classical' negation in nonmonotonic reasoning and logic programming

Journal of Automated Reasoning 20: 107–142, 1998. 107c© 1998 Kluwer Academic Publishers. Printed in the Netherlands.

‘Classical’ Negation in Nonmonotonic Reasoningand Logic Programming

JOSE JULIO ALFERES,1? LUIS MONIZ PEREIRA2? andTEODOR C. PRZYMUSINSKI3??1Dep. Matematica, Univ. Evora, and A.I. Centre, UNL, 2825 Monte da Caparica, Portugal,e-mail:[email protected]. Centre, Univ. Nova de Lisboa, 2825 Monte da Caparica, Portugal, e-mail:[email protected] of Computer Science, University of California, Riverside, CA 92521, U.S.A.e-mail:[email protected]

(Received: September 1996)

Abstract. Gelfond and Lifschitz were the first to point out the need for a symmetric negationin logic programming and they also proposed a specific semantics for such negation for logicprograms with the stable semantics, which they called ‘classical’. Subsequently, several researchersproposed different, often incompatible, forms of symmetric negation for various semantics of logicprograms and deductive databases. To the best of our knowledge, however, no systematic studyof symmetric negation in non-monotonic reasoning was ever attempted in the past. In this paperwe conduct such a systematic study of symmetric negation. We introduce and discuss two natural,yet different, definitions of symmetric negation: one is called strong negation and the other iscalled explicit negation. For logic programs with the stable semantics, both symmetric negationscoincide with Gelfond–Lifschitz’ ‘classical negation’. We study properties of strong and explicitnegation and their mutual relationship as well as their relationship to default negation ‘not’, andclassical negation ‘¬’. We show how one can use symmetric negation to provide natural solutionsto various knowledge representation problems, such as theory and interpretation update, and beliefrevision. Rather than to limit our discussion to some narrow class of nonmonotonic theories, suchas the class of logic programs with some specific semantics, we conduct our study so that it isapplicable to a broad class of non-monotonic formalisms. In order to achieve the desired level ofgenerality, we define the notion of symmetric negation in the knowledge representation frameworkof AutoEpistemic logic of Beliefs, introduced by Przymusinski.

Key words: nonmonotonic reasoning, negation, logic programming, autoepistemic logic of beliefs.

1. Introduction

Logic programs, deductive databases, and, more generally, nonmonotonic theo-ries use various forms of default negation, not F , whose major distinctive featureis that not F is assumed ‘by default’, that is, it is assumed in the absence of suf-ficient evidence to the contrary. The meaning of ‘sufficient evidence’ depends onthe specific semantics used. For example, in Reiter’s Closed World Assumption,CWA [41], not A is assumed for atomic A if A is not provable, or, equivalently,

? Partially supported by JNICT-Portugal.?? Partially supported by the National Science Foundation grant #IRI-9313061.

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108 JOSE JULIO ALFERES ET AL.

if there is a minimal model in which A is false. On the other hand, in Minker’sGeneralized Closed World Assumption, GCWA [23, 17], or in McCarthy’s Cir-cumscription, CIRC [22], not A is assumed only if A is false in all minimalmodels. In Clark’s Predicate Completion semantics for logic programs [8], thisform of negation is called negation-as-failure because not A is derivable when-ever attempts to prove A finitely fail.

The more recent semantics proposed for logic programs and deductive databas-es, such as the stable semantics [14], well-founded semantics [44], partial stableor stationary semantics [33], and static semantics [36], propose even more sophis-ticated meanings for default negation, closely related to more general nonmono-tonic formalisms such as Default Logic, DL [42], Autoepistemic Logic, AEL [24],and Autoepistemic Logic of Beliefs, AEB [37]. Under all of these semantics, how-ever, default negation ‘not’ significantly differs from classical negation ‘¬’. Forexample, the formula

charged(x) ∧ ¬guilty(x) ⊃ acquitted(x)

says that a person charged with a crime should be acquitted if he or she is actuallyproven to be not guilty. On the other hand, the formula

charged(x) ∧ not guilty(x) ⊃ acquitted(x)

says that one should be acquitted of any charges by default unless sufficientevidence of that person’s guilt is demonstrated. In particular, one should beacquitted if there is no information available about one’s guilt or innocence.

1.1. DEFAULT NEGATION DOES NOT SUFFICE

Although default negation proved to be quite useful in various domains andapplication frameworks, it is not the only form of negation that is needed innonmonotonic formalisms. Indeed, while default negation not A of an atomicformula A is always assumed ‘by default’, we often need to be more carefulbefore jumping to negative conclusions. For example, it would make little senseto say

not innocent(x) ⊃ guilty(x)

to express the fact that being guilty is the opposite of being innocent because itwould imply that people should be considered guilty ‘by default’. Similarly, wewould not want to say

crossing(x), not train(x) ⊃ drive(x)

to express the fact that one may safely cross a railway track if (s)he first makessure that there is no train approaching, because such a clause would imply that(s)he should cross the railway even if (s)he did not bother to look around at all.

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‘CLASSICAL’ NEGATION IN NONMONOTONIC REASONING AND LOGIC PROGRAMMING 109

1.2. CLASSICAL NEGATION DOES NOT PROVIDE A SOLUTION

One could argue that the above feature constitutes the main difference betweendefault negation and classical negation, and therefore if default negation not Fdoes not work in some particular context, then classical negation ¬F should beused instead. One problem with such an approach is that classical negation ¬Fis not part of the language of standard logic programs and deductive databasesand thus it is not immediately available without an appropriate extension of theirlanguages and semantics, such as the one proposed in [36]. More important,however, classical negation ¬F has some specific characteristics that often makeit unsuitable or at least insufficient in the context of nonmonotonic reasoning:

– Classical negation ¬F satisfies the law of the excluded middle, F ∨ ¬F , forevery formula F . While it is a natural axiom in the domain of formal logicand mathematics, it is not suitable for commonsense reasoning where we areoften faced with situations in which some things are true, some are false,and yet some others are not (perhaps, yet) determined to be either true orfalse. For example, an employer may be faced with some candidates who areclearly qualified for the job and thus should be hired, some who are clearlynot qualified and thus should be rejected, and yet some other candidateswhose qualifications are not clear and who therefore should be interviewedin order to determine their suitability for the job (see [13]). However, if weattempt to describe the first two statements using real classical negation ‘¬’in the clauses

qualified(x)⊃hire(x), ¬qualified(x)⊃reject(x)

then, by virtue of the law of the excluded middle, we will immediately con-clude that every candidate must either be hired or rejected, without leavingany room for those who need to be further interviewed.The either/or character of the law of the excluded middle presupposes thatwe have sufficient information to determine, at least in principle, whether anygiven property or its opposite is true. However, when dealing with incompleteinformation or with properties involving gradual change, we often encounter‘gray areas’ in which neither the property nor its opposite seems to hold.The law of the excluded middle leads to particularly unintuitive results whenused with reference to nonexisting or nonapplicable properties or objects, forexample, in statements like ‘the pink elephant in my pocket is either old ornot old’ or ‘the chair is either happy or unhappy’. It would be clearly moreappropriate to say ‘the chair is neither happy nor unhappy’.Moreover, the law of the excluded middle is highly nonconstructive and thusdoes not fit well within the somewhat vague ‘spirit’ of logic programmingand deductive databases which seems to rely heavily on directional reason-ing, that is, on first establishing the validity of premises of a clause beforededucing its consequents and not the other way around. As a result of its

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nonconstructive character, the law of the excluded middle is also computa-tionally expensive.

– Classical literals A and ¬A are not treated symmetrically by default negation.More precisely, in the absence of evidence to the contrary, default negationnot A of positive literals (atoms) A is always assumed, while the same doesnot apply to negative literals ¬A. For example, even though GCWA (orcircumscription) applied to the theory

charged(tom)

charged(x) ∧ guilty(x) ⊃ convicted(x)

implies not convicted(tom)?, this is no longer the case if we perform asimple syntactic substitution of ¬innocent for guilty, thus obtaining

charged(tom)

charged(x) ∧ ¬innocent(x) ⊃ convicted(x)

because the new theory has minimal models in which convicted(tom) istrue.As we can see, a simple syntactic substitution of ¬innocent for guiltyturns out to have a significant impact on the semantics of the resultingtheory. As a consequence, default negation, not F , is heavily dependent onthe syntactic form of the formula F , and, in particular, it is not invariantunder the equivalence or renaming of predicates.Often, however, we would like to apply negation by default symmetricallyto both positive and negative literals. For instance, in the preceding exam-ple, given no information about Tom’s qualifications we would like to con-clude that Tom was neither found qualified nor unqualified, that is, that bothnot qualified(Tom) and not (¬qualified(Tom)) hold. This would provideus with an important information that Tom has to be interviewed.

1.3. SYMMETRIC NEGATION

Based on the above discussion, we conclude that, in addition to default nega-tion, ‘not’, and classical negation, ‘¬’, which are already used in nonmonotonicreasoning, we need a new form of negation, denoted here by ‘∼’, which has thefollowing properties:

– Similar to classical negation, it satisfies the basic properties of negationsuch as the double-negation law ∼(∼F ) ≡ F and the distributive laws∼(F ∨G) ≡ ∼F ∧ ∼G and ∼(F ∧G) ≡ ∼F ∨ ∼G;

– However, as opposed to classical negation, symmetric negation ‘∼’ is to betreated symmetrically by the default negation ‘not’; that is, in the absenceof evidence to the contrary, both not A and not (∼A) are to be assumed.

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More generally, the nonmonotonic semantics should be invariant under therenaming of any predicates from A to ∼A and vice versa.

– Moreover, also in contrast to classical negation, ‘∼’ should not be requiredto satisfy the law of the excluded middle, A ∨ ∼A.

– It also should substantially differ from default negation in that the negation∼A of atomic formulae A should not be assumed by default.

In view of the second property, we will call the form of negation ‘∼’ satisfyingthe above three properties symmetric negation. It is important to point out thatthe above conditions are not intended to be jointly sufficient to fully characterizesymmetric negation, and thus they do not uniquely determine this notion butrather represent constraints that may be met by several different notions.

Gelfond and Lifschitz [15] were the first to point out the need for such nega-tion in logic programming and to propose a specific semantics for such nega-tion for logic programs with the stable semantics. Somewhat unfortunately, theycalled their negation classical negation. Independently, Pearce and Wagner in [40]studied Nelson’s strong negation [27] and pointed out the need for using strongnegation in nonmonotonic reasoning. However, their strong negation is differentfrom the strong negation defined in this paper (see Remark 3.2).

Subsequently, several researchers proposed different, often incompatible, formsof symmetric negation for various semantics of logic programs and deductivedatabases [12, 28, 34, 36, 32, 45]. To the best of our knowledge, however, nosystematic study of symmetric negation in nonmonotonic reasoning was everattempted in the past.? In this paper we conduct such a systematic study ofsymmetric negation:

– We introduce and discuss two natural, yet different, definitions of symmetricnegation: one is called strong negation, and the other is called explicit nega-tion. For logic programs with the stable semantics, both symmetric negationscoincide with Gelfond-Lifschitz’s ‘classical negation’.

– We study properties of strong and explicit negation and their mutual rela-tionship as well as their relationship to default negation ‘not’ and classicalnegation ‘¬’.

– We show how one can use symmetric negation to provide natural solutions tovarious knowledge representation problems, such as theory and interpretationupdate and belief revision.

– Rather than limiting our discussion to some narrow class of nonmonotonictheories, such as the class of logic programs with some specific semantics,we conduct our study so that it is applicable to a broad class of nonmonotonicformalisms, which includes the well-known formalisms of circumscription,autoepistemic logic, and all the major semantics recently proposed for logicprograms (stable, well-founded, stationary, static, and others).

? For logic programs, an extensive study was carried out in [2].

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– To achieve the desired level of generality, we define the notion of symmetricnegation in the knowledge representation framework of Autoepistemic Logicof Beliefs, AEB, introduced by Przymusinski in [37], which was shown toisomorphically contain all of the above mentioned formalisms as specialcases. As a result, we automatically provide the corresponding notions ofsymmetric negation for all formalisms embeddable into AEB.

The paper is organized as follows. In the next section we briefly recall thedefinition and basic properties of AEB, as well as the translation of logic programsinto belief theories in AEB. In Section 3 we introduce strong negation (the firstform of symmetric negation), discuss its basic properties, and illustrate its simpleapplication to the problem of interpretation and theory update.

In Section 4 we introduce explicit negation (the second form of symmetricnegation) and establish its basic properties, and in Section 5 we discuss appli-cations of explicit negation to knowledge representation. In Section 6 we brieflydiscuss some issues involved in the implementation of explicit negation. Weconclude with some final remarks.

2. Autoepistemic Logic of Beliefs

First we briefly recall the definition and basic properties of AEB, introduced byPrzymusinski in [37]. Consider a fixed propositional language L with standardconnectives (∨, ∧, ⊃, ¬) and the propositional letter ⊥ (denoting false). Thelanguage of AEB is a propositional modal language LAEB obtained by aug-menting the language L with a modal operator B, called the belief operator. Theatomic formulae of the form BF , where F is an arbitrary formula of LAEB, arecalled belief atoms. The formulae of the original language L are called objective.Observe that arbitrarily deep level of nested beliefs is allowed in belief theories.Any theory T in the language LAEB is called a belief theory.

DEFINITION 2.1 (Belief Theory). By an autoepistemic theory of beliefs, or justa belief theory, we mean an arbitrary theory in the language LAEB, that is, a(possibly infinite) set of arbitrary clauses of the form

B1 ∧ · · · ∧Bk ∧ BG1 ∧ · · · ∧ BGl ∧ ¬BF1 ∧ · · · ∧ ¬BFn ⊃ A1 ∨ · · · ∨Am,

where k, l,m, n > 0, Ais and Bis are objective atoms, and Fis and Gis arearbitrary formulae of LAEB. Such a clause says that if the Bis are true, the Gisare believed, and the Fis are not believed, then one of the Ais is true.

By an affirmative belief theory we mean any belief theory all of whose clausessatisfy the condition that m > 0. 2

We assume the following two simple axiom schemata and one inference ruledescribing the arguably obvious properties of belief atoms:

(D) Consistency Axiom: ¬B⊥.

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(K) Normality Axiom: For any formulae F and G: B(F ⊃ G) ⊃ (BF ⊃ BG).

(N) Necessitation Rule: For any formula F : FBF

The first axiom states that tautologically false formulae are not believed. Thesecond axiom states that if we believe that a formula F implies a formula Gand if we believe that F is true, then we believe that G is true as well. Thenecessitation inference rule states that if a formula F has been proven to be true,then F is believed to be true.

DEFINITION 2.2 (Formulae Derivable from a Belief Theory). For any belieftheory T , we denote by CnAEB(T ) the smallest set of formulae of the lan-guage LAEB that contains the theory T and all the (substitution instances of) theaxioms (K) and (D) and is closed under both standard propositional consequenceand the necessitation rule (N).

We say that a formula F is derivable from theory T in the logic AEB if Fbelongs to CnAEB(T ). A belief theory T is consistent if the theory CnAEB(T )is consistent. 2

In the presence of the axiom (K), the axiom (D) is equivalent to the axiomBF ⊃ ¬B¬F , stating that if we believe in a formula F , then we do not believe inits negation ¬F . Moreover, it is easy to see that the axioms imply that the beliefoperator B obeys the distributive law for conjunctions B(F ∧G) ≡ BF ∧ BG(see [37]). For readers familiar with modal logics it should be clear by now thatwe are, in effect, considering here a normal modal logic with one modality Bthat satisfies the consistency axiom (D) [25].

Let us recall that throughout the paper we view the modal language LAEBsimply as a propositional language extending the underlying objective languageL. In particular, formulae of the form BF are considered to be propositional atomsin the extended language LAEB, and models of belief theories (i.e., models oftheories in the language LAEB) are just classical models of propositional theories.

DEFINITION 2.3 (Minimal Models). [37] By a minimal model of a belief the-ory T we mean a model M of T with the property that there is no smaller modelN of T that coincides with M on belief atoms BF . If a formula F is true in allminimal models of T , then we write T |=min F and say that F is minimallyentailed by T . 2

For readers familiar with circumscription, this means that we are consideringcircumscription CIRC(T ;K) of the theory T in which atoms from the objectivelanguage L are minimized while the belief atoms BF are fixed, that is, T |=min

F ≡ CIRC(T ;L) |= F . In other words, minimal models are obtained by firstassigning arbitrary truth values to the belief atoms and then minimizing objectiveatoms.

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2.1. STATIC AUTOEPISTEMIC EXPANSIONS

The intended meaning of belief atoms BF is based on the principle of predicateminimization:

BF if F is minimally entailed ≡ F is true in all minimal models.

Accordingly, beliefs in AEB can be called minimal beliefs (see [23, 17, and 22]).As in Moore’s Autoepistemic Logic, also in the Autoepistemic Logic of Beliefsthe intended meaning of belief atoms is implemented by defining plausible setsof beliefs that an ideally rational and introspective agent may hold, given a setof premises T .

DEFINITION 2.4 (Static Autoepistemic Expansion). [37] A belief theory T♦ iscalled a static autoepistemic expansion of a belief theory T if it satisfies thefollowing fixed-point equation

T♦ = CnAEB(T ∪ {BF : T♦ |=min F}),

where F ranges over all formulae of LAEB. 2

The definition of static autoepistemic expansions is based on the idea of buildingan expansion T♦ of a belief theory T by closing it with respect to (i) the deriv-ability in the logic AEB, and (ii) the addition of belief atoms BF satisfyingthe condition that the formula F is minimally entailed by T♦. Consequently,the definition of static expansions enforces the intended meaning of belief atomsdescribed above. Note that negations ¬BF of the remaining belief atoms are notexplicitly added to the expansion although some of them will be forced in by theNormality and Consistency axioms.

It turns out that every belief theory T in AEB has the least (in the sense ofset-theoretic inclusion) static expansion T♦ that has an iterative definition as theleast fixed point of the monotonic? belief closure operator

ΨT (S) = CnAEB(T ∪ {BF : S |=min F}),

where S is an arbitrary belief theory and the F ’s range over all formulae ofLAEB.

THEOREM 2.1 (Least Static Expansion). [37] Every belief theory T in AEBhas the least static expansion, namely, the least fixed point T♦ of the monotonicbelief closure operator ΨT .

More precisely, the least static expansion T♦ of a belief theory T can beconstructed as follows. Let T 0 = T , and suppose that Tα has already beendefined for any ordinal number α < β.? Strictly speaking, the operator is monotone only on the lattice of all theories of the form

T ∪Mbel, where T is fixed and Mbel ranges over all belief formulae.

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– If β= α+ 1 is a successor ordinal, then define

Tα+1 = ΨT (Tα) = CnAEB(T ∪ {BF : Tα |=min F} and F ∈ LAEB).

– Otherwise, define T β =⋃α<β T

α.

The sequence {Tα} is monotonically increasing and has a unique fixed pointT♦ = T λ = ΨT (T λ), for some ordinal λ. 2

According to the following recent result established in [7], for any finite belieftheory only one iteration of the operator ΨT is needed.

THEOREM 2.2. [7] For every finite belief theory T , the least static expansionT♦ of T is given by

T♦ = ΨT (T ) = CnAEB(T ∪ {BF : T |=min F}). 2

Observe that the least static autoepistemic expansion T♦ of T contains thoseand only those formulae that are true in all static autoepistemic expansions ofT . It defines the static semantics of a belief theory T . It is easy to verify thata belief theory T either has a consistent least static expansion T♦ or does nothave any consistent static expansions at all. Moreover, least static expansions ofaffirmative belief theories are always consistent [37].

EXAMPLE 2.1. Consider the following belief theory T describing a simple diag-nosis of car problems

B¬Broken ⊃ Runs

B¬Runs ⊃ Fix.

Let us first observe that T |=min ¬Broken, T |=min (Runs⇔ B¬Broken) andT |=min (Fix⇔ B¬Runs). Indeed, to find minimal models of T , we just need toassign arbitrary truth values to the two belief atoms B¬Broken and B¬Runs,and then minimize the objective atoms Fix, Broken and Runs. We easily seethat T has the following four minimal models:?

M1 = {B¬Broken, B¬Runs, ¬Broken, Runs, F ix};M2 = {¬B¬Broken, ¬B¬Runs, ¬Broken, ¬Runs, ¬Fix};M3 = {¬B¬Broken, B¬Runs, ¬Broken, ¬Runs, F ix};M4 = {B¬Broken, ¬B¬Runs, ¬Broken, Runs, ¬Fix}.

Since in all of them the formulae ¬Broken, Runs ⇔ B¬Broken and Fix ⇔B¬Runs are satisfied, we infer that T |=min ¬Broken, T |=min (Runs ⇔B¬Broken) and T |=min (Fix⇔ B¬Runs).? For simplicity, when describing static expansions of this and other examples, we list only

those elements of the expansion that are ‘relevant’ to our discussion. In particular, we usually omitnested beliefs.

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Consequently, since T 1 = ΨT (T ) = CnAEB(T ∪ {BF : T |=min F}), weobtain

{B¬Broken,B(Runs⇔ B¬Broken),B(Fix⇔ B¬Runs)} ⊆ T 1.

Since T 1 |= Runs, from the Necessitation rule it follows that BRuns ∈ T 1. Bythe Consistency axiom we infer that ¬B¬Runs ∈ T 1 and thus also B(¬B¬Runs)∈ T 1. Since

T 1 |= B(¬Fix⇔ ¬B¬Runs),

by the Normality axiom

T 1 |= B(¬Fix)⇔ B(¬B¬Runs),

and therefore B¬Fix ∈ T 1. It is easy to check that T 1 is clearly, the (least) fixedpoint of ΨT and therefore T♦ = T 1 = CnAEB(T∪{B¬Broken,Runs,B¬Fix})is the least static expansion of T . The static semantics of T asserts our beliefthat the car is not broken and thus runs fine and does not need fixing. One easilyverifies that T does not have any other (consistent) static expansions. 2

2.2. EMBEDDABILITY OF CIRCUMSCRIPTION AND AUTOEPISTEMIC LOGIC

One easily sees that propositional circumscription (and thus also CWA, GCWAand ECWA [41, 23, 17]) can be properly embedded into AEB.

PROPOSITION 2.1 (Embeddability of Circumscription). [37] Propositional cir-cumscription, CWA, GCWA, and ECWA are all properly embeddable intothe Autoepistemic Logic of Beliefs, AEB. More precisely, if T is any objectivetheory, then T has a unique static expansion T♦, and any objective formula Fis logically implied by the circumscription CIRC(T ) of T if and only if T♦

logically implies the belief atom BF :

CIRC(T ) |= F ≡ T♦| = BF. 2

Moreover, a simple extension, AELB, of AEB, obtained by adding an addition-al knowledge operator, has been shown [37] to isomorphically contain Moore’sAutoepistemic Logic, AEL.

2.3. LOGIC PROGRAMS AS BELIEF THEORIES

Major semantics defined for normal and disjunctive logic programs are alsoembeddable into AEB [37]. In particular, this is true for the stable semantics [14],well-founded semantics [44], partial stable or stationary semantics [33], andstatic semantics [36] of normal and disjunctive logic programs.

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Suppose that P is a disjunctive logic program consisting of rules

A1 ∨ · · · ∨Ak ← B1 ∧ · · · ∧Bm ∧ notC1 ∧ · · · ∧ notCn.

The translation of P into the affirmative belief theory TB¬(P ) is given by theset of the corresponding clauses

B1 ∧ · · · ∧Bm ∧ B¬C1 ∧ · · · ∧ B¬Cn ⊃ A1 ∨ · · · ∨Ak (1)

obtained by replacing the default negation, not F , by the belief atom, B¬F , andby replacing the rule symbol → by the standard material implication ⊃.

The translation, TB¬(P ), gives therefore the following meaning to the defaultnegation:

not Fdef≡ B¬F ≡ F is believed to be false ≡ ¬F is minimally entailed.(2)

THEOREM 2.3 (Embeddability of Stationary and Stable Semantics). [37] Thereis a one-to-one correspondence between stationary (or, equivalently, partial sta-ble) modelsM of the program P and consistent static autoepistemic expansionsT♦ of its translation TB¬(P ) into a belief theory. Namely, for any objective atomA we have

A ∈M iff A ∈ T♦ iff BA ∈ T♦

¬A ∈M iff B¬A ∈ T♦.

In particular, the well-founded model MWF of the program P corresponds tothe least static expansion of TB¬(P ). Moreover, stable models (or answer sets)M of P correspond to those consistent static autoepistemic expansions T♦ ofTB¬(P ) that satisfy the condition that for all objective atoms A, either BA ∈ T♦or B¬A ∈ T♦, that is, that every atom is either believed or disbelieved in T♦. 2

EXAMPLE 2.2. It is easy to see that the belief theory T considered in Example2.1 can be viewed as a translation TB¬(P ) of the stratified logic program P givenby

Runs ← notBroken

Fix ← notRuns.

The unique consistent static expansion,

T♦ = CnAEB(T ∪ {B¬Broken,Runs,B¬Fix}),

of T corresponds therefore to the unique perfect model,

M = {not Broken,Runs, not Fix}

of P , which is also its unique well-founded and stable model [33]. 2

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3. Strong Negation

In the Introduction we concluded that in addition to default negation, not F , non-monotonic reasoning requires a new type of negation that is similar to classicalnegation ¬F , in the sense that it is not assumed by default, but does not satisfythe law of the excluded middle and ensures a symmetric treatment of positiveand negative information by default negation. In this section we define the so-called strong negation, ∼F , and argue that it is a natural candidate for such anegation. We study properties of strong negation and its relationship to defaultnegation and classical negation. We also illustrate how strong negation can beapplied to provide a natural solution to an important knowledge representationproblem.

We introduce strong negation within the broad framework of the Autoepis-temic Logic of Beliefs, AEB, described in Section 2. As a result, the defini-tion of strong negation applies to all nonmonotonic formalisms embeddable intoAEB, including circumscription, autoepistemic logic, and all the major semanticsrecently proposed for logic programs. The definition is patterned after the origi-nal definition given in [15], and therefore, when applied to logic programs withstable semantics, strong negation coincides with Gelfond-Lifschitz’s so-calledclassical negation.

Strong negation is introduced to belief theories T in the Autoepistemic Logicof Beliefs, AEB, by

(1) Augmenting the underlying objective language L with a set S = {∼A : A ∈S} of new objective propositional symbols, called strong negation atoms,where S is any fixed set of propositional symbols from L. As a result weobtain an extended objective language L and an extended language of beliefsLAEB.

(2) Adding to the belief theory T the following strong negation constraint:

(SA) A ∧ ∼A ⊃ ⊥ or, equivalently, ∼A ⊃ ¬A,

for every strong negation atom ∼A in S.

The strong negation constraint SA states that A and ∼A cannot be both trueand thus ensures that the intended meaning of strong negation ∼A is ‘∼A isthe opposite of A’. For example, a proposition A may describe the propertyof being ‘qualified’, while the proposition ∼A describes the property of being‘unqualified’. The strong negation constraint states that a person cannot be bothqualified and unqualified. We do not assume, however, that everybody is alreadyknown to be either qualified or unqualified.

In practice, given a theory T , the set S is usually defined as S = {∼A :∼A occurs in T}. As a result, theories that do not contain strong negation atoms∼A are not affected by the strong negation constraints in any way. We wantto emphasize that rather than introducing strong negation at the meta-level, by

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including strong negation constraints in the logical closure operator CnAEB,we instead make strong negation part of the object-level language, by mak-ing the constraints part of any belief theory in which the corresponding atoms∼A occur. In the sequel, we will implicitly assume that the strong negationconstraint SA is implicitly included in any belief theory T that contains theatom ∼A.

For the sake of readability, instead of using somewhat cumbersome names,like∼good teacher, to denote the strong negation of the predicate good teacher,we use more readable names, such as bad teacher. Furthermore, even thoughwe only consider propositional languages, we often use variables as a shortcutfor all ground instantiations of a given formula.

EXAMPLE 3.1. Consider a university that periodically evaluates faculty mem-bers based on their research and teaching performance. Faculty members who areknown to be strong researchers and who are believed to be good teachers (as weall know, it is difficult to objectively evaluate teaching) receive a positive evalu-ation. On the other hand, those who are believed not to be strong researchers aswell as those that are believed to be poor teachers receive a negative evaluation.Anyone who received a teaching award is considered to be a good teacher, andanyone who published at least 10 reviewed papers during the evaluation periodis considered a good researcher. The individuals whose evaluation status is notyet determined undergo some further (unspecified) review process. This leads usto the following belief theory in AEB:

good researcher(x) ∧ B good teacher(x) ⊃ good evaluation(x)

B¬good researcher(x) ⊃ bad evaluation(x)

B bad teacher(x) ⊃ bad evaluation(x)

teaching award(x) ⊃ good teacher(x)

many publications(x) ⊃ good researcher(x).

The predicates bad teacher, bad evaluation, and bad researcher are intendedto represent strong negation of the predicates good teacher, good evaluation,and good researcher, respectively, and, therefore, we need to add to our theorystrong negation constraints (SA) for each one of them:

good researcher(x) ∧ bad researcher(x) ⊃ ⊥good teacher(x) ∧ bad teacher(x) ⊃ ⊥good evaluation(x) ∧ bad evaluation(x) ⊃ ⊥.

Suppose that both Ann and Tom published at least 10 papers. Suppose, further,that Ann is a good teacher, both Mary and Keith received teaching awards, but

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Tom is considered to be a bad teacher. Moreover, Keith and Paul have a lot ofjoint publications so at least one of them must be a good researcher:

many publications(Ann)

many publications(Tom)

good teacher(Ann)

teaching award(Mary)

teaching award(Keith)

bad teacher(Tom)

good researcher(Keith) ∨ good researcher(Paul).

The resulting belief theory has one consistent static expansion in which Annreceives a positive evaluation because she is a good researcher and is believedto be a good teacher. Tom receives a negative evaluation because even thoughhe is a good researcher, he is also believed to be a bad teacher. Mary receives anegative evaluation as well because even though she is a good teacher, there isno evidence of good research work in her file (i.e., B¬good researcher(Mary)holds).

Keith’s and Paul’s status is not yet clear, and thus they will have to be furtherreviewed. Indeed, neither of them individually has been shown to be a strongresearcher so they are not eligible for positive evaluation. At the same time, nei-ther of them is believed to be a bad teacher, and there is some evidence supportingeach one of them being a good researcher (i.e., neither B¬good researcher(Paul)nor B¬good researcher(Keith) is true) and therefore they are not subject tonegative evaluation, either.

Observe, however, that if we later find out that Paul is in fact a lousyresearcher:

bad researcher(Paul),

then Paul will receive a negative evaluation, and we will conclude (by virtue ofthe strong negation constraint) that Keith is a good researcher, and thus Keithwill receive a positive evaluation. 2

Remark 3.1. The correctness of the above example has been verified by usingthe prototype interpreter for static semantics developed by Stefan Brass and basedon the characterizations of static semantics obtained in [7]. The interpreter isavailable from ftp : //ftp.informatik.uni−hannover.de/software/static/static.html via FTP and WWW. 2

Even though the strong negation operator ∼F is defined so far only for atomicformulae F of the language L, we can easily extend it to arbitrary formulae F of

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the extended language L. This is accomplished by adding the following axiomschemas for all formulae F and G of the language L:

∼(∼F ) ≡ F

∼(¬F ) ≡ ¬(∼F )

∼(F ∨G) ≡ ∼F ∧ ∼G∼(F ∧G) ≡ ∼F ∨ ∼G.

By using the above axioms, strong negation ∼F of an arbitrary formula F ofthe language L can be shown to be equivalent to a formula of the language Litself. Moreover, if two formulae F and G of the language L are equivalent, thenso are their strong negations ∼F and ∼G. Readers interested in the details ofthis extension of the strong negation operator ∼F to arbitrary formulae F andin its properties should consult the paper [38]. In the remainder of this section,whenever the operator ∼F is used with reference to a nonatomic formula F ,it is implicitly assumed that the above axiom schemas are part of the belieftheory.

Remark 3.2. The underlying logic with strong negation, ∼F , considered inthis paper does not coincide with the so-called classical logic with strong nega-tion [43, 40], that is, with classical logic augmented with Nelson’s strong nega-tion [27]. The difference lies in the mutual relationship between strong and clas-sical negations. In classical logic with strong negation, which is known to beequivalent to three-valued Lukasiewicz’s logic, the two negations ¬ and ∼ mutu-ally cancel each other because of the axiom ∼(¬F ) ≡ F . On the other hand,the logic presented in this paper contains the axiom ∼(¬F ) ≡ ¬(∼F ), ensuringthat the two negations commute and thus are mutually symmetric.

The consequences of this seemingly small difference are profound. It is wellknown that in the classical logic with strong negation, two formulae F and Gmay be tautologically equivalent and yet their strong negations ∼F and ∼G notbe equivalent. For example, A and ¬¬A are tautologically equivalent and yet∼A is not equivalent to ∼(¬¬A) because the latter formula is equivalent to ¬A.This problem can be avoided only if ∼A is always equivalent to ¬A, that is,when strong negation actually coincides with classical negation. On the otherhand, in the logic presented in this paper, one can prove that if two formulae Fand G of the language L are equivalent, then so are their strong negations ∼Fand ∼G. Readers should consult the paper [38] for a more detailed treatment ofthis subject. In there the strong negation operator is also extended to all formulaeof the modal language with beliefs LAEB. 2

The above extension preserves the basic property of strong negation, namely,the fact that it is stronger than classical negation. However, this is true only forpositive formulae, namely, those that do not use classical negation ¬.

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PROPOSITION 3.1 (Strong Negation vs. Classical Negation). Suppose that T isa belief theory and F is a positive formula of L. Then

T |= (∼F ⊃ ¬F ).

Proof. If F = A is atomic, then, because of the strong negation constraints,(∼F ⊃ ¬F ) ∈ T . Suppose that the claim is true for positive formulae G and H .

If F = ∼G, then T |= (∼G ⊃ ¬G), and therefore T |= (G ⊃ ¬∼G) whichimplies that T |= (∼F ⊃ ¬F ).

If F = G ∨H then T |= (∼G ⊃ ¬G) and T |= (∼H ⊃ ¬H). Since T |=∼F ≡ (∼G ∧ ∼H), we obtain that T |= (∼F ⊃ (¬G ∧ ¬H) and thereforeT |= (∼F ⊃ ¬F ). The proof for conjunction is completely analogous. 2

From the above proposition one easily derives an important relationship be-tween the strong negation operator and the belief operator:

PROPOSITION 3.2 (Strong Negation vs. Beliefs). Suppose that T is a belief the-ory. All static autoepistemic expansions T♦ of T are closed under the inferencerule

∼FB¬F ,

where F is any positive formula of L. In particular, this is true for the leaststatic expansion of T .

Proof. By the previous proposition, if ∼F belongs to T♦, then so does ¬F .By Necessitation, also B¬F belongs to T♦. 2

This result confirms the intuitively desirable fact that (positive) formulae Fwhose strong negation∼F is known to hold are also believed to be false. Observethat for any positive formula F of L , both F and ∼F are assumed false bydefault, that is, both B¬F and B¬(∼F ) are true, because, in the absence ofany other information, F and ∼F are false in all minimal models. This is theintended behavior for symmetric negation.

Remark 3.3. However, whenever necessary, for any given positive formulaF , one can easily ensure that one of the formulae, F or ∼F , is true by default,and thus that only the other is false by default. This can be accomplished byassuming one of the following default axioms:

B¬F ⊃ ∼F or B¬(∼F ) ⊃ F

which say that if we disbelieve F (respectively, ∼F ), then ∼F (respectively, F )can be assumed to be true. For example, the first default axiom causes ∼F to betrue by default, thus forcing strong negation to behave in a way that is similar

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(but not identical) to classical negation. This will prove useful later on when wediscuss the application of strong negation to theory and interpretation update.Moreover, assuming both by default makes them both undefined, in the absenceof other information.

One can also prevent any minimization of F and ∼F by assuming the lawof the excluded middle, F ∨∼F , for a given formula F , which causes preciselyone of F or ∼F to be true at all times. 2

As we mentioned in the Introduction, nonmonotonic formalisms based onsome form of predicate minimization, such as CWA, circumscription, and mostsemantics of logic programs, do not treat atomic formulae A and their classicalnegations ¬A symmetrically, and thus they are not invariant under a simplerenaming substitution replacing atoms by their negations and vice versa.

The Autoepistemic Logic of Beliefs, a superset of such formalisms, naturallysuffers from the same problem. For example, the belief theory B¬A ⊃ C hasa unique static expansion in which C is true, and, yet, after substituting A′

for ¬A, the resulting theory BA′ ⊃ C has a unique expansion that no longercontains C. However, since strong negation atoms ∼A are introduced as regularobjective atoms and since renaming of atoms has no effect on the semantics, weimmediately conclude that beliefs in AEB are invariant under renaming of anypredicates from A to ∼A and vice versa.

PROPOSITION 3.3 (Strong Negation Is Symmetric). Static semantics of belieftheories in AEB is invariant under the renaming of any predicates from A to∼A and vice versa.

More precisely, suppose that S is a subset of the set of all objective predicatesfrom L, and let Θ be the operator simultaneously replacing all occurrences ofthe atom ∼A by A and all occurrences of the atom A by ∼A, for all atoms Ain S. Then T♦ is a static expansion of a belief theory T in AEB if and only ifΘ(T♦) is a static expansion of the theory Θ(T ). 2

The above simple proposition illustrates the most basic difference betweenclassical and strong negation.

3.1. STRONG NEGATION AND LOGIC PROGRAMMING

Since logic programs under major semantics, including stable semantics [14],well-founded semantics [44], partial stable or stationary semantics [33], and stat-ic semantics [36], can be translated into belief theories in AEB via the embeddingdefined in Section 2.3 [36, 35], the introduction of strong negation into belieftheories immediately introduces strong negation into logic programs with thesesemantics.

In particular, it follows from [15, 33] that stable semantics augmented withstrong negation is equivalent to stable semantics with the so-called classical

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negation, originally introduced by Gelfond and Lifschitz [15]. We assume herethat ‘classical negation’ of an atom A is translated into its strong negation ∼A.

THEOREM 3.1 (Strong Negation Extends ‘Classical’ Negation). There is a one-to-one correspondence between stable models M of a logic program P with‘classical negation’ and consistent static autoepistemic expansions T♦ of itstranslation TB¬(P ) into belief theory that satisfy the condition BA ∨ B¬A, forall objective atoms A. 2

From Proposition 3.2 and the fact that default negation, not F , is translatedinto B¬F , one immediately derives an important relationship between strongnegation and default negation in logic programs:

PROPOSITION 3.4 (Strong Negation vs. Default Negation). Suppose that P isa logic program. The inference rule

∼Fnot F

is satisfied by all semantics of P obtained by the embedding of P into AEB viathe translation TB¬(P ) and for all positive formulae F of L. 2

Remark 3.4. It is also worth noting that, for atomic objective formulae A, thedefault axioms discussed above have a particularly simple translation into logicprogramming rules

∼A← not A and A← not ∼A. 2

3.2. APPLICATION OF STRONG NEGATION TO KNOWLEDGE REPRESENTATION

Applications of various forms of symmetric negation to knowledge representationhave been studied in a number of papers and proved to be quite extensive (see,e.g., [6, 16, 20, 21, 29, 30, 31, 37]). In particular, in [37] strong negation is usedto provide a simple embedding of Gelfond’s epistemic specifications [13] intoAELB, and in [6] strong negation is applied to propose a solution to the problemof belief revision. Here we show how one can apply strong negation to providea natural solution to an important knowledge representation problem, namely, tothe problem of theory update. This generalizes to arbitrary belief theories theapproach to theory update proposed earlier in [26, 39].

Katsuno and Mendelzon [19] have distinguished two abstract frameworks forreasoning about change: theory revision and theory update. As opposed to theoryrevision, which involves a change in knowledge or belief with respect to a staticworld, theory update involves a change of knowledge or belief in a changingworld. Winslett [46] showed that reasoning about actions should be done ‘one

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model at a time.’ That is, when reasoning about the outcome of an action, wemust consider its effect in each one of the states of the world that are consistentwith our (possibly incomplete) knowledge of the current state of the world. Thisinsight is reflected in the general definition of theory update due to Katsuno andMendelzon, which can be formulated as follows.

Let T and S be sets of propositional formulae. A set S′ of formulae is a‘theory update’ of S by T if

Models(S′) = {I ′ : ∃I ∈ Models(S) . I ′ is ‘an update of I by T ’}.

From this definition we infer that in order to determine ‘theory update’ it sufficesto define when an interpretation I ′ is an update of an interpretation I by a theoryT . Below we illustrate how strong negation can be used to define interpretationupdate by an arbitrary belief theory T in AEB. As we already mentioned, thisis a generalization to arbitrary belief theories of the approach to interpretationupdate proposed earlier in [26, 39].

Suppose that T is the theory discussed in Example 3.1 describing a univer-sity that periodically evaluates its faculty based on their research and teachingperformance and consisting of formulae

good researcher(x) ∧ B good teacher(x) ⊃ good evaluation(x)

B¬good researcher(x) ⊃ bad evaluation(x)

B bad teacher(x) ⊃ bad evaluation(x)teaching award(x) ⊃ good teacher(x)

many papers(x) ⊃ good researcher(x)

together with strong negation constraints for all the atoms appearing in the theory,for example,

good researcher(x) ∧ bad researcher(x) ⊃ ⊥, . . . , etc.

Moreover, suppose that I is an interpretation of the objective language L givenby I (with the obvious abbreviations):

{good eval(Jack), good eval(Lisa), good teacher(Scott)} ∪∪ {many papers(Lisa), t award(Jack),many papers(Scott)}.

To produce an interpretation I ′ of the objective language L, which can be con-sidered an update of I by the belief theory T , we first augment T with the defaultaxioms B¬(∼A) ⊃ A, for every atom A in I , for example,

B¬bad evaluation(Jack) ⊃ good evaluation(Jack), . . . , etc.

These axioms can be viewed as natural inertia or frame axioms stating thatpredicates that are initially true in I remain true unless we have some evidenceto the contrary. We will denote the set of these default axioms by DI .

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The augmented belief theory TI = T ∪DI has a unique static expansion inwhich Jack receives a negative evaluation because he does not have any researchrecord and Scott receives a positive evaluation because he is a good teacher anda good researcher. Even though Lisa’s teaching record is unclear, she keeps herpositive evaluation because of the default axiom

B¬bad evaluation(Lisa) ⊃ good evaluation(Lisa).

It is therefore natural to consider the interpretation I ′

{bad eval(Jack), t award(Jack), good eval(Scott)} ∪∪ {good res(Scott), good res(Lisa), good eval(Lisa)} ∪∪ {good t(Scott),many p(Scott),many p(Lisa), good t(Jack)},

consisting of all the atomic formulae from the language L that belong to theunique static expansion T♦I of TI , to be the update of I by T . In general, wemay have more than one static expansion of TI , thus resulting in more than onepossible interpretation update of I .

It follows from the results proved in [39] that revision programming, intro-duced in [26], is a special case of this approach obtained when the theory T isa positive logic program. However, the approach proposed here is more gener-al because the updating theory T can be an arbitrary belief theory in AEB. Inparticular, T can be an arbitrary propositional theory, or it can represent any ofthe nonmonotonic formalisms embeddable into AEB, such as CWA, circumscrip-tion, autoepistemic logic, and all the major semantics recently proposed for logicprograms. In [39] it was also shown how to define update in the framework ofdefault logic.

4. Explicit Negation

In this section we introduce an alternative weaker notion of symmetric negationin the Autoepistemic Logic of Beliefs, which we call explicit negation. Theresulting extension of AEB, called the Autoepistemic Logic of Beliefs with ExplicitNegation, AEBX, demonstrates the flexibility of the AEB framework in expressingdifferent forms of negation.

After providing motivation and formal definitions, we show that static expan-sions in AEBX are sufficiently expressive to characterize the well-founded seman-tics with explicit negation, WFSX, a semantics of logic programs introduced ear-lier in [28]. We then contrast explicit negation with strong negation. Section 5illustrates an application of explicit negation showing how belief revision inAEBX can directly capture the contradiction removal techniques used for logicprograms [3].

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4.1. THE ISSUE OF RELEVANCE

In logic programming, clauses are seen as inference rules rather than materialimplications. However, for definite and normal programs this distinction is imma-terial. Because of the absence of negative facts in such programs, the additionof a contrapositive ¬Head→ ¬Body of a program rule Head← Body has nobearing on the rest of the program.

On the other hand, in normal logic programs extended with a symmetricnegation (hereafter, simply called extended logic programs or ELPs), some careis needed if one wants to preserve the procedural reading of logic program rules.According to this reading, as in a procedure, the truth value of the head of arule is solely determined by the truth value of its body. Thus, the proceduralreading indicates that computing the truth value of a literal can be done byrelying solely on the procedural call graph implicitly defined by the rules forliterals. In other words, literals that are not (transitively) called by the rules foranother literal should not influence its truth value. This well-known property,called relevance [9], is essential to guarantee the availability of (strictly) top-down evaluation procedures for a semantics. It is worth noting that the well-founded semantics for normal logic programs [18] obeys relevance (see [9]).On the other hand, neither the Stable Models semantics nor AEB when appliedto theories resulting from logic programs with the strong negation constraintssatisfies this principle.

One paramount motivation for introducing AEBX is the desire to capturethose semantics of logic program that satisfy relevance and thus permit efficient,top-down implementations. As shown in Example 4.1, queries for nonrelevantsemantics cannot, in general, be evaluated in a top-down fashion by using stan-dard logic programming implementation procedures, that is, by simply follow-ing the program’s call graph. The other was the desire to ensure that AEBX ismore expressive than AEB by providing a meaningful semantics to those con-sistent programs that appear to have a well-defined intended meaning and yetdo not have a consistent semantics when strong negation is used (see Exam-ple 4.2).

EXAMPLE 4.1. Take the following theory T and corresponding logic programP , where T = TB¬(P )

B¬∼god exists ⊃ god exists god exists ← not ∼god existsB¬god exists ⊃ ∼god exists ∼god exists ← not god exists

B¬god exists ⊃ ∼go to church ∼go to church ← not god exists

go to church go to church

where the first two rules represent two conflicting default axioms, which read ‘Iconclude God exists if I disbelieve Its nonexistence’ and ‘I conclude God doesn’texist if I disbelieve Its existence’. The third rule states that ‘If I disbelieve the

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existence of God, then I do not go to church’, and the fourth that ‘I go tochurch’.

If ∼ is understood as strong negation, that is, if strong negation constraints areadded to T , then the theory has one expansion, where god exists because I go tochurch. Indeed, go to church ⊃ ¬∼go to church (strong negation constraint),which by contraposition of the third clause entails ¬B¬god exists. Therefore,since ∼god exists only appears in the second clause, ¬∼god exists holds inall minimal models, and by necessitation rule (N), B¬∼god exists. Hence, bythe first clause of T , god exists holds in all expansions of the theory.

Note that this conclusion is not relevant. Looking at the program P makes itclear that only the first two rules are (transitively) called by god exists. Howev-er, the reader may check that the theory consisting solely of the first two clausesdoes not have god exists in all its expansions.

With explicit negation (to be defined below) we have different conclusions,and the corresponding least expansion includes {go to church,B¬∼go tochurch} but not god exists nor B¬∼god exists (cf. Example 4.3). 2

EXAMPLE 4.2. Take now the following theory and corresponding logic pro-gram:

B¬shave(x, x) ⊃ shave(John, x)

shave(y, x) ⊃ go dine out(x)

∼shave(Peter, Peter)∼go dine out(John)

shave(john,X) ← not shave(X,X)

go dine out(X) ← shave(Y,X)

∼shave(peter, peter)∼go dine out(john)

The first rule states that ‘John shaves everyone not believed to shave themselves’.The second says that ‘If x has been shaved (by anyone) then x will go out todine’. The third states that ‘Peter does not shave himself’, and the fourth that‘John has not gone out to dine’. We would like to know whether we believe Johnhas shaved himself given that he has not gone out to dine. Note that believinghe has not shaved himself leads to a contradiction, and that the conclusion thathe has shaved himself is not true in all minimal models.

If the strong negation constraints are added to this theory, then AEB assignsno consistent meaning to it (there is no consistent expansion). On the contrary,if explicit negation, to be defined below, is used instead, the theory has oneexpansion that includes {∼go dine out(john),B¬go dine out(john)} but not

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shave(john, john). The absence of consistent expansions by using strong nega-tion is brought about by the use of the contrapositive ¬go dine out(X) ⊃¬shave(X,X) to conclude ¬shave(john, john),B¬shave(john, john), andhence shave(john, john) (contradiction). 2

As we have seen, the reason strong negation does not directly capture anysemantics for extended logic programs complying with relevance, such as WFSX,is because of its very definition. Strong negation constraints, ∼A ⊃ ¬A, statethat strongly negated facts or conclusions entail classically negated ones, therebypermitting the use of the contrapositives of the material implications resultingfrom the translation of the logic program rules.

What the above two paradigmatic examples have in common is the backpropagation of truth values by strong negation, against the logic program rulearrow, into a loop of otherwise undefined literals (i.e., such that neither L nornot L holds). In Example 4.1 we have an even loop over default negation, andin Example 4.2 an odd one. In the first example, the back propagation decidesthe loop one way, and in the second it comes up against the impossibility ofresolving the loop by imposing a truth value. However, as we shall see below,explicit negation does not affect either loop.

To conclude, because of its use of classical negation contrapositives, strongnegation leads both to logic programs without a semantics and to logic programtheories with unwarranted (non-relevant) conclusions, that is, conclusions notsolely based on the procedural call graph of the logic program. To be able tocapture relevant logic program semantics a weaker notion of symmetric negationis needed. Theories with explicit negation that are translatable into extended logicprograms can be efficiently queried by the top-down procedural implementationtechnology of logic programs.

4.2. INTRODUCING EXPLICIT NEGATION

Explicit negation is added to AEB by means of an explicit negation operator. Forthe sake of simplicity, and since we will never consider theories with both explcitand strong negation, we use the same negation symbol for both. We thus definingthe Autoepistemic Logic of Beliefs with Explicit Negation, AEBX. Specifically,this is accomplished by

– Augmenting the underlying objective language L with a set S = {∼A : A ∈S} of new objective propositional symbols, called dual orexplicit negationatoms, where S is any fixed set of propositional symbols from L. As aresult we obtain an extended objective language L and an extended languageof beliefs LAEB. The extension of explicit negation to arbitrary positiveobjective formulae can be done in the same way as for strong negation.

– Extending the logical closure operator CnAEB with the following Coherenceinference rule

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∼AB¬A

for every objective propositional symbol A?.

A Coherence inference rule says that if one derives the dual, one has to believeits negation, that is, ∼A serves as evidence against A. Since the Coherenceinference rules have no effect on belief theories that do not include explicitnegation atoms, in the sequel we will assume them as part of the operator CnAEBwithout further mention, whenever explicit negation is used.

EXAMPLE 4.3. The details of this example show the essence of how explic-it negation treats both previous examples, and the way it differs from strongnegation.

The theory

T♦ = CnAEB(T ∪ {Bgtc,B¬∼gtc})

is, with the obvious abbreviations, an expansion of the AEBX theory in Exam-ple 4.1 (where strong negation is replaced by explicit negation). Its minimalmodels are

M1 = {Bgtc, B¬∼gtc, Bge, B∼ge, ¬ge, ¬∼ge, ¬∼gtc, gtc}M2 = {Bgtc, B¬∼gtc, Bge, B¬∼ge, ge, ¬∼ge, ¬∼gtc, gtc}M3 = {Bgtc, B¬∼gtc, B¬ge, B∼ge, ¬ge, ∼ge, ∼gtc, gtc}M4 = {Bgtc, B¬∼gtc, B¬ge, B¬∼ge, ge, ∼ge, ∼gtc, gtc}

B¬∼gtc follows in T♦ from gtc and the Coherence inference rule.This example illustrates why explicit negation does not affect the theory’s

even loop and, for the same reason, why it does not affect the odd loop ofExample 4.2. Indeed, in the general case, the explicit negation of the head ofa program rule may be true even though its body is undefined (i.e., such thatneither BBody nor B¬Body hold in an expansion). In other words, explicit nega-tion allows the overriding to false of a rule’s head when its body is undefined.Because of this feature, there is no backward propagation of falsity of the headinto the rule’s body. On the other hand, when the rule’s body is true, then itshead must necessarily be true, which, however, represents a forward, rather thanbackwards, propagation of truth values.

Notice that an explicit negation model may have evidence for both go tochurch and ∼go to church. Not so for strong or classical negations. However,when there is overwhelming evidence for ∼L (i.e., in all models), but not forL (i.e., only in some of the models), then explicit negation, via the Coherenceinference rule, decides in favor of ∼L, and vice versa. 2

? If A = ∼F , then we have F

B¬F.

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As noted above, the definition of explicit negation, contrary to that of strongnegation, does not prevent the existence in a model of both A and ∼A, for someatom A. However, this kind of ‘paraconsistency’ in models does not spill overto AEBX expansions.

PROPOSITION 4.1. Let T♦ be a consistent expansion of a belief theory T inAEBX. Then, for no atom A

T♦ |= A ∧ ∼A.

Proof. By contradiction, assume that T♦ is a consistent expansion of T suchthat T♦ |= A ∧ ∼A, for a given arbitrary atom A. Since T♦ is closed underthe necessitation rule (N), and by hypothesis A ∈ T♦, also BA ∈ T♦. Giventhat, by hypothesis, ∼A ∈ T♦ and that T♦ is, by definition of AEBX, closedunder the Coherence axiom, also BA ∈ T♦. Thus, according to the distribu-tive law for conjunctions mentioned in Section 2, B(A ∧ ¬A) ∈ T♦, that is,B⊥ ∈ T♦. Accordingly, given the consistency axiom (D), T♦ is inconsistent(contradiction). 2

In other words, if there is overwhelming evidence both for and against someatom A, then there are no consistent expansions.

The following result shows that the relevant logic program semantics WFSX(defined in [28]) is embeddable in AEBX. This embeddability result requires,besides the translation defined in Section 2.3, a preliminary WFSX-semanticspreserving transformation of the logic program. This transformation consists inthe complete elimination of objective literal goals from rule bodies by meansof unfolding, that is by successively replacing them by their various alternativerule definitions. The obtained programs are said to be in ‘semantic kernel form’.Due to the length of the proof, and given that it requires a significant body ofdefinitions and results not needed elsewhere in this paper (such as the definitionsof WFSX and of ‘semantic kernel transformation’, plus the result showing thatthis transformation is WFSX-semantics preserving [4]), the proof of the followingtheorem is omitted here.

THEOREM 4.1 (Explicit Negation and WFSX). Let P be an extended logic pro-gram in the semantic kernel form. There is a one-to-one correspondence betweenthe partial stable models M of P , as defined in [28], and the consistent stat-ic autoepistemic expansions T♦ of its translation TB¬(P ) into an AEBX belieftheory, where ‘explicit negation’ of an atom A is translated into ∼A. 2

Up to now, we have mainly considered AEBX theories resulting from the trans-lation of (nondisjunctive) logic programs. This is so because we have motivated

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explicit negation by contrast with strong negation on such programs. Neverthe-less, there is motivation to extend AEB with explicit negation to general theories,and not just logic programs.

The theory of Example 3.1, with strong negation replaced by explicit negationeverywhere, illustrates such general use. In fact, the results will be the same,except when later one adds ∼good researcher(Paul), expressing that there isevidence against Paul’s being a good researcher.? As shown in Example 3.1,by using strong negation, one additionally concludes that Paul will receive anegative evaluation and that Keith is a good researcher and so receives a positiveevaluation. When explicit negation is used instead, we still conclude that Paulwill receive a negative evaluation, but we no longer surmise that Keith is a goodresearcher. Instead, a milder conclusion follows, that Keith is believed to be agood researcher. This conclusion is not enough to give Keith a good evaluation.See Example 4.4 for a detailed explanation of how these results are obtained.

4.3. EXPLICIT AND STRONG NEGATION COMPARED

Explicit and strong negation share some similarities. The (derived) inference rulefrom strong negation to beliefs shown in Proposition 3.2, also holds for explicitnegation. Explicit negation is in fact characterized by taking this inference ruleas primitive, since it is no longer derivable in the absence of the strong negationaxiom.

Another important similarity between both regards the symmetry betweenatoms and their negations. The result of Proposition 3.3 still holds if strongnegation is replaced by explicit negation. In other words, explicit negation isalso symmetric.

Because of the similarities, strong and explicit negation are even equivalent forthe class of theories whose expansions capture the semantics of logic programsunder Stable Models. Indeed, the result in Theorem 3.1 remains true regardlessof whether ‘classical’ negation is translated into strong or into explicit negation.

For logic programs, explicit negation can be seen as an approximation tostrong negation, in the sense that any relevant consequence of a belief theory withstrong negation remain true in AEBX after strong negation is replaced everywhereby explicit negation:

PROPOSITION 4.2. Let Ts be an AEB theory with strong negation obtainedfrom an extended logic program P via the translation described in Section 2.3,Tx be the AEBX theory obtained from Ts by replacing strong by explicit negationand deleting all the strong negation constraints in Ts, and T♦s be an expansion ofTs. Then there exists an expansion T♦x of Tx such that, for every objective atomA:

– BA ∈ T♦s if and only if BA ∈ T♦x ,

? In Example 3.1 we added instead bad researcher(Paul), standing for∼good researcher(Paul).

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– B¬A ∈ T♦s if and only if B¬A ∈ T♦x . 2

Proof. It is easy to check that, for theories resulting from (nondisjunctive)logic programs, T♦s can be expressed as

T♦s = CnAEB(Ts ∪ {BA : T♦s |= A} ∪ {B¬A : T♦s |=min ¬A}),where A ranges over all objective atoms. Moreover:

– BA ∈ T♦s if and only if T♦s |= A,– B¬A ∈ T♦s if and only if T♦s |=min ¬A.

Thus, it is enough to prove that

T♦x = CnAEBX(Ts ∪ {BA : BA ∈ T♦s } ∪ {B¬A : B¬A ∈ T♦s })is an expansion of Tx, that is,

T♦x = CnAEBX(Tx ∪ {BA : T♦x |= A} ∪ {B¬A : T♦x |=min ¬A}),where CnAEBX is the closure operator with the Coherence inference rules.

Let T ′x = Tx∪{BA : BA ∈ T♦s }∪{B¬A : B¬A ∈ T♦s }, and let T ′s be obtainedfrom T ′x by adding the strong negation constraints. Clearly, T♦s = CnAEB(T ′s)and, by Proposition 3.2, T♦s = CnAEBX(T ′s). We now prove that the minimalmodels of T ′s exactly coincide with the minimal models of T ′x, thus proving that,given that CnAEBX(T ′s) is an expansion, CnAEBX(T ′x) = T♦x is an expansionof Tx.

Let M1, . . . ,Mn, . . . be all the minimal models of T ′s. Since T ′x ⊆ T ′s all thesemodels are models of T ′x. Moreover, since the strong negation constraints deleteonly models M such that, for some objective atom A, both A and ∼A are truein M , then all the Mis are minimal models of T ′x.

It remains to be proven that T ′x has no other minimal models. This is provenin two steps: first we prove that (4.3) for a given fixed combination of beliefpropositions in some Mi, the only minimal model of T ′x is Mi; then we provethat (4.3) no new combinations of belief propositions are possible.

(1) Since T ′s results from the translation of a nondisjunctive program, for a givenfixed combination of belief propositions, there exists a single minimal modelMi. By removing the strong negation constraints, new models appear where,for some objective atom A, both A and ∼A hold. These models differ fromMi at least in that one of A or ∼A became true. Since T ′x resulted from thetranslation of a nondisjunctive logic program, the addition of atoms can onlyhave as consequence the addition of new atoms.? Thus all these models arestrictly greater than Mi, and so they are not minimal.

? Note that this is not the case for a disjunctive program where the addition of an atom mightcause a new minimal model to appear. For example, the theory T = {∼a, a ∨ b, ∼a ⊃ ¬a}has the minimal model {∼a,¬a, b}. By removing the strong negation constraint (the last clause),a new minimal model appears {∼a, a,¬b}. In this case the addition of a causes the deletion of b.

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(2) By contradiction, assume that T ′x has a minimal model N with a combinationof belief propositions that does not exist in any of the Mis, that is, N differsfrom any of the Mi in that there exists at least one literal L such thatBL ∈Mi and ¬BL ∈ N , or ¬BL ∈Mi and BL ∈ N .

– Assume that BL ∈Mi and ¬BL ∈ N . If additionally T ′s |=min L, then, byconstruction of T ′x, BL ∈ T ′x, and N is not a model of T ′x (contradiction).Otherwise, that is, if T ′s 6|=min L, there exists a minimal model of T ′swith ¬BL, and so the combination of belief propositions in N is not new(contradiction).

– Assume that ¬BL ∈Mi and BL ∈ N . If T ′s |=min L; then, since T ′s is anexpansion, all its minimal models have BL, and so Mi is not a model ofT ′s (contradiction). If T ′s |=min ¬L then, by construction of T ′x, B¬L ∈ T ′x,and N is not a model of T ′x (contradiction). Otherwise, that is, T ′s 6|=min Land T ′s 6|=min ¬L, there exists a minimal model of T ′s with BL, and so thecombination of belief propositions in N is not new (contradiction). 2

COROLLARY 4.1.1 (Explicit Negation Approximates Strong Negation). Let Tsand Tx be as in Proposition 4.2. If some formula F of the form BL, where L iseither an objective atom A or its negation ¬A, holds in every expansion of Tx,then F also holds in every expansion of Ts. 2

Consequently, query evaluation procedures for explicit negation in logic pro-grams can be used as sound query evaluation procedures for strong negation inlogic programs.? However, those procedures are not complete for strong nega-tion, since, as shown by Examples 4.1 and 4.2, the converse of Corollary 4.1.1is not true in general.

Another result contrasting explicit and strong negation is as follows.

THEOREM 4.2 (Explicit Negation Extends Strong Negation). There is a one-to-one correspondence between expansions of a theory Ts with strong negation andexpansions of the AEBX theory Tx obtained by replacing in Ts strong by explicitnegation, and by adding, for every atom A, the clause ∼A ⊃ ¬A. 2

Proof. The only possible difference between expansions of Tx and those of Tswould be that the former are closed under Coherence inference rules. However,as shown by Proposition 3.2, all expansions of Ts are also closed under thoserules. 2

Disjunctive syllogism is a derived inference rule that is applicable to classicaland strong negations, but that is not enjoyed by explicit negation. This ruletypically allows one to conclude A on the strength of A ∨B and ¬B (or ∼B).

? In Section 6, the reader can find a brief discussion on the implementation of explicit negation.

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EXAMPLE 4.4. Consider the theory

∼good researcher(Paul)good researcher(Keith) ∨ good researcher(Paul)

stating that ‘there is evidence against Paul being a good researcher’ and that ‘atleast one of Keith or Paul is a good researcher’.

Whereas with strong negation good researcher(Keith) follows from thetheory by virtue of disjunctive syllogism, with explicit negation it is not so.Instead, a milder conclusion obtains.

First, note that good researcher(Keith)∨good researcher(Paul) is equiv-alent to the formula ¬good researcher(Paul) ⊃ good researcher(Keith).Now, applying Necessitation, B(¬good researcher(Paul) ⊃ good researcher(Keith)). Then, by applying axiom (K), we conclude B¬good researcher(Paul)⊃ Bgood researcher(Keith). Since the premise of this implication followsfrom ∼good researcher(Paul) by Coherence, we extract the milder conclusionBgood researcher(Keith).

Intuitively, having evidence against Paul being a good researcher does notensure that Paul is not a good researcher, and so does not warrant Keith’s beinga good researcher. However, this information is enough for believing Paul is nota good researcher, and consequently to believe Keith is. Contrast this readingwith the one of strong negation, where ∼good researcher(Paul) means ‘Paulis a bad researcher’. In this case, and knowing that either Paul or Keith is a goodresearcher, it is expected that Keith is a good researcher. 2

5. Application of Explicit Negation

Applications of the logic programming semantics WFSX have been studied invarious domains, including hypothetical reasoning [29], model-based diagno-sis [31], declarative debugging of logic programs [30], and updates [5]. Manyof these applications require belief revision methods, via contradiction removaltechniques [3]. Here we show how to capture belief revision in AEBX, and weillustrate how to extend the above mentioned applications from the class of logicprograms to the significantly broader class of belief theories.

Unlike normal programs, extended logic programs with symmetric negationmay be contradictory, that is, might not have a (consistent) semantics. While forsome programs this seems reasonable (e.g., for a program with the two facts aand ∼a), for some others this may not be justified.

EXAMPLE 5.1. Consider the program

runs ← not broken∼runs

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stating that ‘if we assume that a car is not broken, then it runs’ and that ‘the cardoes not run’.

None of the logic programming semantics mentioned in this paper assignsa consistent meaning to this program. Indeed, since there are no rules in theprogram for broken, all of the semantics assume not broken true (¬broken istrue in all minimal models of the corresponding AEB theory), and so both runsand ∼runs hold. If ∼runs in the logic program is understood as the strongnegation of runs, then, since runs ∧ ∼runs ⊃ ⊥, no consistent expansion ofthe corresponding belief theory exists. The same happens if ∼runs is understoodas the explicit negation of runs; cf. Proposition 4.1.

But one can argue that, since the car does not run, it follows by reductio adabsurdum that our assumption that the car is not broken must be incorrect andthus has to be revised. 2

The Contradiction Removal with Explicit Negation (CRSX) technique of [3]removes contradiction from logic programs under WFSX (wherever possible), byrevising any default assumption that would otherwise lead to contradiction. Todo so, it adds to programs a rule of the form

L← not L

for each such assumption not L, with the effect that no model of the programcan now contain not L. The alternative minimal contradiction removing sets ofsuch ‘inhibition rules’ can be added to the original program in order to obtainthe alternative minimally revised programs.

EXAMPLE 5.2. The only revision of the program in Example 5.1 is obtained byadding to it

broken← not broken.

The revised program is no longer contradictory: its WFSX is {∼runs, not runs}.Furthermore, this addition is minimal. 2

The addition of inhibition rules will be allowed for only some pre-designatedset of literals – the revisables. These are the literals that can be independentlyrevised. Other literals may have their truth value revised but only as a con-sequence of changes in the revisables. Restricting the revisables is crucial forcontrolling the causal level at which assumption withdrawal may be performed.Indeed, in an application domain such as diagnosis, revising the functional nor-mality assumption about a component may be conditional upon revising thefunctional normality assumption about some subcomponent. Conversely, it maysuffice to hypothesize the abnormality of a subcomponent to put in question thenormality of the component containing it. We deal next with both issues.

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EXAMPLE 5.3. Consider now the (contradictory) program

runs ← not broken

∼runsbroken ← flatT ire

broken ← badBattery

If addition of inhibition rules is allowed for any literal, then one such minimaladdition removing the contradiction is {broken ← not broken}. This revisioncan be seen as a diagnosis of the car that simply states the car might be broken.However, in this case one would like the diagnosis to delve deeper into thecar’s functional structure and obtain two minimal diagnoses: one suggesting apossible problem with a flat tire, and another suggesting a possible problem witha bad battery. This is achieved by declaring as revisables only badBattery andflatT ire. 2

This declaration of revisables is akin to the declaration of abducible literals inabductive logic programming. There, a solution to an abductive query is obtainedby minimally, and consistently, adding to the theory facts needed in order to provethe abductive query literal. Moreover, addition of facts is allowed only for literalsdeclared abducibles.?

EXAMPLE 5.4. Consider the program obtained from Example 5.3 by removingthe fact ∼runs:

runs ← not broken

broken ← flatT ire

broken ← badBattery

If every literal is abducible, then one abductive solution for not runs is {broken}(i.e., this is one possible explanation of the fact that the car is not running).To obtain only the deeper diagnosis of the car, simply declare as abducibles{flatT ire, badBattery}: the abductive solutions to that query become then{flatT ire} and {badBattery}. 2

This technique can easily be imported into AEBX. Define the revisions of atheory T which has no expansions (a contradictory theory), as the noncontradic-tory theories obtained by minimally adding clauses of the form

B¬L ⊃ L

which disallow (consistent) expansions in which B¬L is true.??

? A formal comparison of contradiction removal and abduction in logic programs can be foundin [11].?? An expansion with B¬L would also contain L and BL, since expansions are closed under

Necessitation, and would thus be inconsistent.

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It is worth emphasizing that, as opposed to logic programs, in AEBX there isno need for recourse to a meta-linguistic declaration of revisables. The languageitself is sufficiently expressive to handle the definition of revisables. To realizethis, let us get back to the car example, which in AEBX is rendered as the belieftheory T :

B¬broken ⊃ runs∼runsbadBattery ⊃ brokenflatT ire ⊃ broken,

where broken is nonrevisable. In CRSX this declaration of nonrevisability meansthat broken may have its truth value revised only indirectly, as a consequenceof changes in the revisables, but never directly on account of the addition of aninhibition rule for it. In AEBX it means that beliefs about broken may changeonly indirectly, as a consequence of changes in beliefs about revisables, but neverdirectly because of the addition of an inhibition clause for it. To achieve this, wemust require that belief changes about broken be solely determined by the beliefchanges about the revisables, so that adding a single inhibition clause, for brokenonly, is insufficient because the resulting theory has no consistent expansions.

By Necessitation and the axiom (K), the closure of T contains

BflatT ire ∨ BbadBattery ⊃ Bbroken.

Thus, belief in the truth of broken follows from belief in flatT ire or belief inbadBattery. For the falsity of belief in broken to be determined by the falsityof beliefs in flatT ire and badBattery, we need an additional statement, whichensures that if both flatT ire and badBattery are disbelieved, then broken mustbe disbelieved as well, that is,

B¬flatT ire∧ B¬badBattery ⊃ B¬broken. (3)

EXAMPLE 5.5. The belief theory T above, augmented with clause (3), has tworevisions

T ∪ {B¬badBattery ⊃ badBattery}T ∪ {B¬flatT ire ⊃ flatT ire}

each corresponding to one of the desiredly deeper diagnoses.Observe that T ′ = T ∪ {B¬broken ⊃ broken} is not a revision. Indeed,

all minimal models of the theory have ¬flatT ire and ¬badBattery, becausethere are no clauses defined for either flatT ire or badBattery. Thus everyexpansion of T ′ must contain {B¬flatT ire,B¬badBattery}. Now, by clause(3), every expansion of T ′ includes B¬Broken, and thus, by the inhibition clausefor broken and by Necessitation, is inconsistent. 2

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Notice the similarity between clause (3) and Clark’s completion [8] of broken.The latter implies that if both flatT ire and badBattery are false, then brokenis false, while (3) states the same about the corresponding beliefs. For this reason(3) is called the belief completion clause for broken. The formal definition ofbelief completion rules can be found in [6].

The specification of revisables is captured in the language of AEBX by addingfor each literal which is not a revisable the corresponding belief completionclause. This language level declarative specification achieves the same effectas the declaration of revisables for logic programs, which in the latter casecan only be made meta-linguistically. Though not detailed here, it can also beused to explain the intuitive notion that a literal is abducible unless the rulesdefining it are closed by a completion clause. Moreover, it has greater generality,by allowing conditions to be added to belief completion clauses. For example,instead of simply stating that broken is not a revisable, we might want to saythat broken is not revisable only if the atom deeper is a consequence of thetheory. In such a case we would just add

deeper ∧ B¬flatT ire∧ B¬badBattery ⊃ B¬broken.

This allows for flexible control over the level of revision: adding the fact deeper,or concluding it, results in only causally deeper faults being obtained. Other-wise, superficial faults can be obtained. Diverse complex diagnosis preferencesand strategies can be encoded in AEBX (see [10] to find out how this can beaccomplished by changing the revisables in logic programs).

We conclude by remarking that either one of symmetric negation in AEB canbe used to obtain a solution to belief revision not relying on a theory transfor-mation: instead of the addition of minimal sets of inhibition rules, alternativerevisions can be obtained directly by modifying the definition of expansions,leading to the notion of Careful Autoepistemic Expansions (see [6]).

6. On the Implementation of Explicit Negation

As we argued above, one advantage of explicit negation is that theories that aretranslatable into logic programs can be implemented and queried by efficient(strictly) top-down methods based on standard logic programming techniques.

Top-down query evaluation procedures for explicit negation in extended log-ic programs can be obtained by adapting existing procedures for well-foundedsemantics of normal programs. One such adaptable derivation procedure is definedin [1]. It relies on the definition of two kinds of derivation: one, called verityderivation, proving truth, and another proving nonfalsity of literals (where non-falsity of L means that not L 6∈ WFM ). Shifting from one kind of derivationto the other is required when trying to prove a default literal not L: the verityderivation of not L succeeds if and only if the nonfalsity derivation of L fails;the nonfalsity derivation of not L succeeds if and only if the verity derivation

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of L fails. Apart from the switch between derivation types at default literals, thederivation procedure relies on the usual SLD-like rewrtting techniques.

Two adaptations are required to incorporate explicit negation (described in [1],where soundness and completeness wrt. WFSX are proven). The first incorporatesone extra method for deriving truth of a default literal not L – the verity deriva-tion of not L succeeds, additionally, if the verity derivation of ∼L succeeds. Thesecond restricts nonfalsity derivations of objective literals – a nonfalsity deriva-tion of L fails if a verity derivation of ∼L succeeds. The first adaptation directlyreflects the Coherence inference rule that defines explicit negation. Also as aconsequence of this rule, if some ∼L is true then not L must be also true andL cannot be false. This justifies the failure required by the second adaptation innonfalsity derivations.

7. Conclusion

We have shown that, in order to represent and reason about knowledge, onerequires a form of negation that is not automatically assumed by default andthat is symmetric with respect to default negation. Neither classical negation nordefault negation satisfies these conditions.

We then introduced, within the broad framework of the Autoepistemic Logicof Beliefs, two types of symmetric negation, strong and explicit, and illustratedtheir application on a number of knowledge representation examples.

Although similar, strong negation and explicit negation differ in their use ofcontrapositives and their amenability to strictly top-down querying procedures.They are alike in that they both capture the answer-sets semantics [15], differfrom classical negation, and enjoy symmetry. Of the two, explicit negation isweaker and easier to implement. It can be viewed and used as an approximationto strong negation.

ACKNOWLEDGMENTS

The authors are grateful to D. Vakarelov and D. Pearce for their helpful com-ments.

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classical' negation in nonmonotonic reasoning and logic programming

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