operational concepts of nonmonotonic logics part 2: autoepistemic logic

Artificial Intelligence Review12: 431–443, 1998. 431c 1998Kluwer Academic Publishers. Printed in the Netherlands.

Operational Concepts of Nonmonotonic LogicsPart 2: Autoepistemic Logic

GRIGORIS ANTONIOUSchool of Computing and Information Technology, Griffith University, QLD 4111, AustraliaE-mail: [email protected]

VOLKER SPERSCHNEIDERDepartment of Mathematics and Computer Science, University of Osnabruck, 49069Osnabruck, GermanyE-mail: [email protected]

Abstract. The subject of nonmonotonic reasoning is reasoning with incomplete information.One of the main approaches is autoepistemic logic in which reasoning is based on introspection.This paper aims at providing a smooth introduction to this logic, stressing its motivation andbasic concepts. The meaning (semantics) of autoepistemic logic is given in terms of so-calledexpansions which are usually defined as solutions of a fixed-point equation. The present papershows a more understandable, operational method for determining expansions. By improvingapplicability of the basic concepts to concrete examples, we hope to make a contribution to awider usage of autoepistemic logic in practical applications.

Key words: autoepistemic logic, knowledge representation, nonmonotonic reasoning

1. Motivation

Complete information is difficult to come by and generally not available evenin simple database applications. Consequently, an intelligent system mustbe capable of making plausible conjectures, which may be retracted whenfound to be incorrect according to new information that becomes available.Nonmonotonic Reasoning provides mechanisms for an intelligent reasoningsystem to make such conjectures when its knowledge is incomplete.

A well–known example for making conjectures is the use of Negation AsFailure in logic programming: for a ground atomic formulaA,notA succeedsin case no proof forA can be found; that means, we concludenotA by defaultwhenA is not currently known.

One of the most prominent methods for nonmonotonic reasoning is auto-epistemic logic, a formalism that was developed by Moore (Moore 1985). Itsmain idea is to give a formal account of an agent reasoning about his ownknowledge or beliefs. Consider the following dialog:

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432 GRIGORIS ANTONIOU & VOLKER SPERSCHNEIDER

Are the Rolling Stones giving a concert in Newcastle next week?No, because else I would have heard of it.

We can illustrate some interesting points using this example. First, it is clearthat I have no definite knowledge that the Rolling Stones are not giving aconcert in Newcastle next week. In this sense, my knowledge is incomplete,and, by giving a negative answer, I am making a conjecture. This conjectureis based on reflection upon the knowledgeI have (if something so importantis happening in my city, then I will know). The wordautoepistemicmeansreflection upon self-knowledge.

To proceed with the example, suppose that I buy the Newcastle Herald themorning after the conversation took place, and read the headlines:

The concert of the century: The Rolling Stones in Newcastle next week!

Now situation has changed. I know that the Stonesaregiving the concert, somy answer to the question above would now be “Yes”. This means that theold conclusion I had drawn from introspection is no longer valid and mustbe revised. In this sense, my reasoning is nonmonotonic (new informationhas invalidated a previous conclusion). Note, though, that my long-termknowledge has not changed: I can still argue“If something so important istaking place in my city, then I will know”. The only difference is that now Ido know that the concert will take place, so I cannot conclude the contrarybased on my own knowledge.

Autoepistemic logicformalizes this kind of reasoning. The approach ittakes is to introduce a so-called modal operatorL that is applied to first ordersentences (i.e. formulae without free variables).

L'

has the meaning

I know' (or I believe that').

Our concert example is formulated as follows:

concert! Lconcert (“If a concert takes place then I know it”):Lconcert (“I don’t know that a concert will take place”)

The first rule is equivalent to:Lconcert ! :concert, which says that Iconcludenot concert if I do not explicitly knowconcert. This is similar tothe idea of negation as failure, in that my failing to find a proof forconcert

based on my current knowledge leads to the conclusion thatconcert is wrong.

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NONMONOTONIC LOGICS PART 2: AUTOEPISTEMIC LOGIC 433

The L-operator may be applied in a nested way in the sense that, forexample, I may know that I don’t know something. Thus, we may write downLLp, L:Lq, :LL(:p _ Lr) etc.

The semantics of autoepistemic logic is given in terms of so-calledexpan-sions, pieces of knowledge defining a sort of “world views” compatible withand based on the given knowledge. One of the main properties of expansionsis stability: a setE is calledstableif the following conditions hold:� ' 2 E ) L' 2 E� ' 62 E ) :L' 2 E.The concept of stability clearly reflects introspection, i.e. autoepistemic

reasoning: if A is in my knowledge, then I know A, I know that I know A etc.On the other hand, if I do not know B, then I know that I do not know B etc.

The formal definition of expansions is given by means of a fixed-pointequation. Though mathematically elegant, such a definition is not adequate foractuallyusingthe concepts of the logic, for example by applying them to con-crete applications. The following, admittedly early, statement by Moore, thedeveloper of autoepistemic logic, illustrates that the problem of determiningexpansions is not only restricted to novices in the topic:“One of the problemswith our original presentation of autoepistemic logic was that, since bothlogic and semantics were defined nonconstructively, we were unable to easilyprove the existence of stable expansions of nontrivial sets of premises...”.

This paper is a sequel to (Antoniou and Sperschneider 1994) which pro-vided a tutorial–style presentation of default logic based on an operationalmodel. It describes a simple method for determining expansions of givenautoepistemic theories (pieces of knowledge given in autoepistemic logic).As we intend to give a tutorial on the subject, we have concentrated onmotivation, description of the method, and discussion of examples, and haveleft out the formal proofs that can be found in (Antoniou and Sperschneider1993). Nevertheless, a basic understanding of first order logic is necessaryto fully understand the following sections. In cases of discomfort, the readermay refer to (Sperschneider and Antoniou 1991) which also describes thenotation we are using.

The paper is organized as follows: section 2 presents the syntax (language),section 3 the semantics (meaning) of autoepistemic logic. Section 4 is centraland introduces an operational interpretation of the logic. Section 5 discussessome examples, while in section 6 we give some historical and bibliographicalremarks.

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2. The language of autoepistemic logic

As already said in the previous section, the language of first order logic isextended by a modal operatorL to give the language of autoepistemic logic.

Definition 2.1The set ofautoepistemic formulae(AE-formulae) is defined asthe smallest set satisfying the following:� Each closed first order formula is an AE-formula.� If ' is an AE-formula, thenL' is also an AE-formula.� If ' and are AE-formulae, then so are the following::', (' _ ),(' ^ ), ('! ).

For is the set of all AE-formulae. Anautoepistemic theory(AE-theory) is aset of AE-formulae, i.e. a subset ofFor.

Let us point out that the definition above allows application of the L-operatorto a quantified first order formula (if it is closed!), but not vice versa: it isimpossible to quantify over beliefs. So, whereasL8X9Y X < Y (“I knowthat for all natural numbers X there is a greater number Y”) is an autoepis-temic formula,8XL9Y X < Y (“For all natural numbers X, I know thatthere is a greater number Y”) is not an AE-formula, because9Y X < Y

is not closed. The reason for this restriction is that it is difficult to definethe meaning of quantification over knowledge or beliefs; current research istrying to erase this restriction (see also section 6).

Definition 2.2 The kernelT0 of an AE-theoryT is defined as the set ofall first order formulae that are members ofT .

So, ifT = fp; Lp;:Lq;L:Lr; rg, thenT0 = fp; rg. Also, if T = fp^Lq!rg thenT0 = ;. The concept of kernel will be extensively used in subsequentsections. We conclude this section with a further technical definition whichwill be needed in the presentation of our method of determining expansions.

Definition 2.3 Given an AE-theoryT , sub(T ) is the union ofsub('), forall ' 2 T , wheresub(') is defined as follows:� sub(') = ; for first order formulae'� sub(:') = sub(')

� sub(' _ ) = sub(' ^ ) = sub('! ) = sub(') [ sub( )

� sub(L') = f'g.sub(') consists of the subformulae of' that occur immediately after theoccurrence of the modal operatorL. Furthermore, the last item in this defini-tion lays down that we do not go further into the structure of a formula after

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the outmost occurrence ofL. So, if T = fL:Lq;L(Lp _ r);:Lr; sg, thensub(L) = f:Lq; (Lp _ r); rg.

3. Expansions of autoepistemic theories

What is the knowledge an agent with introspection would have, given a setof facts (AE-formulae)T? It will be a setE of AE-formulae that� includesT .� allows introspection, in the sense that I may reason about what I know

and what I do not know.� is grounded inT in the sense that the knowledge inE must be

reconstructable usingT , belief in (knowledge of)E, and non-beliefin (non-knowledge of)EC . That means, givenT , knowledge ofE, andnon-knowledge ofEC (the complement ofE), it must be possible tologically derive all the information included in the expansionE.

Definition 3.1 Let T andE be sets of AE-formulae. We denote the setfL'j' 2 Eg asLE, and the setf:L j 62 Eg as:LEC . j= denotes theusual first order entailment.

DefineT (E) := f'jT [ LE [ :LEC j= 'g. E is anexpansionof T iffE = T (E).

How does this definition realize the requirements just given? Well, ifE = T (E), thenE includesT becauseT (E) includesT . E allowsintrospection in the sense of a stable set: if' 2 E, thenL' 2 T (E), soL' 2 E. Similarly, if ' 62 E, then:L' 2 T (E), so:L' 2 E. The thirdrequirement is given by the definition ofT (E).

The only thing we must additionally say is that each AE-formulaL' isconsidered as a new atom of the logical language, and it is not necessaryor allowed to go into the structure of'. So, it is impossible to use logicalreasoningwithin the scope of anL-operator. Here are some simple examples:

fp! Lp; pg j= Lp

f(p _ q)! Lq; pg j= Lq

fL(p _ q)! Lq;Lpg 6j= Lq

The latter relationship illustrates what we were saying above.Lp andL(p_q)are treated as different atoms, so we cannot deriveL(p _ q) from Lp eventhough(p _ q) follows fromp.

Let us look at the definition of an expansion. What it essentially says isthe following: imagine that you decide to believe (include in your knowl-edge) a set of AE-formulaeT . Given this decision, a set of AE-formulae

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can be deduced from the given theoryT and the belief we adopted (repre-sented asLE [ :LEC). If in this way we turn out to obtain the belief setE we adopted (“Truth implies belief (knowledge)”), and onlyE (“Belief(knowledge) implies truth”), thenE is an expansion ofT .

Expansions arestable setsby definition (see section 1); therefore, they areoften calledstable expansions. So, if an expansionE containsp, then alsoLp,LLp,LLLp etc.; sinceE is deductively closed and(p_ q) follows fromp in classical logic,E must also contain(p _ q), L(p _ q) (by stability) etc.Similarly, if an expansion does not includeq, then it contains:Lq, L:Lq,(r _ LL:Lq) etc.

There is a close relationship between expansions (and more generally stablesets) and kernels (first order portions of AE-theories). Indeed, the followingtheorem should not be surprising, if one looks carefully at the properties ofstability.

Theorem 3.2 If, for two stable setsE and F , E0 = F0, thenE = F .Conversely, for each deductively closed first order theoryT there is a stablesetF such thatF0 = T .

Example 3.3The following theory is borrowed from (Brewka 1991).

(german ^ :L:drinks-beer)! drinks-beergerman

There is only one expansion of this theory. There is no possibility of deriv-ing:drinks-beer, therefore:L:drinks-beer is contained in the expansion.Thus, the first rule is applicable and derivesdrinks-beer. The only expansionof this theory has the kernelCn(fgerman; drinks-beerg). But if we extendthe theory by adding

(eats-pizza ^ :Ldrinks-beer)! :drinks-beereats-pizza

then the extended theory has two expansions: as before we may concludedrinks-beer, so the kernel of one expansion containsgerman; eats-pizza,anddrinks-beer. Alternatively we may decide to use the new rule regardingpizza eaters; since we do not knowdrinks-beer, we may conclude:drinks-beer, so the kernel of the other expansion containsgerman; eats-pizza and:drinks-beer.

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The reader is encouraged to prove the claims about the expansions of thetwo theories. But even if this task is carried out, the question remains whetherthere are more expansions! Even if the reader is quite uneasy by now, weencourage him/her to continue reading. The next section will show how todetermine all expansions of an autoepistemic theory. We shall return to thisexample in section 5.

4. An operational interpretation of autoepistemic logic

In this section we present a method for computing expansions of autoepistemictheories. LetT be such a theory. Since all expansions ofT are stable, we mustinclude in each of themLT , LLT; : : :, (' _ ) for each' 2 T and each ,etc. The approach so far is monotonic.

As autoepistemic logic is a nonmonotonic formalism, we are allowed tomake conjectures and include some formulae or their negation in an expansioneven if we are not forced to. In this case, some other formulae must beincluded, too. For example, if we decide not to believe in�, then:L�,L:L� etc. will be included in the expansion. Different conjectures may leadto different expansions.

Now, what are the obstacles in determining the expansions ofT?1. Nested occurrences of theL-operator make potential expansions hard to

deal with.2. There are potentially infinitely many conjectures that can be made, so

how can we compute all expansions?The first problem can be treated using Theorem 3.2 from the previous section:we turn our attention to potentialkernels of expansions, since the expansionsare determined by their kernels.

The second problem is eased by establishing aCoincidence Lemmawhichessentially says that it suffices to consider belief or non-belief in formulaefrom sub(T ) in order to determine the expansions ofT . As sub(T ) is finite,we have solved the second problem, too ; see (Antoniou and Sperschneider1993) for details. The following approach may be summarized as follows:� Partitionsub(T ) into a partE(+) you believe in, and a partE(�) you

do not believe in.� Compute the corresponding kernelE(0) of a potential expansion, usingT , belief inE(+), and non-belief inE(�).

� Check whether the stable set determined byE(0) is indeed an expansion(this test is carried out in a simpler way; see the following descriptionfor details).

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Example 4.1Consider the autoepistemic theoryT = fLp ! pg. There are two possibledecompositions ofsub(T ) = fpg: either we decide to believe inp or not.These two possibilities correspond exactly to the two lines of the followingtable. The tests carried out succeed, and we have as result thatT has exactlytwo expansions, one with kernelCn(;) (Cn(S) denotes the deductive closureofS) and one with kernelCn(fpg) (it is the information on the third column).The general method will be described in a moment, where the approach ofthis table is formally introduced and informally explained.

E(+) E(�) E(0) E(+) � E(0) E(�) \E(0) = ; expansion

fpg ; Cn(fpg) yes yes yes; fpg Cn(;) yes yes yes

To simplify the situation, we treat first autoepistemic theories without nestedoccurrences of theL-operator. In this case,sub(T ) consists, by definition,only of first order formulae.

Procedure for AE-theories withoutL-nesting

Expansions := ;FORALL partitionsE(+) andE(�) of sub(T ) DO

BEGINE(0) := f' 2 For0jT [ LE(+) [ :LE(�) j= 'gIF E(+) � E(0) AND E(�) \E(0) = ;THENExpansions := Expansions [ fE(0)gEND

END

Some short comments on this procedure:E(0) is the set of first order formulaethat follows fromT , belief inE(+) and non-belief inE(�). The conditionE(+) � E(0) tests whetherE(0) includes what we decided to believe in,while the conditionE(�) \ E(0) = ; tests whetherE(0) excludes every-thing we decided not to believe in. If both conditions are passed, then wehave determined an expansion (more precisely, the kernel of an expansion).Finally, we turn to the general case.

Example 4.2LetT = fLp! p;:L:Lpg. Then,sub(T ) = fp;:Lpg. The only differencein the table below is the following: becauseT contains nested occurrences oftheL-operator, the decomposition setsE(+) andE(�) are no longer sets of

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first order formulae. Therefore, it is impossible to compare them withE(0)which is still a potential kernel of an expansion. So, in columns 4 and 5 wehave to useE instead ofE(0);E is hereby defined as the uniquely determinedstable set with kernelE(0).

E(+) E(�) E(0) E(+) � E E(�) \E = ; expansion

fp;:Lpg ; For0 yes yes yesfpg f:Lpg Cn(fpg) yes yes yesf:Lpg fpg For0 yes no ?; fp;:Lpg Cn(;) yes no ?

Unfortunately, we have to pay a price for allowingL-nesting. Whereas it waseasy to read and understand the table in Example 4.1 (everything referred tofirst order, actually even to propositional logic), here we have to take stabilityinto consideration when testing columns 4 and 5. We give some hints tounderstand the answers in these columns:� 1. line: E(0) is inconsistent sinceL:Lp 2 LE(+) and:L:Lp 2 T .

Three times “yes” is clear then.� 2. line: T [ LE(+) [ :LE(�) = fLp ! p;:L:Lp;Lpg, thereforeE(0) = Cn(fpg). To establish the answer in the fifth column, notethatp 2 E, andE is stable and consistent. So,Lp 2 E and therefore:Lp 62 E.

� 3. line:T[LE(+)[:LE(�) contains bothL:Lpand:L:Lp, thereforeE(0) = For0.

� 4. line: T [LE(+)[:LE(�) = fLp! p;:L:Lp;:Lpg, soE(0) =Cn(;). Sincep is not member ofE, it follows :Lp 2 E and thereforeE(�) \E 6= ;.

The reader may ask why we have used “?” in the last column instead of“no”. The reason is the following: all we obtain from a failed test is thatthisdecomposition ofsub(T ) (represented by the current line in the table) is notsuccessful and does not justify thatE(0) is the kernel of an expansion. It maybe the case, though, thatanother decomposition yields exactly the same setE(0) and passes both tests in columns 4 and 5. Example 5.2 illustrates this.But, what we are sure about is the following: for each expansionE of T thereis a line in the table such that the tests are passed and the computed setE(0)is the kernel ofE. So, after completing the whole table, question marks areinterpreted as negative answers if there is not another line with the sameE(0)which passes the tests.

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Procedure for general autoepistemic theories

Expansions := ;FORALL partitionsE(+) andE(�) of sub(T ) DO

BEGINE(0) := f' 2 For0jT [ LE(+) [ :LE(�) j= 'gLetE be the unique stable set with kernelE(0)IF E(+) � E AND E(�) \E = ;THENExpansions := Expansions [ fEgEND

END

5. Some examples

Here we will apply the method of the previous section to a number of simpleexamples. The AE-theories have been chosen so that some interesting proper-ties of autoepistemic logic can be illustrated. All of them include noL-nestingin order to keep discussion as simple as possible.

Example 5.1Consider the theoryT = fp ! L:pg. The following table shows that noneof the decompositions ofsub(T ) = f:pg passes both tests. Therefore, thistheory does not possess any expansions.

E(+) E(�) E(0) E(+) � E(0) E(�) \ E(0) = ; expansion

f:pg ; Cn(;) no yes ?; f:pg Cn(f:pg) yes no ?

Example 5.2Let T = f:Lp ! q; Lq ! p; Lq ! qg; sub(T ) = fp; qg. The followingtable shows thatCn(fp; qg) is the kernel of the only expansion ofT . Notethat this set occurs twice in the table below, in line 1 and in line 3. Line 3 fails,but we write “?” and not “no” which would be wrong:E(0) is the kernel ofan expansion, obtained from line 1!


fp; qg ; Cn(fp; qg) yes yes yesfpg fqg Cn(;) no yes ?fqg fpg Cn(fp; qg) yes no ?; fp; qg Cn(fqg) yes no ?

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Example 5.3Let T = f:Lpg. The following table shows thatT has two expansions, oneof which is inconsistent (i.e. the set of all AE-formulae). The example showsthat a consistent theory may have an inconsistent expansion.


fpg ; For0 yes yes yes; fpg Cn(;) yes yes yes

Example 5.4Let T = fLp ! q; Lq ! pg. The table below shows thatT has twoexpansions. This is quite clear on the intuitive level: if I decide to believein either p or q, the rules inT force me to include the other atom in myknowledge, too. So, either I believe in both or in none of them.


fp; qg ; Cn(fp; qg) yes yes yesfpg fqg Cn(fqg) no no ?fqg fpg Cn(fpg) no no ?; fp; qg Cn(;) yes yes yes

Example 5.5

Finally we reconsider Example 3.3. LetT = fg; e; g ^ :L:d ! d; e ^:Ld! :dg (whereg stands forgerman, e for eats-pizza andd for drinks-beer). sub(T ) = fd;:dg. The following table shows thatT has exactlytwo expansions, as claimed in the discussion of Example 3.3 (For0 is theinconsistent set of all formulae).


fd;:dg ; Cn(fg; eg) no yes ?fdg f:dg Cn(fg; e; dg) yes yes yesf:dg fdg Cn(fg; e;:dg) yes yes yes; fd;:dg For0 yes no ?

6. Historical and bibliographical remarks

Developed by Moore (Moore 1984, 1985), Autoepistemic Logic is one of themain nonmonotonic formalisms, and is treated in books like (Brewka 1991,

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Marek and Truszczynski 1993). Its origin lies in the nonmonotonic modallogics presented in (McDermott and Doyle 1980, 1982). AE-logic has beenextensively studied in literature (e.g. Gelfond 1987, Marek 1989, Gelfond andPrzymusinska 1989). A thorough presentation of the mathematical propertiesis found in (Gelfond and Przymusinska 1992).

The main contribution of this paper is the method for determining expan-sions. It was presented in (Antoniou and Sperschneider 1993), where theproofs can be found, but is based on ideas that have been around in theliterature for a while.

One interesting question is the relationship between autoepistemic anddefault logic (Reiter 1980). In general, autoepistemic logic allows more free-dom in that expansions are more loosely grounded in the given theory thandefault extensions. Thus, the two methods are not equivalent. (Konolige1988a) shows that default theoryT can be represented as an AE-theoryT 0

such that the kernels of the so-calledstrongly grounded expansionsof T 0

correspond exactly to the extensions ofT ; see also (Marek and Truszczynski1989).

One of the main limitations of autoepistemic logic is that quantification intothe scope of theL-operator is not allowed. (Konolige 1991) extends the logicin this direction. The same author developed the Hierarchic AutoepistemicLogic (Konolige 1988b) to deal with some deficiencies of AEL: to allowpriorities among formulae, and to increase computational efficiency. The mainidea is to consider a collection of subtheories linked together in a hierarchy,and to use a set ofL-operators with restricted applicational scope. Thoughnew representational problems arise, (Appelt and Konolige 1988) shows aninteresting application.

References

Antoniou, G. & Sperschneider, V. (1993). Computing Extensions of Nonmonotonic Logics. InProc. 4th Scandinavian Conference on Artificial Intelligence. IOS Press.

Antoniou, G. & Sperschneider, V. (1994). Operational Concepts of Nonmonotonic Logics –Part 1: Default Logic.Artificial Intelligence Review8: 3–16.

Appelt D. E. & Konolige, K. (1988). A Nonmonotonic Logic for Reasoning about SpeechActs and Belief Revision. In Reinfrank et. al. (eds.),Nonmonotonic Reasoning, Proc. 2ndInternational Workshop. Springer LNAI 346.

Brewka, G. (1991).Nonmonotonic Reasoning: Logical Foundations of Commonsense.Cambridge University Press.

Gelfond, M. (1987). On Stratified Autoepistemic Theories. InProc. American National Confer-ence on Artificial Intelligence.

Gelfond, M. & Przymusinska, H. (1989). Formalization of Inheritance Reasoning in Auto-epistemic Logic.Fundamenta Informaticae13(4): 403–444.

Gelfond, M. & Przymusinska, H. (1992). On Consistency and Completeness of AutoepistemicTheories.Fundamenta Informaticae16: 59–92.

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Konolige, K. (1988a). On the Relation Between Default and Autoepistemic Logic.ArtificialIntelligence35: 343–382; see also Errata,Artificial Intelligence41: 115, 1989.

Konolige, K. (1988b). Hierarchic Autoepistemic Theories for Nonmonotonic Reasoning:Preliminary Report. In Reinfrank et al. (eds.),Nonmonotonic Reasoning, Proc. 2nd Inter-national Workshop. Springer LNAI 346.

Konolige, K. (1991). Quantifying in Autoepistemic Logic.Fundamenta Informaticae15(3–4).Marek, W. (1989). Stable Theories in Autoepistemic Logic.Fundamenta Informaticae12:

243–254.Marek, W. & Truszczynski M. (1989). Relating Autoepistemic Logic and Default Logic.Proc.

1st International Conference on Knowledge Representation and Reasoning.Marek, W. & Truszczynski, M. (1993).Nonmonotonic Logic – Context-Dependent Reasoning.

Springer.McDermott, D. & Doyle, J. (1980). Nonmonotonic Logic I.Artificial Intelligence13: 41–72.McDermott, D. & Doyle, J. (1982). Nonmonotonic Logic II: Nonmonotonic Modal Theories.

Journal of the ACM29(1): 33–57.Moore R. C. (1984). Possible-World Semantics for Autoepistemic Logic. InProc. Non-

Monotonic Reasoning Workshop. New Paltz.Moore, R. C. (1985). Semantical Considerations on Nonmonotonic Logic.Artificial Intelli-

gence25: 75–94.Reiter, R. (1980). A Logic for Default Reasoning.Artificial Intelligence13: 81–132.Sperschneider, V. & Antoniou, G. (1991).Logic: A Foundation for Computer Science. Addison-

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operational concepts of nonmonotonic logics part 2: autoepistemic logic

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