with them. FDLpreserves and even enhances the non-monotonic character of DL by removing the needfor a separate truth maintenance system for non-monotonic belief revision. It isalso tolerant to contradictions within the belief set. A number of interesting aspectsrelated to degrees of belief, consistency and knowledge management arise in FDL.These topics along with several issues and their solutions have been discussed here indetail. It is also interesting to find that FDL is a generalization of DL, much the sameway as FL is a generalization of classical bi-valued logic.

As a practical example, FDL is very suitable for recognition of objects in a scene(real or synthetic) where under occlusion many features or important properties ofobjects may not be observable, though we want (and can afford) to jump to someconclusion about the scene. If at a subsequent stage, we find our recognition erroneous,we may go for belief revision. In real environments such features of objects are mostlyimprecise/vague/fuzzy in nature due to noise and other external disturbances, andhence FDL would be very effective. This is a specific application domain that has beenfound suitable for FDL, and further work on it would be reported in the subsequentfuture. Another specific target area could be a military application where we try tostrategize according to the incomplete and possibly imprecise data coming in fromdifferent regions of the battlefield. Using default rules, we can make approximatepredictions about overall enemy troop movements and engage our own forcesaccordingly. Since FDL can be made to handle different overlapping versions of reality,we can look for possible critical situations and take corrective measures, even if thepossibility is less. Different degrees of belief can be used to define thresholds for takingactions of varying severity.

Such real-world applications can benefit in several ways from the differentcharacteristics of FDL that we will try to bring out. This paper draws a lot fromAntoniou’s work in the domain of non-monotonic reasoning (Parsons and Hunter,1998), both in content and presentation. The syntax and semantics of our proposedformalism have been given first in Sections 2 and 3, followed by some special kinds ofdefaults in FDL – including those that are equivalent to traditional defaults. Section 5discusses properties of resulting knowledge sets (termed extensions) that we considerdesirable from the reasoning perspective. Section 6 then moves on to processes in FDL,which give us an operational model for defining extensions. Parameters thatcharacterize processes and extensions have been discussed in Section 7. A separatesection containing examples that illustrate the concepts introduced in this papercomes next. This is followed by Section 11 in which we give a set of results regardingextensions in FDL that make it sound and complete. The text concludes withsuggestions on how to obtain crisp knowledge by defuzzyfying the resultant belief set.

2. SyntaxWe shall first introduce the mathematical syntax of FDL, and define the basicsemantics associated with it.

Definition 1. Let:. U be our universe of discourse containing all possible pieces of knowledge;

its elements being predicate logic formulae.

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. W be a fuzzy set based on U with a membership function mW: x ! [0, 1] definedfor all x [ U; it represents the set of facts or axioms that are known initially.

. E be our “current” belief-set, also fuzzy, and initially equivalent to W,i.e. mE(x) ¼ mW(x);x [ U.

Now, a fuzzy default theory (FDT), T is defined as a pair (W, D) where W is the fuzzyset of initial facts, and D is a countable set of fuzzy defaults.

A fuzzy default d has the form:

ðw;aÞ : ðc1;b1Þ; ðc2;b2Þ. . .ðcn;bnÞ

ðx;gÞð1Þ

where, w, c1, c2, . . . cn [ U are closed predicate logic formulae; a, g [ [0,1] and b1,b2, . . . bn [ (0,1).

The term (w, a) is known as the pre-requisite of d, and is also denoted by pre(d).Similarly, (c1, b1), (c2, b2) . . . (cn, bn) are known as the justifications of d, and their setis denoted by just(d). And, (x, g) is known as the consequent of d, denoted by cons(d).

Note: If we consider our Universe of Discourse U to contain only ground formulae,then the formulae present in a fuzzy default (w, ci, and x) must also be ground. Openfuzzy defaults (that may have free variables) can then be explicitly represented byfuzzy default schema of the form:

ðws;aÞ : ðc1s;b1Þ; ðc2s;b2Þ. . .ðcns;bnÞ

ðxs; gÞð2Þ

Here, s represents any ground substitution that assigns values to all free variablesoccurring in the schema; and thus, a fuzzy default schema generates a set of fuzzydefaults, one for each ground substitution. In this paper, we use the term “fuzzy default”to refer to open fuzzy defaults, in general. Some specific examples may deal with groundformulae only, in which case, (w, ci, and x) must also be considered ground.

3. Basic semanticsUsing fuzzy terminology, the basic meaning of a fuzzy default d of the form defined inthe previous section can be expressed as the following rule.

If w is at least a-known (or a-true) and it is bi-consistent to assume ci (; i ¼ 1. . . n),Then conclude that x is at least g-known (or g-true).

Let us now mathematically define the condition for applicability of a fuzzy default,i.e. when it can be “fired”.

Definition 2 (applicability of a fuzzy default). A fuzzy default d ¼ (w, a):(c1, b1),(c2, b2). . . (cn, bn)/(x, g) is applicable to a deductively closed fuzzy set of formulae E iff:

. mE(w) $ a – [Satisfying the pre-requisite].

. mE( : ci) # (1 2 bi) ; i ¼ 1. . . n – [Consistency of justifications].

When a fuzzy default is applicable to the current knowledge-base E, its consequent is addedto E itself, thus “extending” our knowledge-base. The precise details of the process of beliefinduction in FDL shall be discussed in later sections. Let us first define some special kinds offuzzy defaults and study their analogy with defaults in traditional default logic.


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4. Some special fuzzy defaultsNormal defaults are a very important class of defaults in traditional default logic.A major part of our reasoning can be captured using normal defaults alone. In FDL too,there is a fuzzy form of a normal default.

Definition 3 (normal fuzzy default). A normal fuzzy default is of the form:

ðw;aÞ : ðc;bÞ

ðc;bÞð3Þ

i.e. the justifications and consequent of a normal fuzzy default are equivalent.As we have seen, the applicability of a fuzzy default is governed by the values of a

and b. Let us have a look at some special kinds of defaults, which exhibit interestingproperties due to restrictions placed on their a and b values.

Definition 4. Some more special fuzzy defaults are:

. Safe fuzzy default: a ¼ 1.

. Fully safe fuzzy default: a ¼ 1, and b1 ¼ b2 ¼ . . . ¼ bn ¼ 12 .

. Confident fuzzy default: g ¼ 1.

. Fully safe confident fuzzy default: a ¼ g ¼ 1, and b1 ¼ b2 ¼ . . . ¼ bn ¼ 12 .

Note:. In fully safe fuzzy defaults, we see that b1 ¼ b2 ¼ . . . ¼ bn ¼ 12 (the left-hand

limit of 1). This is because, all bi [ (0, 1) as per FDL’s syntax. Intuitively,condition c that needs to be 1-consistent is not really a justification, rather it is aprerequisite for the rule, and hence deserves to be included in w. On the otherhand, being 0-consistent implies that the condition is not important for that rule,and is as good as not being there.

The technical reasons why any b should not be allowed to assume a value ofexactly 0 or 1 arise from the fact that we want to keep track of all thejustifications using a fuzzy set – this will help us generalize certain notions moreeasily. As we shall see later in the section on processes, b ¼ 0 or 1 can causeproblems.

. In a FDT T ¼ (W, D), the set D may contain all kinds of defaults including fullysafe confident fuzzy defaults, which are equivalent to traditional defaults. Hence,D can have both fuzzy as well as non-fuzzy defaults. In this sense, FDL may beviewed as a generalization of classical default logic.

5. A discussion of extensions in fuzzyfied default logicExtensions are maximal possible world views that are based upon the given FDTs.Some desirable properties of extensions are listed below:

. An extension E should include the initial set of facts W since it contains explicitknowledge. As W is a fuzzy set, it is implied that E must also be fuzzy (asdeclared in the syntax section); and initially, mE(x) ¼ mW(x) ; x [ U, where U isour universe of discourse.

. E should be deductively closed. Thus, E ¼ Th(E), where Th denotes deductiveclosure, as described below.

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Just like closure operators in topology, closure operators in algebra, partially orderedsets, etc. we can have a closure operator in logic. Suppose, we have some logicalformalism that contains certain rules allowing us to derive new formulae from givenones.

Consider the set F of all possible formulae and let P be the power set of F. For a set offormulae X [ P, let T h(X) be the set of all formulae that can be derived from X. Thus,“T h” is a closure operator on P, since T h(X) [ P. Also, T h is maximal in the sensethat Th(X) ¼ Th(Th(X)), and clearly inclusive such that X # Th(X). A generalapproach towards logic based on closure operator theory is proposed in (Brown andSuszko, 1973) and (Tarski, 1956). Similar ideas in logic programming (Lloyd, 1987), FL(Gerla, 2000) and default logic (Etherington, 1987) have also appeared.

In our present work, we interpret the deductive closure Th(E) in the same sense(Antoniou, 1999) uses it for classical logic. So E ¼ Th(E) means everything deducible fromE in the sense of fuzzy reasoning – simply because we want to draw more conclusions.The term fuzzy reasoning is a generalization of classical reasoning:

. E should be closed under the application of the fuzzy defaults in D; i.e.If:

d ¼ðw;aÞ : ðc1;b1Þ; ðc2;b2Þ. . .ðcn;bnÞ

ðx; gÞ [ D;

mEðwÞ $ a and;

mEð: ciÞ # ð1 2 biÞ;i ¼ 1. . .n

Then:mE(x) $ g.

Now we have an idea about what extensions in FDL should be like. But before definingextensions formally, we need to study the concept of a “Process”. This will help us arriveat an operational definition of extensions, which would be much more comprehensiveand will lend itself more directly to the development of practical systems as compared toa fix-point notation. The approach we take to characterize extensions using desirableproperties, and then construct an operational definition for them, follows the structureand style of Antoniou’s tutorial paper (Antoniou, 1999).

6. Processes of fuzzy default theoriesGiven a FDT T ¼ (W, D), let P ¼ (d0, d1, d2, . . .) be a finite or infinite sequence of fuzzydefaults from D without multiple occurrences.

Two fuzzy sets of formulae In(P) and Out(P) can be associated with P:. In (P) – Fuzzy set of formulae that should be in our knowledge base:

InðPÞ ¼ ThðW < {consðdÞjd [ P}Þ ð4Þ

where cons(d) ¼ (x, g), the fuzzy consequent of d, and W < {cons(d)} denotesmIn(P)(x) ¼ max(mIn(P)(x), g), i.e. an update of the truth-value.

In (P) collects all the knowledge gained by the application of defaults in P. It alsocorrects previous conjectures by revising the degree of truth associated with factswhen newer information is available. And as clear from the definition above, it is


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deductively closed, i.e. In(P) ¼ Th(In(P)). Thus, In(P) represents our currentknowledge base.

In traditional default logic, In(P) ¼ Th(W < {cons(d)jd [ P}) means a simpleunion of the new consequents with the existing set of world knowledge. In FDL, theknowledge sets are fuzzy. As a result we apply fuzzy union, which corresponds totaking the maximum of the existing truth value and the new consequent’s value in eachcase. Since we are only updating the truth-value, and not explicitly adding a new fact toour belief set, we can say that FDL performs belief revision somewhat implicitly.

The choice of fuzzy union (maximum of truth values) instead of just replacing oldtruth values with new ones has the advantage of guaranteeing that all the initial worldknowledge in W is included in the current belief set at all times.

. Out(P) – Fuzzy set of formulae that must be kept out:

OutðPÞ ¼ ð: c; 1 2 bÞjðc;bÞ [d[P< justðdÞ

� �: ð5Þ

This set collects all the formulae ( : c) that should not turn out to be true – i.e. shouldnot become a part of In(P) – beyond a certain degree (1 2 b), otherwise the set In(P)would become inconsistent.

Recall that a fuzzy default d ¼ (w, a):(c1, b1), (c2, b2). . . (cn, bn)/(x, g) is applicableonly when m(w) $ a and m( : ci) # (1 2 bi) ; i ¼ 1. . .n.

If a justification (c, b) is assumed to be true during the application of some default,and at a later stage, the application of some other default results in the addition of a fact( : c, b0) to In(P) and b0 . (1 2 b) then it would invalidate our previous conjecture bycontradicting a justification we assumed earlier. Such contradictions can be identified,and possibly prevented, using the set Out(P).

Since Out(P) is a fuzzy set, all justifications (c, b) must have a b-value such that(1 2 b) lies within 0 and 1, which implies 0 # b # 1. Now b ¼ 1 means that c is ofutmost importance, but then : c will have degree (1 2 b) ¼ 0 in Out(P) which iscounterintuitive ( : c will never be added to Out(P) since a membership value of 0 in afuzzy set implies no membership at all). And b ¼ 0 is useless, as (1 2 b) ¼ 1 makesthe justification always consistent with the current belief set: mE(c) # 1 ; c [ E.

So essentially, we need 0 , b , 1, as noted earlier in Section 4.We have seen till now that In(P) and Out(P) are two fuzzy sets of formulae

associated with a sequence of defaults P ¼ (d0, d1, d2. . .). Let P[k] be a segment of P oflength k starting from d0 onwards.

Hence, P[0] ¼ (), P[1] ¼ (d0), P[2] ¼ (d0, d1). . . and so on. This leads us to aninteresting definition of a Process.

Definition 5. P is a Process of a FDT T ¼ (W, D) iff dk is applicable to In(P[k]) forevery k such that dk occurs in P.

Applicability of dk is governed by the familiar fuzzy default conditions with respectto the set In(P):

mlnðP½k�ÞðwÞ $ a and mlnðP½k�Þð: ciÞ # ð1 2 biÞ;i ¼ 1. . .n :

On applying default dk, belief revision takes place in two ways. First, its con-sequent isadded to our knowledge base: In(P[k þ 1]) ¼ Th(In(P[k]) < {cons(dk)}). Note thatIn(P[0]) ¼ T h(W). The deductive closure ensures that our knowledge base is maximal.

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Second, its justifications are collected in the Out-set. Let the set of justificationsaccumulated from previous defaults (before applying dk) be recursively defined as:

JUSTðP½k�Þ ¼B if k ¼ 0

JUSTðP½k2 1�< just ðdk21Þ if k $ 1Þ:

(

An alternative definition would be JUSTðP½k�Þ ¼ {ðc;bÞ [ <dk[P½k� justðdÞ}.The revised Out-set can then be defined as:

OutðP½k þ 1�Þ ¼ {ð: c; 1 2 bÞjðc;bÞ [ ðJUSTðP½k þ 1�Þ¼ JUSTðP½k�Þ< justðdkÞÞ}:

Taking the union of justifications ensures that we consider the maximum b value forany c that occurs in more than one justification. Then we store the negation : c witha degree 1 2 b in the Out-set, which is the minimum or most constrained degreethat must not be crossed in the In-set. One important thing to note here is that we donot take the deductive closure when defining the Out-set. This is because the Out-set isnot our knowledge base, and therefore should not be made subject to any inference. It isonly a collection of forbidden facts, maintained as a fuzzy set. The reason why it mustbe a fuzzy set will be clear when we consider parameters to qualify processes. It helpsin showing that FDL is a generalization of the bi-valued case.

So, a process P represents a possibly applicable sequence of fuzzy defaults. Thisdoes not, however, guarantee that the justifications of a default shall not be invalidatedby the result of a subsequent default (when adding a consequent to the In-set, we do notcheck whether it is forbidden as per the Out-set). Also, a process by itself need notnecessarily be closed under the application of defaults in D. Hence, we come to identifytwo important classes of processes.

Definition 6 (closed processes). A process P is closed if every d [ D that isapplicable to In(P) as already present in P.

Definition 7 (successful processes). A process P is said to be successful whenmIn(P)(u) # mOut(P)(u) for every u such that mOut(P)(u) . 0. Otherwise, the process P istermed as failed.

Semantically, this means that a process P should only be called successful whenevery fact “forbidden” from our knowledge base (mOut(P)(u) . 0) has a degree of truthin In(P) that is less than the threshold set in Out(P). It is natural at this point to expressa desire for processes in which all forbidden facts are completely absent from ourknowledge base In(P). Such strictly successful processes can be characterized by thecondition: mIn(P)(u) ¼ 0 for every u such that mOut(P)(u) . 0.

Since both In(P) and Out(P) are fuzzy sets, the above condition is equivalent to:

minðmInðPÞðuÞ;mOutðPÞðuÞÞ ¼ 0;u

which is in fact, the fuzzy intersection condition:

mInðPÞðuÞ> mOutðPÞðuÞ ¼ B:

Thus, a strictly successful process corresponds to the notion of a “successful process”as per traditional default logic. On further contemplation we realize that theblack-and-white classification of FDL processes as simply successful or failed


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seriously limits the possibilities that fuzzyfication presents us with. To realize theadvantages of using fuzzy facts, we must define several “fuzzy” parameters to betterqualify FDL processes.

7. Parameters to qualify FDL processesDefinition 8 (degree of success). Given a process P, the degree of success of a fact u isdefined as

Su ¼ mOutðPÞðuÞ2 mInðPÞðuÞ if mOutðPÞðuÞ $ mInðPÞðuÞ . 0

¼ 1 if mOutðPÞðuÞ . mInðPÞðuÞ ¼ 0

¼ undefined otherwise:

In this sense, “success” represents how far a forbidden fact has been kept away fromour current knowledge base. Using Su as our basic building block, we can define thefollowing quantities that represent the parameter “degree of success” of an entireprocess P.

For all u [ P such that mOut(P)(u) $ mIn(P)(u) and mOut(P)(u) . 0:. Minimum degree of success: Smin ¼ min(Su).. Maximum degree of success: Smax ¼ max(Su).. Average degree of success: Savg ¼ S(Su)/NS.

where NS is the number of us in P with a defined degree of success.Other statistical measures may also be employed to further analyze the success of

a process. The following fuzzy set can be used for this purpose:

{ðu; SuÞju [ P andmOutðPÞðuÞ $ mInðPÞðuÞ andmOutðPÞðuÞ . 0}:

Definition 9 (degree of failure). The previous (degree of success) takes into account onlythose forbidden facts that have not been included in In(P) beyond the threshold set inOut(P), i.e. for all u such that mIn(P)(u) # mOut(P)(u). What about the other facts thatviolate this condition? They have been left out of our calculation of degree of success forreasons that shall be justified later. But the impact of such facts on our knowledge basecannot be neglected, and thus they deserve a separate parameter degree of failure:

Fu ¼ mInðPÞðuÞ2 mOutðPÞðuÞ if mInðPÞðuÞ . mOutðPÞðuÞ . 0


Here, “failure” represents the degree by which the truth-value of a fact u in In(P) hassurpassed its forbidden threshold set in Out(P). Note that the degree of failure isundefined for those facts that do not break this threshold. Failed processes in traditionaldefault logic are of almost no value as they contain serious contradictions between newconjectures and previously assumed justifications. But in FDL, a slightly failed processmay still be of interest and may even assume a successful status upon belief revision bysubsequent defaults.

The overall degree of failure of a process P can be represented by quantitiesanalogous to those used for the degree of success.

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For all u [ P such that mIn(P)(u) . mOut(P)(u) . 0:. Minimum degree of failure: Fmin ¼ min(Fu).. Maximum degree of failure: Fmax ¼ max(Fu).. Average degree of failure: Favg ¼ S(Fu)/NF.

Where NF is the number of us in P with a defined degree of failure.Definition 10 (degree of contention). Unlike traditional logic in which the truth of a

fact implies the falsehood of its negation, facts in FDL and their negations may existsimultaneously with independent degrees of membership in the current knowledge set.The difference between the degrees of truth of two opposite facts indicates how closelythey contend for a superior position. Lesser the difference, closer the competition.Obviously, the truth with a higher degree must be favoured or believed, but thenearness of its opposite can give us a clue as to how “delicate”, “precarious” or“ambiguous” that belief is.

The degree of contention for a single fact u can be expressed as follows, consideringIn(P) to be our current belief set:

Cu ¼ mInðPÞðuÞ2 mInðPÞð: uÞ if mInðPÞðuÞ $ mInðPÞð: uÞ . 0


Notice that Cu is defined for pairs of opposite facts taking the positive differencebetween their truth-values. This takes into account all the facts in our currentknowledge set that face contradiction, while also preventing negative values.

The overall degree of contention of a process P can be measured using the followingquantities.

For all u [ P such that mIn(P)(u) $ mIn(P)( : u) . 0:. Minimum degree of contention: Cmin ¼ min(Cu).. Maximum degree of contention: Cmax ¼ max(Cu).. Average degree of contention: Cavg ¼ S(Cu)/NC.

where NC is the number of us in P with a defined degree of contention.

Justification for defining three different parametersIt can be argued that there was no need for separate parameters “degree of success” (S)and “degree of failure” (F), as they are like two sides of the same coin; instead, a singleoverall degree of success (say, S

0

) of a process could have been defined, intuitivelyrelated to S & F as: S

0

¼ S 2 F.To see if a single parameters would be sufficient or not, let us consider two

processes P1 & P2 with their average measures of success and failure:

S1avg ¼ 0:3;F1avg ¼ 0; S2avg ¼ 0:4;F2avg ¼ 0:1:

Now, the overall average degree of success for each process would be:

S01 ¼ 0:3 2 0 ¼ 0:3andS0

2 ¼ 0:4 2 0:1 ¼ 0:3:

Thus, we see that a combined degree of success S0 qualifies both P1 & P2 to be ofan equal degree and loses the essential information that P1, though a little uncertain,is still consistent (successful), whereas P2 is inconsistent (a failed process) even if someconjectures resulting from it have a greater degree of certainty.


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Table I shows which kinds of facts are included in the degree of success and whichones in the degree of failure. Notice that their domains are mutually exclusive. It is alsoimportant to note that neither of the two parameters are defined for facts which havenot been forbidden to any degree (mOut(P)(u) ¼ 0). This agrees with our interpretationof success/failure, which we have defined in this text to consider only those facts thathave some degree of membership in Out(P).

The need for a third parameter “degree of contention” arises from the observationthat the previous two parameters only deal with consistencies/contradictions betweenthe in-set and out-set. They say nothing about contradictions within our currentknowledge base, the In-set. As mentioned earlier, fuzzy sets allow a fact and itsnegation to exist simultaneously, and the difference between their membership/truthvalues can be interpreted as a measure of how close the two opposing facts are to beingbelieved as true. Lesser the degree (difference), higher the contention; and when thedifference is zero, it indicates perfect ambiguity. In this sense, Degree of contention isdifferent from the other two in which a higher value implies a greater degree.

Suggestion for a fourth/alternative measureApart from the three measures (minimum, maximum and average) for each parameter, theneed for a fourth measure – one that represents a weighted value/weighted mean – can beanticipated. For example, consider the degrees of contention of two facts u1 & u2:

LetmInðPÞðu1Þ ¼ 0:2; mInðPÞð: u1Þ ¼ 0:1

and;mInðPÞðu2Þ ¼ 0:9; mInðPÞð: u2Þ ¼ 0:8:

[Cu1¼ 0:2 2 0:1 ¼ 0:1

and; Cu2¼ 0:9 2 0:8 ¼ 0:1:

We see that this measure indicates bothu1 andu2 are equally contradicted truths. But if wethink logically, u1 is a fact that has not been conjectured with much certainty (mIn(P) (u1) isonly 0.2); ambiguity about a “weak” fact like this is not of much importance. Whereas, u2 isa fact which has been strongly supported (mIn(P)(u2) is 0.9, quite high, and the same is truefor its negation (mIn(P)( : u2) ¼ 0.8). u2 represents a peculiar situation, and its degree ofcontention Cu2

is clearly worth more attention and weight-age than Cu1.

One possible way of defining this fourth weighted measure for a given fact u is tomultiply the difference measure (difference between two truth values) with the greaterof the two values. For degree of failure, this weighted measure, say WFu, would looklike:

Condition Success/failure Expression

mOut(P)(u) . mIn(P)(u) . 0 Success Su ¼ mOut(P)(u) 2 mIn(P)(u)mOut(P)(u) ¼ mIn(P)(u) . 0 Success Su ¼ mOut(P)(u) 2 mIn(P)(u) ¼ 0mOut(P)(u) . mIn(P)(u) ¼ 0 Success Su ¼ 1mIn(P)(u) . mOut(P)(u) . 0 Failure Fu ¼ mIn(P)(u) 2 mOut(P)(u)mOut(P)(u) ¼ 0 None Both Su & Fu undefined

Table I.Domains of the degrees ofsuccess and failure

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WFu ¼ ½mInðPÞðuÞ2 mOutðPÞðuÞ� £ mInðPÞðuÞ if mInðPÞðuÞ . mOutðPÞðuÞ . 0


Similarly, we can define WSu and WCu. Since the original definition of Su creates adiscontinuity when mOut(P) . 0 and mIn(P) ¼ 0, we can define a weighted successmeasure as follows to take it continuous:

WSu ¼ ½mOutðPÞðuÞ2 mInðPÞðuÞ�=mOutðPÞðuÞ if mOutðPÞðuÞ $ mInðPÞðuÞ . 0


In case of the degree of contention, we must take into account the fact that a lesser valueof Cu represents greater contention. To preserve this relationship for the weightedmeasure WCu, we should complement the value that acts as a weighing factor:

WCu ¼ ½mInðPÞðuÞ2 mInðPÞð: uÞ� £ ð1 2 mInðPÞðuÞÞ if mInðPÞðuÞ $ mInðPÞð: uÞ . 0


The aim here is to keep all these measures for the various parameters (degrees) withinthe interval [0, 1] so that pairs of facts and their corresponding parameters can be used toconstruct fuzzy sets for further analysis. A weighted measure could then be averagedover the entire set to give a weighted mean which would be better representative of theparameter under consideration for an entire process.

8. Process tree of a fuzzy default theoryAll possible processes of a FDT T ¼ (W, D) can be arranged in a tree-like form thatbranches upon the application of each default. The root of the tree would represent ourinitial state, when none of the defaults have been applied. Each node would representour current knowledge base, labeled with the In-set to the left and the Out-set to theright. A path from the root to any node represents a process containing the defaultsindicated by the edges (branches) traversed along the path, and ordered according tothe sequence of traversal from root to node. Each node may be labeled with one or moreof the parameters (degrees) to trace their changing values along the sequence ofdefaults belonging to a process.

The process tree for a given FDT T ¼ (W, D) would have the structure shown inFigure 1.

9. An operational definition of extensions in FDLAfter having dealt with processes at length, we can arrive at an operational definitionof extensions in FDL.

Definition 11 (extension). A fuzzy set of formulae E is an extension of the FDTT ¼ (W, D) if there is some closed process P of T such that E ¼ In(P).

When a maximal possible process tree has been constructed, the leaf nodes denoteclosed processes (closed under the application of defaults in D), which correspond toextensions of T. These processes, and corresponding extensions, may or may not besuccessful.


15

In fact, extensions can possess varying degrees of failure, success and contention –parameters that are inherited from the processes, they are associated with. Anextension E ¼ In(P) associated with a strictly successful process P can be called astrict ex-tension, which corresponds to the notion of an extension in traditional defaultlogic. A strict extension has a degree of success equal to 1, and no degree of failure.Degree of contention, however, may vary.

10. Some examplesExample 1 (illustrating the degrees of failure, success and contention)Let us consider the now classic problem of conjecturing whether Tweety can fly or not,based upon the following FDT:

T ¼ ðW;DÞ

where:

D ¼ {ðbirdðXÞ; 0:75Þ : ðfliesðXÞ; 0:3Þ=ðfliesðXÞ; 0:8Þ}; ðd1Þ

ðpenguinðXÞ; 0:8Þ; : ð: fliesðXÞ; 0:1Þ=ð: fliesðXÞ; 0:9Þ} ðd2Þ

and:

W ¼ {ðbirdðTweetyÞ; 0:8Þ; ðpenguinðTweetyÞ; 0:8Þ}:

This FDT has two extensions corresponding to the two closed processes:

(1) P1 ¼ (d1, d2)E1 ¼ In(P1) ¼ W < {(flies(Tweety), 0.8), ( : flies(Tweety), 0.9)}Out(P1) ¼ {( : flies(Tweety), 0.7), (flies(Tweety), 0.9)}.

(2) P2 ¼ (d2)E2 ¼ In(P2) ¼ W < {( : flies(Tweety), 0.9)}Out(P2) ¼ {(flies(Tweety), 0.9)}.

Here both processes are closed, but P1 is a failed process, whereas P2 is not. Thepresence of d2 in P1 makes it closed (since it is applicable); and the presence of d1 does

Figure 1.The process tree of a FDT

W

In (Π1) In (Π2)Out (Π1) Out (Π2)

d1

∅

d2

d2 d3 d1 d3

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not affect this property. In case of P2, d2 is applied first – it fulfills the closed propertysince d1 is no longer applicable, and it also prevents failure. It is interesting to note thatin both extensions, m( : flies(Tweety)) is equal (0.9); the difference being that E1 showssome confusion regarding the fact. The following from P1, that also characterize E1:

For u1 ¼: fliesðTweetyÞ;mInðP2Þðu1Þ . mOutðP1Þðu1Þ . 0:

Hence;Degree of Failure : Fu1 ¼ mInðP1Þðu1Þ2 mOutðP2Þðu1Þ ¼ 0:9 2 0:7 ¼ 0:2:

This represents the extent or margin by which the truth-value of the forbidden fact u1

in our belief set In(P1) has surpassed the threshold set in Out(P1):

For u2 ¼ fliesðTweetyÞ;mOutðP1Þðu2Þ . mInðP1Þðu2Þ . 0:

Hence;Degree of Success : Su2 ¼ mOutðP1Þðu2Þ2 mInðP1Þðu2Þ ¼ 0:9 2 0:8 ¼ 0:1:

This represents the extent or margin by which we have successfully kept the forbiddenfact u2 from being included in our belief set In(P1).

Similarly, the degree of contention for u1 is:

Cu1¼ mInðP1Þðu1Þ2 mInðP2Þð: u1Þ

¼ mInðP1Þð: fliesðTweetyÞÞ;2mInðP2ÞðfliesðTweetyÞÞ

¼ 0:9 2 0:8 ¼ 0:1:

In case of P2, there is no inconsistency between the In(P2) and Out(P2) sets, thusmaking it a successful process. This implies that the degree of failure is not defined forany fact, and that the degree of success Su ¼ 1 (for u ¼ f lies(Tweety)). Also, sincethere is no contradiction between facts within the In(P2) set, the degree of contentionalso remains undefined.

We can learn some intriguing aspects of the extensions of a FDT from this example.First, though P1 is a failed process, its corresponding extension E1 indicates a closecontention between two opposing beliefs, f lies(Tweety) & : flies(Tweety), out ofwhich the latter emerges stronger. With this we can draw meta-level inferences aboutthe fact that Tweety does not fly. It is not the case that the fact itself is weak in ourknowledge base (it is 0.9-true); but, since its negation also has a high degree of truth(0.8), we can think of it as saying: “We must conclude Tweety does not fly, though thereis strong support for the contrary”. A small difference in value of either of the factsmay have led us to believe the opposite – the purpose of having a degree of contentionis to preserve such small yet important details.

Second, we can see that both P1 and P2 ultimately lead to the same overallconclusion (that Tweety does not fly) with the same degree of truth (0.9), and the failedprocess P1 is actually better able to capture the inherent complexity of reasoning inthis example that is lost in P2. This reinforces the need to give due consideration tofailed extensions in FDL, unlike traditional default logic in which extensions, bydefinition, cannot correspond to failed processes. In fact, a failed extension can increasethe expressiveness of our conclusions, as we have already seen above.

Let us now move on to an example that will illustrate revision of initial knowledgein FDL.


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Example 2 (belief revision characteristic of FDL)We are considering here, the problem of deciding if it is daytime based upon ourobservations from within a room. The basic idea being expressed here is that we canconjecture to a certain extent that it is daytime if we sense some light in the room. But ifwe spot a lamp (or some other artificial object) as the chief source of light, then we mustretract our knowledge (conjecture) about it being daytime. Then again, if we spot lightcoming from the sun (say, through a window) then we can be pretty sure its daytime.

Following a set of three fuzzy defaults that together express this logic:

D ¼ ðLight; 0:5Þ :ðDay; 0:3Þ

ðDay; 0:8Þ

�; . . .d1

ðLightFromLamp; 0:8Þ :ð: Day; 0:15Þ

ðDay; 0Þ; . . .d2

ðLightFromSun; 0:8Þ :ðDay; 0:1Þ

ðDay; 0:9Þ

�: . . .d3

Now we shall study several cases with different initial sets of facts (W’s). Thecorresponding processes (P), extension sets (E ¼ In(P)) and the Out(P) sets are shownin Table II.

Case #2 is worth special mention here. The fact (Day, 0.6) has been included in ourinitial set of facts W as a preliminary piece of information. Its truth value increasesfrom 0.6 to 0.8 on the application of d1. Thus, initial knowledge in the system is revisedaccording to new information. Now d2 is still applicable, but its application does notchange truth value of day. The effect of d2 is not lost, however. It drives the degree ofsuccess of day down to 0.05. In this manner, FDL manages to retain all initialknowledge in its belief set, while also preserving finer details regarding the inferenceprocess. Note that, this kind of complex belief revision takes place in FDL withoutrequiring the help of any other external knowledge-management framework.

Case #3 models the situation where d1 is applied first, and then d3. This helps usfirst conclude that it is a day with a truth value of 0.8, then finally with a value of0.9 ðmInðP2Þ ðDayÞ ¼ 0:9Þ. It consequently blocks d2, whose pre-requisite is satisfied by(LightFromLamp, 0.9) but the justification requires:

mInðP3Þð: ð: DayÞÞ # ð1 2 0:15Þi:e:; mInðP3ÞðDayÞ # 0:85

W E ¼ In(P) Out(P)

Case #1:P1 ¼ (d1)

{(Light, 0.7)} {(Light, 0.7), (Day, 0.8)} {( : Day, 0.7)}

Case #2:P1 ¼ (d1, d2)

{(Light, 0.9), (Day, 0.6),(LightFromLamp, 0.9)}

{(Light, 0.9), (Day, 0.8),(LightFromLamp, 0.9)}

{( : Day, 0.7),(Day, 0.85)}

Case #3:P1 ¼ (d1, d3)

{(Light, 0.9), (LightFromLamp,0.9), (LightFromSun, 0.9)}


{( : Day, 0.7)}

Case #4:P1 ¼ (d1, d2, d3)



{( : Day, 0.7),(Day, 0.85)}

Table II.Extensions found usingdifferent initial sets

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Thus, (d1, d3, d2) is not a valid process. This effectively conveys the reasoning thatonce we have spotted sunlight we are sure it is day, no matter if we spot a lamp later onor not.

Case #4 includes the application of all three defaults. It is another possible way inwhich inference could be made starting with the same set of initial facts as case #3.The truth value of Day changes from 0.8 to 0.9, as in case #3. Interestingly, theintermediate application of d2 does not affect that value (because of fuzzy union), butagain introduces some lack of success. This finally results in failure of the processwhen d3 is applied (Note that there are other possible processes with the same initialknowledge set as cases #3 and #4, but they all result in one of these two extensions).

E1 and E3 are strict extensions (degree of success ¼ 1, and no failure). E2 has areduced degree of success, as mentioned earlier, but it is still not failed. E4, however, isa failed extension with degree of failure:

Fu ¼ mInðP4ÞðDayÞ2 mInðP4ÞðDayÞ ¼ 0:9 2 0:85 ¼ 0:05 for u ¼ Day:

As mentioned in the earlier example, failed extensions like this should not bediscarded. In fact, E4 indicates a slight confusion in the course of reasoning whichcorrectly represents our intermediate ambiguity about the truth of day.

Example 3 (A trivial FDT)Now we shall consider a FDT which does not result in any useful extensions. In thisexample we try (unsuccessfully) to model the notion that when we are told a politicianis “Good”, if there is a possibility that the information came from some unreliablesource, then we should be cautious and reduce the truth value:

Fuzzy default theory T ¼ ðW;DÞ

where:

D ¼ ðGoodPolitician; 0:5Þ :ð: ReliableSource; 0:5Þ

ðGoodPolitician; 0:2Þ

� �

W ¼ {ðGoodPolitician; 0:7Þ}:

The sole default in this FDT gets fired because there is no reason not to believe that theinformation came from an unreliable source, but its consequent has less degree ascompared to the initial world knowledge. Thus, it does not add any useful information,and E ¼ W. Notice that our intended goal has not quite been realized, since extensionsof a FDT must contain the initial world knowledge. A simple conclusion can be drawnfrom here: If the prerequisite and consequent have identical facts, and the degree of theconsequent is less, then such a default will not add anything to the knowledge base; butthe Out-set could still be affected.

Such situations are rare and easy to detect; it is safe to assume that most FDT’s willhave at least one non-trivial extension. Still, this point requires further contemplation,and is part of our next discussion.


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11. A closer look at extensionsIn Section 5, we discussed some desirable properties of extensions. Now that we havedefined extensions using the notion of processes, let us show that these extensionsactually possess these characteristics. Consider a FDT T ¼ (W, D).

Theorem 1. If E is an extension of T, then W # E.Proof. LetP be a process associated with E such that E ¼ In(P). Recall the definition:

InðPÞ ¼ Th W < consðdÞjdf g [ P� �

:

Regardless of whetherP is an empty sequence or not, all the facts inWwill have greater orequal degree in E than they had in W. Let X ¼ W < {cons(d)jd [ P}. By definition offuzzy union,mX(g) $ mW (g);(x,g) [ {cons(d)jd [ P}, soW # X (see Zadeh, 1965 forthe notion of fuzzy set containment). Now deductive closure impliesX # Th(X) and sinceE ¼ In(P) ¼ Th(X), we have X # E. Thus, W # X # E. A

Theorem 2. Any extension of T is deductively closed: E ¼ Th(E).Proof. Consider again a process P such that E ¼ In(P). And In(P) ¼ Th(X) where

X ¼ W < {cons(d)jd [ P}, as per the definition noted in the previous proof. SinceTh denotes the maximal set of formulae that can be derived from X, we haveTh(X) ¼ Th(Th(X)). Therefore, E ¼ In(P) ¼ Th(X) ¼ Th(Th(X)) ¼ Th(E). A

Theorem 3. Any extension E of T is closed under the application of defaults in D.Proof. By Definition 11, E ¼ In(P) implies P is a closed process. And from

Definition 6 (closed processes), we know that every d [ D that is applicable to In(P) isalready present in P. Hence, E is closed under the application of defaults in D.

The above results show that extensions form a sound and complete foundation forreasoning in FDL. Now, we shall study the question of existence of extensions.Consider a non-trivial FDT T ¼ (W, D) such that among W is non-empty; i.e. W – Ø.

Now, let us enlist possible cases according to the contents of D:

Case A. If D ¼ Ø

Then E ¼ Th(W) since there are no defaults to be applied.

Case B. If D – Ø

Then we have two sub-cases:

Case B1. If ’d [ D such that d is applicable to Th(W)

Then X ¼ W < {cons(d)} such that X # E where E is an extension of T.

Case B2. Else E ¼ Th(W).

We can see that in each case there exists a non-empty extension E. In case B1 (whichshould be most common), we can have multiple extensions. So we can conclude fromthe above that most FDT’s will have extensions (with varying degrees of success andfailure); only in certain trivial cases, when our initial belief-set is empty do we gettheories that would not produce any extensions. A

12. Defuzzyfication of extensionsThe ultimate goal of an artificially intelligent system implementing FDL might be toobtain crisp facts that it can assert as true or false. As with any other fuzzy system,these results can be extracted by defuzzyfication after applying all our “rules”

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(defaults) on the initial knowledge-set. In FDL, this would mean defuzzyfying anextension that we obtain after processing an entire FDT.

Given an extension E of a FDT T ¼ (W, D), the simplest and most exhaustive crispset of beliefs that we can obtain is the support of E, denoted by supp E. But choosing thisset has inherent problems. For example, consider an extension E ¼ {(p, 0.8), ( : p, 0.4),(q, 0.1)}.

Taking the support of E, we have our crisp belief set F ¼ supp E ¼ {p, : p, q}.This clearly violates our notion of a crisp belief set in which two opposing facts

(p & : p) cannot co-exist. Moreover, we lose information regarding which one out ofp & : p was stronger in our fuzzy extension; hence, we cannot rationally eliminateeither one. Also, the fact q has been included in F, even though it had a very low truthvalue (0.1) in E. It is natural to think that we may want to exclude such weak facts fromour final belief set.

These considerations lead us to two distinct problems and their subsequent solutions.

Eliminating weak factsThe simplest approach will be to take an a-cut (Fa) of the fuzzy extension set (E) withthe value of a determining at least how strong a fact must be in order to remain in thecrisp belief set:

Fa ¼ {wjmEðwÞ $ a}:

Specifying an absolute a-value as above might be too strict in certain cases. Onesolution to this issue is to take an a-cut of the normalized extension, which can beobtained by scaling the truth values of all facts such that the maximum value becomesunity (i.e. the extension becomes a normal fuzzy set):

EN ¼ {ðw;bÞjðw;aÞ [ E andb ¼ a=maxðmEÞ}

or;mENðPÞ ¼ mEðwÞ=gwhere g $ mEðwÞ;w:

Taking the a-cut of the normalized extension would keep only those facts in our crispbelief-set that are at least a-times as true as the fact that is most true:

FNa ¼ {wjmENðwÞ $ a}

¼ {wjmENðwÞ $ ða £ gÞ} whereg $ mEðwÞ;w

Several such a-cuts may be obtained from a single extension E representing variouslevels of credulousness.

Handling co-existence of negationsAs shown in our previous example, mutually negating facts (such as p & : p) cannotbe kept in a single crisp belief set. This issue poses a problem considerably greaterthan that of eliminating weak facts, and may warrant the adoption of a radicalapproach. We may solve this problem by obtaining an intermediate fuzzy set thateliminates all such facts, whose negation is stronger than the fact itself:

E0 ¼ {ðw;aÞjðw;aÞ [ E and mEð: wÞ , a}:

Proceeding with a simple or normalized a-cut of E0, we can obtain a consistent crispbelief-set. But even this solution has a drawback. We entirely lose our knowledge of


21

contention between opposing facts. If we wish to retain such knowledge, we may haveto partition the extension E into two fuzzy sets: STRONGE & WEAKE, where STRONGEwill hold the stronger or un-challenged facts and WEAKE shall contain the weakercontenders:

STRONGE ¼ {ðP;aÞjðw;aÞ [ E andmEð: wÞ , aÞ:

WEAKE ¼ {ðw;aÞjðw;aÞ [ E andmEð: wÞ . a}:

Yet another set might be required to hold the facts that are equivocal inE: EQE ¼ {(w, a)j(w, a) [ E and mE(w) ¼ mE( : w) ¼ a} where w is required to be anon-negated formula (to prevent duplicates).

Note that STRONGE, WEAKE and EQE are mutually exclusive, and exhaustive withrespect to E, i.e.:

E ¼ ðSTRONGE < WEAK E < EQ EÞ; and ðSTRONGE > WEAK E > EQ EÞ ¼ B:

One can use only STRONGE for simple and straightforward reasoning; or STRONGE alongwith WEAKE for reasoning with greater expressivity. EQE can be used to showambiguous conjectures.

The next step after partitioning E would be to take a-cuts to obtain one or morecrisp belief-sets. A word of caution: if normalized a-cuts are desired, then thenormalization step must be carried out before partitioning. Also, the same value of amust be used to cut all three partitions (STRONGE, WEAKE & EQE), otherwise the threecrisp sets obtained would not really be meaningful.

It is easy to see that the a-cuts of a partitioned extension (simple or normalized) willnecessarily be consistent crisp belief-sets, and that they will be able to preserve theexpressivity of the original fuzzy extension to an acceptable degree. With this, we cansatisfactorily conclude our discussion over defuzzyfication of the results obtained froma FDT.

13. ConclusionFDL, a new framework for reasoning in an imprecise environment, has been introducedin this paper. Intriguing new concepts that were not applicable to traditional defaultlogic, such as degrees of success, failure and contention, the importance of failedextensions, the implicit nature of belief revision, and the ability of FDL to exhibitcomplex reasoning have been studied. Detailed examples have been given to illustratethese characteristics.

Further work on FDL is necessary, especially in studying the extensions of aFDT for better and more meaningful inference. A software implementation ofFDL is highly desirable as computing the extensions of large FDTs is tedious. Othersystem-dependent parameters may then be designed to qualitatively evaluateprocesses and extensions, and be examined for applicability. Such software will alsohelp us properly investigate priorities among defaults, by illustrating the exact effectsof changing the order of their application.

One particular improvement that is being planned as a future addition to FDL is theability to handle multiple contending facts. As of now, FDL only recognizes a fact and itsnegation ( p& : p) as contending or contradicting facts. Consider the facts: Indian(John),

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English(John), and Japanese(John). Given that John can belong to only one nationality,it is clear to a human that these facts contradict each other. But it is not possible atthis time to represent, in FDL, more than two facts as mutually contradicting, withoutintroducing rules of the form: Indian(John) ! : English(John) ^ : Japanese(John).

As such explicit rules are somewhat against the spirit of default reasoning, we arelooking for more intuitive ways to express multiple contention, perhaps using specialdefault rules.

References

Antoniou, G. (1999), “A tutorial on default logics”, ACM Computing Surveys, Vol. 31 No. 3,pp. 337-59.

Benferhat, S., Dubois, D. and Prade, H. (1999), “Towards fuzzy default reasoning”,Proceedings of the 18th International Conference of NAFIPS, New York, NY, USA,pp. 23-7.

Brown, D.J. and Suszko, R. (1973), “Abstract logics”,DissertationesMathematicae, Vol. 102, pp. 9-42.

Etherington, D.W. (1987), “A semantics for default logic”, Proceedings of the IJCAI, Milan, Italy,pp. 495-8.

Gerla, G. (2000), Fuzzy Logic: Mathematical Tools for Approximate Reasoning, Kluwer Academic,Dordrecht.

Lloyd, J.W. (1987), Foundations of Logic Programming, Springer, Berlin.

Parsons, S. and Hunter, A. (1998), “A review of uncertainty handling formalisms”, Applications ofUncertainty Formalisms, pp. 8-37.

Reiter, R. (1980), “A logic for default reasoning”, Artificial Intelligence, Vol. 13, pp. 81-132.

Tarski, A. (1956), “Methodology of deductive sciences”, Logic, Semantics, Metamathematics,Oxford University Press, New York, NY.

Yager, R.R. (1987), “Using approximate reasoning to represent default knowledge”, ArtificialIntelligence, Vol. 31 No. 1, pp. 99-112.

Zadeh, L.A. (1965), “Fuzzy sets”, Information and Control, Vol. 8 No. 3, pp. 338-53.

Further reading

Antoniou, G. (1997), Nonmonotonic Reasoning, MIT Press, Cambridge, MA.

Brule, J.F. (1985), “Fuzzy systems – a tutorial”, available at: www.austinlinks.com/fuzzy/tutorial.html

Dubois, D. and Prade, H. (1996), “What are fuzzy rules and how to use them”, Fuzzy Sets andSystems, Vol. 84 No. 2, pp. 169-89.

Fuller, R. and Werners, B. (1991), “The compositional rule of inference: introduction, theoreticalconsiderations, and exact calculation formulas”, Working Paper No. 1991/7, Institut FurWirtschaftswissenschaften, RWTH Aachen.

Jian, Z., Yong, D. and Jian, G. (2003), “Intrusion detection system based on fuzzy default logic”,Fuzzy Systems, Vol. 2, pp. 1350-6.

Ray, K.S. and Chakraborty, A. (2007), “Fuzzyfied default logic”, Technical Report (TR-ECSUNo. 1/07), Indian Statistical Institute, New Delhi.

Zadeh, L.A. (1975), “The concept of a linguistic variable and its application to approximatereasoning-I”, Information Sciences, Vol. 8 No. 3, pp. 199-249.


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About the authors

Kumar S. Ray received the BE degree in Mechanical Engineering in 1977 fromCalcutta University, Kolkata, India, the MSc degree in Control Engineering in1980 from the University of Bradford, Bradford, UK, and the PhD degree inComputer Science in 1987 from Calcutta University. He is currently a Professor inthe Electronics and Communication Sciences Unit, Indian Statistical Institute,Kolkata. He was Visiting Faculty Member of the University of Texas at Austinunder a United Nations Development Programme (UNDP) fellowship in 1990. He

has 50 journal publications to his credit. He is the co-author of two edited volumes ofNorth-Holland. Dr Ray was a member of the task force committee of the Government of India,Department of Electronics (DoE/MIT), for the application of AI in power plants. He is the foundermember of the Indian Society for Fuzzy Mathematics and Information Processing (ISFUMIP) andmember of the Indian Unit for Pattern Recognition and Artificial Intelligence (IUPRAI). In 1991,he was the recipient of the K.S. Krishnan memorial award for the best system-oriented paper incomputer vision. His fields of interests are control theory, computer vision, AI, fuzzy reasoning,neural networks, genetic algorithms, qualitative physics and DNA computing. Kumar S. Ray isthe corresponding author and can be contacted at: [email protected]

Arpan Chakraborty studied at Vishwa Bharati Public School, India. At presenthe is a Graduate Assistant at the Department of Computer Science, NorthCarolina State University. His fields of interest include intelligent systems,computer vision and robotics.

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a fuzzy version of default logic

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