1 Introduction
Due to their applications in knowledge representation and, more specifically, in commonsense reasoning, abduction, diagnosis, belief revision, planning and reasoning about action, default and autoepistemic logics are among the most extensively studied nonmonotonic formalisms^{1}^{1}1For a detailed discussion of these two formalisms and of their applications, and for additional references, the reader is referred to [MT93].. Still, after almost two decades of research in the area several key questions remain open.
The first of them is the question of the relationship between default and autoepistemic logics. Default logic was introduced by Reiter [Rei80] to formalize reasoning about defaults, that is, statements that describe what normally is the case, in the absence of contradicting information. Autoepistemic logic was proposed by Moore [Moo84] to describe the belief states of rational agents reflecting upon their own beliefs and disbeliefs. Although the motivation and syntax of both logics are different, it has been clear for a long time that they are closely related. However, despite much work [Kon88, MT89a, Tru91, MT93, Got95] no truly satisfactory account of the relationship was found. Konolige [Kon88] related default logic to a version of autoepistemic logic based on the notion of a strongly grounded expansion — a concept that depends on a syntactic representation of a theory. Marek and Truszczynski related default logic to two modal nonmonotonic logics related but different from autoepistemic logic: the nonmonotonic modal logic N [MT89a] and nonmonotonic modal logic S4F [Tru91]. Finally, Gottlob [Got95] found a relationship between default and autoepistemic logics but the translation he used was not modular. In fact, he proved that a modular translation of default logic into autoepistemic logic does not exist. These results seem to point to some misalignment between extensions of default theories and expansions of modal theories. Our results in this paper finally clarify the picture.
Another problem is related to the fixpoint definitions of extensions (in the case of default logic) and expansions (in the case of autoepistemic logic). They provide no insights into constructive processes agents might use to build their belief sets based on default or modal theories describing base facts. Finally, there is a problem of high computational complexity of reasoning with extensions and expansions. The problems to decide the existence of an extension (expansion) is complete, the problem to compute the intersection of extensions (expansions) — it is needed for skeptical reasoning — is hard.
In this paper we develop a unifying semantic treatment of default and autoepistemic logics and use it to address the three issues discussed above. For each logic we define a family of 2, 3 and 4valued semantics and show that they include all major semantics for default and autoepistemic logics. Within our framework we define semantics that generalize KripkeKleene and wellfounded semantics for logic programs. These semantics allow us to approximate skeptical default and autoepistemic reasoning and they can be computed faster than extensions and expansions (assuming that the polynomial hierarchy does not collapse). Most importantly, we show that the unified semantic picture of default and autoepistemic logics described here allows us to pinpoint the exact nature of how they are related.
Our approach is motivated by the algebraic approach proposed by Fitting in his analysis of semantics for logic programs with negation [Fit99], and extends our earlier work on 3valued semantics for autoepistemic logics [DMT98]. It relies on the concept of a belief pair, a pair , where and are sets of 2valued interpretations. The concept of a belief pair generalizes the notion of a possibleworld structure, that is, a set of 2valued interpretations, often used to define the semantics of the modal logic S5 and also used by Moore and Levesque [Moo84, Lev90] in their work on autoepistemic logic. Specifically, each possibleworld structure can be identified with a belief pair . Belief pairs allow us to approximate the state of beliefs of an agent whose beliefs are represented by a possibleworld structure.
Given a possibleworld structure or a belief pair , it is often possible to describe a way in which (or ) could be revised to more accurately reflect the agent’s beliefs. Such a revision procedure can be formally described by an operator. Fixpoints of such operators are often used to specify semantics of nonmonotonic formalisms as they represent those belief states of the agent that cannot be revised away. Sometimes fixpoints that satisfy some minimality conditions are additionally distinguished.
These intuitions underlie the original definition of an expansion of a modal theory as a fixpoint of a certain operator on the set of all possibleworld structures [Moo84]. In [DMT98], it is shown that the operator can be extended to the set of belief pairs. The resulting operator, , yields a multivalued generalization of expansions and a semantics that approximates skeptical autoepistemic reasoning. We refer to it as the KripkeKleene semantics as it generalizes KripkeKleene semantics for logic programs. In this paper, we derive from the operator two new operators and show that their fixpoints give rise to semantics for autoepistemic theories under which circular dependence of beliefs upon themselves, present in autoepistemic logic of Moore, is eliminated. One of these semantics is shown to correspond to Reiter’s semantics of default logic. The other one can be viewed as a generalization of the wellfounded semantics.
While possibleworld semantics were used in the study of autoepistemic logic, they had only a marginal effect on the development of default logic. In this paper, we show that possibleworld semantics approach can be extended to the case of default logic. Namely, in a close analogy with autoepistemic logic, for every default theory , we introduce an operator defined on the set of belief pairs. We show that the operator gives rise to three other operators and that the fixpoints of these operators yield several semantics for default theories. Among these semantics are the semantics of extensions by Reiter, the stationary semantics [PP94], the wellfounded semantics for default logic [BS91] (it approximates skeptical reasoning under extensions), the semantics of weak extensions [MT89a] and the KripkeKleene semantics (it approximates skeptical reasoning under weak extensions).
Our results settle the issue of the relationship between autoepistemic and default logics. We show that the operators and are closely related if a default theory is interpreted as a modal theory given by the translation proposed by Konolige [Kon88]. Our results show that under the Konolige’s translation, the families of semantics for default and autoepistemic logics are isomorphic and default logic can be viewed, as has long been expected, as a fragment of autoepistemic logic. But this correspondence does not relate extensions and expansions! The semantics corresponding to these concepts occupy different locations in their respective families of semantics and have different properties.
We also point out that the KripkeKleene and wellfounded semantics studied in the paper have better computational properties than the semantics of expansions and extensions. We conclude with comments about further generalizations and open problems.
2 Preliminaries
The formal language for the semantic study developed in this paper is that of lattices, operators and fixpoints. The key result is that by Tarski and Knaster [Tar55] stating that a monotone operator on a complete lattice has a least fixpoint.
By we denote the set of atoms of the propositional language under consideration. The set of all 2valued interpretations of will be denoted by . Any set is called a possibleworld structure and can be viewed as a universal Kripke model [Che80]. Possibleworld structures are a basic tool in semantic studies of modal logics. They were also used in the context of autoepistemic logic [Moo84, Lev90, MT93].
A collection of all possibleworld structures will be denoted by . This set can be ordered by the reverse set inclusion: for , if (the smaller the set of possible worlds, the bigger the theory it determines). Clearly, is a complete lattice. To study formalisms based on the modal language, we define the truth function inductively as follows ( is a possibleworld structure, is an interpretation):

, if is an atom

Boolean connectives are handled in the standard Tarskian way

, if for every interpretation , , and , otherwise.
The value of a modal atom given by does not depend on . Thus, it is determined only by the possibleworld structure in question. It is clear that the metaknowledge specified by a possibleworld structure is complete: all modal atoms are either true or false with respect to .
For every modal theory , Moore [Moo84] defined an operator on by:
Moore called the theory of a fixpoint of an expansion^{2}^{2}2In the paper, we will frequently use the same term to denote the fixpoint of an operator and its theory. So, fixpoints of will be referred to as expansions, as well..
In [DMT98], Moore’s approach was extended to the 3valued case, in which a possibility of incomplete metaknowledge is admitted. A key concept is that of a belief pair, that is, a pair of possibleworld structures. In a belief pair , can be viewed as a representation of a conservative (pessimistic) view on what is believed while can be regarded as a representation of a liberal (gullible) view. If , formulas believed in according to the conservative view captured by , are believed in according to the liberal view represented by . Consequently, a belief pair such that is called consistent
. However, the ways in which the agent establishes the estimates
and may be independent of each other and, thus, we allow belief pairs such that is not a subset of . We refer to them as inconsistent^{3}^{3}3In [DMT98] only consistent belief pairs were considered.. In applications we are mostly interested in consistent belief pairs. Constructive techniques for building belief pairs given a base theory, which are described in the paper, result in consistent belief pairs. However, admitting inconsistent belief pairs completes the picture, leads to simple intuitions behind mathematical arguments and results in more elegant algebraic structures.With a belief pair and a 2valued interpretation we associate a 2valued truth function defined on a modal language. The idea is to define so that it provides a conservative estimate to the truth value of with respect to a belief pair . Since represents a conservative point of view and a liberal one, to get a conservative estimate we use to evaluate positive occurrences of modal atoms and to evaluate negative ones. The definition is inductive:

, for every atom

Conjunction and disjunction are treated in the standard Tarskian way


if for all . , otherwise.
We stress that when evaluating the negation of a formula the roles of and are switched, which ensures that modal literals appearing positively in a formula are evaluated with respect to while those appearing negatively are evaluated with respect to .
Clearly, to construct a liberal estimate for the truth value of with respect to a belief pair we can proceed similarly and use () to evaluate modal literals appearing positively (negatively) in . It is easy to see, however, that the resulting truth function can be expressed as (we reverse the roles of and ).
Conservative and liberal estimates of truth values of formulas can be combined into a single 4valued estimate. We say that the logical value of a formula is true, (false, ), if both conservative and liberal estimates of its truth value are equal to t (f). We say that the logical value is unknown, u, if the conservative estimate is f and the liberal estimate is t. Finally, the logical value of is inconsistent, i, if the conservative estimate is t and the liberal estimate is f. Thus, the estimates given by and yield a 4valued truth function
where , , and .
We define the metaknowledge of a belief pair as the set of all formulas such that both conservative and liberal estimates of the truth value of the epistemic atom coincide (are both true or are both false). Clearly, the metaknowledge of a belief pair need not be complete. For some modal atoms the two estimates may disagree ( may be assigned value u or i). However, it is easy to see, that for a complete belief pair , , for every interpretation . In other words, belief pairs and the truth function generalize possibleworld structures and the truth function . In addition, for a consistent belief pair , never assigns value i and coincides with the 3valued truth function defined in [DMT98].
The set of all belief pairs is denoted by . It can be ordered by the knowledge ordering : if , and . The set with the ordering forms a complete lattice and, consequently, a monotone operator on is guaranteed to have a least fixpoint by the result of Tarski and Knaster [Tar55]. The ordering coincides with the ordering of increasing metaknowledge (decreasing metaignorance): if and only if the set of modal atoms with the same conservative and liberal estimates with respect to is contained in the set of modal atoms for which conservative and liberal estimates with respect to are the same.
Let be a belief pair. Following [DMT98], we set
where
and
Fixpoints of the operator will be called partial expansions.
Speaking intuitively, the operator describes how an agent might revise a belief pair . The possibleworld structure is replaced by the structure consisting of those interpretations for which all formulas from are true according to the liberal estimates of truth values (given ). A liberal criterion for selecting possible worlds (interpretations) to results in a possibleworld structure capturing a conservative point of view. By duality between conservative and liberal approaches, can be replaced by consisting of all interpretations such that all formulas from are true according to the conservative estimates of truth values (given ).
Let us recall that stands for the operator introduced by Moore. The operator allows us to reconstruct the semantic approach to autoepistemic logic proposed by Moore.
Theorem 2.1
Let be a modal theory. Then, for every possibleworld structure , . Consequently, a belief pair is a fixpoint of if and only if is a fixpoint of .
Theorem 2.1 implies that there is a natural onetoone correspondence between the complete partial expansions of and the expansions of . Thus, partial expansions (consistent partial expansions) can be viewed as 4valued (3valued) generalizations of Moore’s expansions.
The key property of the operator is its monotonicity. It follows that has a unique least fixpoint. We denote it by and refer to it as the KripkeKleene fixpoint (or semantics) for ^{4}^{4}4There is a close analogy between the least fixpoint of the operator and the KripkeKleene semantics for logic programs.. KripkeKleene fixpoint has a clear constructive flavor (it can be obtained by iterating the operator , starting at the least informative belief pair, ). It approximates all partial expansions and, in particular, approximates the skeptical reasoning with expansions.
Theorem 2.2
Let be a modal theory.

The fixpoint is consistent.

For every partial expansion of , .

If is true with respect to then belongs to every expansion of . If is false with respect to then belongs to no expansion of .
Deciding the truth value of a modal atom with respect to the KripkeKleene fixpoint is in the class [DMT98]. Thus, unless the polynomial hierarchy collapses, it is a simpler problem than the problem of computing expansions or their intersection.
We can also use the KripkeKleene semantics as a test for the uniqueness of an expansion. Namely, we can first compute and check if is complete. If it is, has a unique expansion. This method is computationally better (again, if the polynomial hierarchy does not collapse) than the straightforward one which computes all expansions of . However, it is incomplete — there are theories with a unique expansion and such that is not complete.
3 Autoepistemic Logic
The operator allows us to define two additional operators: the operator defined on the lattice , and the operator defined on the lattice . They give rise to new semantics for autoepistemic logic that are closely related to the semantics of extensions for default logic. One of them is a perfect match to Reiter’s semantics of extensions for default logic, an object long sought after in the autoepistemic logic.
Let us recall that the operator associates with each belief pair its revised variant . The way in which is obtained is described by the operator . Namely, . If we fix , this operator becomes a monotone operator on (it follows from the fact that is monotone). Hence, its least fixpoint can be viewed as the preferred revision of , given a fixed . Let us define, then,
Similarly, we can argue that can be regarded as a preferred revision of , given . It turns out that . Thus, we define an operator on belief pairs as follows:
Clearly, is an operator on . The fixpoints of the operator (and also their theories) will be referred to as extensions. The choice of the term is not arbitrary. We show in Section 5 that extensions of modal theories can be regarded as generalizations of extensions of default theories. We have the following property relating fixpoints of the operators and .
Theorem 3.1
For every modal theory , a possibleworld structure is a fixpoint of if and only if a belief pair is a fixpoint of .
It follows that the semantics of fixpoints of can be viewed as a 4valued version of the semantics of extensions. Similarly, consistent fixpoints can be thought of as a 3valued generalizations of extensions. Consequently, we refer to the fixpoints of the operator as partial extensions.
The circular dependence allowing the agent to accept to the belief set just on the basis of this agent believing in , allowed under the semantics of expansions, is eliminated in the case of extensions. For instance, the theory has two expansions. One of them is determined by the possibleworld structure consisting of all interpretations, the other one — by the possibleworld structure consisting of all interpretations in which is true. It is this second expansion that suffers from circularargument problem: the belief in is the only justification for having in this expansion. In the same time, the theory has exactly one extension, the one given by the possibleworld structure consisting of all interpretations. The atom is not true in it and, hence, circular arguments are not used in the construction of this expansion.
The operator is antimonotone. It follows that the operator is monotone. Thus, by the result of Tarski and Knaster, it has the least fixpoint. We will denote this fixpoint by and refer to it as the wellfounded fixpoint (or semantics) of . Our choice of the term is again not accidental. This semantics is closely related to the wellfounded semantics of default logic [BS91] and logic programming [VRS91]. We have the following result indicating that wellfounded semantics can be used to approximate all partial extensions and, in particular, all extensions of a modal theory. It also shows that the wellfounded semantics provides a sufficient condition for the uniqueness of an extension.
Theorem 3.2
Let be a modal theory and let . Then:

The fixpoint is consistent.

For every partial extension , .

If , then belongs to every extension of . Similarly, if , then does not belong to any extension.

If is complete (that is, ) then has a unique extension corresponding to the possibleworld structure .
Based on the approach developed in [DMT98] for computing the KripkeKleene semantics, we can establish computational properties of the wellfounded semantics. We have the following result.
Theorem 3.3
The problem of computing the wellfounded semantics is the class .
Thus, assuming that the polynomial hierarchy does not collapse, computing the wellfounded semantics of a theory is easier than computing the intersection of extensions of (that is, the set of skeptical consequences of ). Since the wellfounded semantics of a modal theory approximates all extensions, it can be used to speed up the computation of their intersection.
The next result connects expansions and the KripkeKleene semantics with extensions and the wellfounded semantics. It shows that the wellfounded semantics is stronger than the KripkeKleene semantics and that (partial) extensions of are (partial) expansions of satisfying some minimality condition.
Theorem 3.4
Let be a modal theory. Then:

.

Every extension of is a minimal expansion of .

Every partial extension of is a minimal partial expansion of in the following sense: for every partial expansion , if and , then and .
We conclude this section with a schematic illustration of the panorama of semantics for autoepistemic logic. The central position is occupied by the operator . Its fixpoints yield the semantics of partial expansions and its least fixpoint yields the KripkeKleene semantics. Restriction of the operator to complete belief pairs leads to the operator , originally introduced by Moore, and results in the semantics of expansions. The operator also gives rise to the operators and that yield new semantics for autoepistemic logic: the semantics of extensions, the semantics of partial extensions and the wellfounded semantics.
4 Default Logic
While possibleworld semantics played a prominent role in the study of autoepistemic logics [Moo84, Lev90, DMT98] they have not, up to now, had a similar impact on default logic. In this section we will introduce a comprehensive semantic treatment of default logic in terms of possibleworld structures and belief pairs. Our approach will follow closely that used in the preceding sections.
We observed earlier that autoepistemic logic can be viewed as the logic of the operator . Its fixpoints, and fixpoints of the operators that can be derived from , determine all major semantics for autoepistemic logic. We will now develop a similar treatment of default logic.
As before, we start with a 2valued truth function that gives a conservative estimate of the logical value of a formula or a default with respect to a belief pair and an interpretation . For a propositional formula , we define . For a default , we set if at least one of the following conditions holds:

there is such that

there is , such that for every ,

.
We set , otherwise. Clearly, the definition of agrees with the intuitive reading of a default : it is true, according to a conservative point of view, if its prerequisite is false (even with respect to a liberal view captured by ) or if at least one of its justifications is definitely impossible (it is false according to a conservative point of view captured by ) or if its consequent is true (in ). As before, we can also argue that provides a liberal estimate for a truth value of with respect to (the roles of and are reversed).
Let be a default theory. We use the truth function to define an operator on the lattice of belief pairs:
where
and
This definition can be justified similarly as that of the operator in Section 3.
We will now define a 2valued version of the operator . To this end, we use the following result.
Theorem 4.1
Let be a default theory. If is complete then is also complete.
Let be a possibleworld structure. We define
where is a possibleworld structure such that (its existence is guaranteed by Theorem 4.1).
To the best of our knowledge, the operator has not appeared explicitly in the literature before. Its fixpoints, however, did. In [MT89a], the concept of a weak extension of a default theory was introduced and studied (the approach used there was prooftheoretic). It turns out that fixpoints of the operator correspond precisely to weak extensions of . Thus, the semantics given by the operator is precisely the semantics of weak extensions.
Theorem 4.2
Let be a default theory. Then:

A propositional theory is a weak extension of a default theory according to [MT89a] if and only if for a fixpoint of .

A possibleworld structure is a fixpoint of if and only if a belief pair is a fixpoint of .
In view of Theorem 4.2(1), we call fixpoints of the operator weak extensions. Theorem 4.2(2) implies that complete fixpoints of the operator are in onetoone correspondence with the fixpoints of the operator . Thus, we call fixpoints of the operator — partial weak extensions (they can be regarded as a 4valued generalization of weak extensions, consistent fixpoints can be regarded as 3valued generalizations).
The key property of the operator is its monotonicity. Thus, has a least fixpoint. We call it the KripkeKleene fixpoint and denote it by . We refer to the corresponding semantics as the KripkeKleene semantics for .
The KripkeKleene semantics can be obtained by iterating the operator starting with the least informative belief pair . Thus, it has a constructive flavor. Second, it approximates the skeptical reasoning with weak extensions and provides a test for uniqueness of a weak extension.
Theorem 4.3
Let be a default theory and let .

The fixpoint is consistent, that is, .

If for every , then belongs to every weak extension. If for some , does not belong to any weak extension.

If (that is, if is complete) then has a unique weak extension corresponding to the possibleworld structure .
The KripkeKleene semantics is computationally attractive. Deciding whether a formula is in the intersection of the weak extensions of a default theory is complete. In contrast, by adapting the methods developed in [DMT98] to the case of default logic we can show that the problem of computing (specifically, assuming , deciding whether for every , or whether for some ) is in the class .
So far we have not yet reconstructed the concept of an extension. In order to do so, we will now derive from two other operators related to default logic. Let us consider a belief pair . We want to revise it to a belief pair . We might do it by fixing and taking for a preferred revision of , and by fixing and taking for a preferred revision of .
It is easy to see that monotonicity of implies that the operator is monotone operator on . Consequently, it has a least fixpoint. This fixpoint can be taken as the preferred way to revise given . Thus, we define
As in the case of autoepistemic logic, one can see that also specifies the preferred way to revise given , that is . Thus, we define the operator on as follows:
It turns out that the concept of extension as defined by Reiter can be obtained from the operator . It is known that Reiter’s extensions are theories of fixpoints of the operator introduced by Guerreiro and Casanova [GC90] (see also [Lif90, MT93]). One can show that the operator coincides with the operator . Thus, we have the following result.
Theorem 4.4
A theory is an extension of a default theory if and only if for some fixpoint of .
In view of Theorem 4.4, we refer to the fixpoints of as extensions. We have the following result relating fixpoints of the operators and .
Theorem 4.5
Let be a default theory. For every possibleworld structure , is a fixpoint of if and only if is a fixpoint of .
It follows that the fixpoints of can be regarded as 4valued (3valued, in the case of consistent fixpoints) generalizations of an extension of a default theory. We will therefore call them partial extensions. It turns out that partial extensions coincide with stationary extensions defined in [PP94].
Our next result describes monotonicity properties of the operators and .
Theorem 4.6
Let be a default theory. Then, the operator is antimonotone and the operator is monotone.
Theorem 4.6 implies that the operator has a least fixpoint. We will denote it by and refer to it as the wellfounded fixpoint of . We will call the semantics it implies a wellfounded semantics of . The wellfounded semantics of coincides with the wellfounded semantics of default logic introduced by Baral and Subrahmanian [BS91]. The wellfounded semantics allows us to approximate skeptical reasoning with extensions and yields a sufficient condition for the uniqueness of an extension.
Theorem 4.7
Let be a default theory and let . Then:

The fixpoint is consistent.

For every partial extensions of , .

If for every , then belongs to every extension. If for some , does not belong to any extension.

If (that is, if is complete) then has a unique extension corresponding to the possibleworld structure .
Wellfounded semantics has a constructive flavor. It can be obtained by iterating the operator over the belief pair . In addition, by extending the approach described in [DMT98], one can show that the problem of computing the wellfounded semantics is in the class .
Finally, let us note connections between (partial) weak extensions and (partial) extensions, and between the KripkeKleene and wellfounded semantics for default logic.
Theorem 4.8
Let be a default theory. Then:

.

Every extension of is a minimal weak extension of .

Every partial extension of is a minimal partial weak extension of in the following sense: for every partial weak extension , if and , then and .
In summary, default logic can be viewed as the logic of the operator . Its fixpoints define the semantics of partial weak extensions. The least fixpoint of defines the KripkeKleene semantics. The operator gives rise to the operator , which yields the semantics of weak extensions. KripkeKleene semantics provides an approximation for the skeptical reasoning under the semantics of weak extensions. The operator also leads to the operator . Consistent fixpoints of this operator yield stationary extensions. Fixpoints of a related operator , defined on the lattice , correspond to extensions by Reiter. The least fixpoint of the operator results in the wellfounded semantics for default logic and approximates the skeptical reasoning under the semantics of extensions. The relationships between the operators of default logic are illustrated in Figure 2.
5 Default Logic Versus Autoepistemic Logic
The results of the paper shed new light on the relationship between default and autoepistemic logics. The nature of this relationship was the subject of extensive investigations since the time both systems were introduced in early 80s. Konolige [Kon88] proposed to encode a default by the modal formula and to represent a default theory by a modal theory . Despite the fact that the encoding is intuitive it does not provide a correspondence between default logic as defined by Reiter and autoepistemic logic as defined by Moore. Consider a default theory with and , where and are two different atoms. Then has exactly one extension, . Applying the translation of Konolige to yields the theory . The theory has two expansions. One of them is generated by the theory and corresponds to the only extension of . The other expansion is generated by the theory . Thus, the Konolige’s translation does not give a onetoone correspondence between extensions of default theories and expansions of their modal encodings.
This mismatch can be explained within the semantic framework introduced in the paper. Konolige’s translation does not establish correspondence between extensions and expansions because they are associated with different operators. Expansions are associated with fixpoints of the operator . Its counterpart on the side of default logic is the operator . Fixpoints of this operator are not extensions but weak extensions of . Extensions turn out to be associated with the operator . Its counterpart on the side of autoepistemic logic is the operator , introduced in Section 3. This operator, to the best of our knowledge, has not appeared in the literature and properties of its fixpoints and the relationship to McDermottDoyle style logics [MD80, McD82, MT93] are not known.
Once we properly align concepts from default logic with those from autoepistemic logic, Konolige’s translation works! This alignment is illustrated in Figure 3 and is formally described in the following theorem.
Theorem 5.1
Let be a default theory and let . Then the following pairs of operators coincide and, thus, have the same fixpoints:

(that is, weak extensions correspond to expansions).

(that is, partial weak extensions correspond to partial expansions; KripkeKleene semantics for and KripkeKleene semantics for coincide).

and (that is, extensions correspond to strong expansions).

and (that is, partial extensions correspond to partial strong expansions, wellfounded semantics for and wellfounded semantics for coincide).
6 Discussion and Future Work
We presented results uncovering the semantic properties of default and autoepistemic logics. In each case, a whole family of semantics can be derived from a single operator by purely algebraic transformations. Most importantly, the translation of Konolige establishes a perfect correspondence between the families of semantics of default and autoepistemic logics. This elegant picture can be further extended to the case of logic programming. As discovered by Fitting, all key semantics for logic programs can be similarly obtained from a single operator, the 4valued van EmdenKowalski onestep provability operator [vEK76]. The resulting semantic structure for logic programming is shown in Figure 4.
In addition, the translation of logic program clauses into default rules proposed in [BF91, MT89b] establishes an embedding of logic programming into default logic that precisely aligns the corresponding semantics (Figure 5).
Let us further note that the approach to semantics of nonmonotonic logics presented here can also be extended to the case of reflexive autoepistemic logic by Schwarz [Sch95]. As in all other cases discussed in this paper, all major semantics for the reflexive autoepistemic logic can be obtained from a single approximating operator closely related to the operator .
Our work also points to an interesting open problem in the area of modal nonmonotonic logics. The fixpoints of the operator have not, to the best of our knowledge, been studied in the literature. In particular, it is not known if they can be described in the McDermottDoyle scheme [MD80, McD82] as expansions for some modal logic .
Finally, let us mention that it is possible to develop an abstract, purely algebraic treatment of the concept of approximations. It generalizes the approach presented here and the work of Fitting on logic programming semantics. An account of this abstract treatment of approximations can be found in [DMT00].
Acknowledgements
This work was partially supported by the NSF grants CDA9502645 and IRI9619233.
References
 [BF91] N. Bidoit and C. Froidevaux. Negation by default and unstratifiable logic programs. Theoretical Computer Science, 78(1, (Part B)):85–112, 1991.
 [BS91] C. Baral and V.S. Subrahmanian. Dualities between alternative semantics for logic programming and nonmonotonic reasoning (extended abstract). In A. Nerode, W. Marek, and V.S. Subrahmanian, editors, Logic programming and nonmonotonic reasoning (Washington, DC, 1991), pages 69–86, Cambridge, MA, 1991. MIT Press.
 [Che80] B.F. Chellas. Modal logic. An introduction. Cambridge University Press, CambridgeNew York, 1980.

[DMT98]
M. Denecker, V. Marek, and M. Truszczyński.
Fixpoint 3valued semantics for autoepistemic logic.
In
Proceedings of the Fifteenth National Conference on Artificial Intelligence (AAAI98)
, pages 840 – 845. MIT Press, 1998.  [DMT00] M. Denecker, V. Marek, and M. Truszczyński. Unified semantic treatment of default and autoepistemic logics. In Principles of Knowledge Representation and Reasoning, Proceedings of the Seventh International Conference (KR2000). Morgan Kaufmann Publishers, 2000.
 [Fit99] M. C. Fitting. Fixpoint semantics for logic programming – a survey. Theoretical Computer Science, 1999. To appear.
 [GC90] R. Guerreiro and M. Casanova. An alternative semantics for default logic. Preprint. The Third International Workshop on Nonmonotonic Reasoning, South Lake Tahoe, 1990.
 [Got95] G. Gottlob. Translating default logic into standard autoepistemic logic. Journal of the ACM, 42(4):711–740, 1995.
 [Kon88] K. Konolige. On the relation between default and autoepistemic logic. Artificial Intelligence, 35(3):343–382, 1988.
 [Lev90] H. J. Levesque. All I know: a study in autoepistemic logic. Artificial Intelligence, 42(23):263–309, 1990.
 [Lif90] V. Lifschitz. On open defaults. In J. Lloyd, editor, Proceedings of the symposium on computational logic, pages 80–95. Berlin: SpringerVerlag, 1990. ESPRIT Basic Research Series.
 [McD82] D. McDermott. Nonmonotonic logic II: nonmonotonic modal theories. Journal of the ACM, 29(1):33–57, 1982.
 [MD80] D. McDermott and J. Doyle. Nonmonotonic logic I. Artificial Intelligence, 13(12):41–72, 1980.
 [Moo84] R.C. Moore. Possibleworld semantics for autoepistemic logic. In Proceedings of the Workshop on NonMonotonic Reasoning, pages 344–354, 1984. Reprinted in: M. Ginsberg, ed., Readings on nonmonotonic reasoning, pp. 137–142, Morgan Kaufmann, 1990.
 [MT89a] W. Marek and M. Truszczyński. Relating autoepistemic and default logics. In Proceedings of the First International Conference on Principles of Knowledge Representation and Reasoning (Toronto, ON, 1989), Morgan Kaufmann Series in Representation and Reasoning, pages 276–288, San Mateo, CA, 1989. Morgan Kaufmann.
 [MT89b] W. Marek and M. Truszczyński. Stable semantics for logic programs and default theories. In E.Lusk and R. Overbeek, editors, Proceedings of the North American Conference on Logic Programming, pages 243–256. MIT Press, 1989.
 [MT93] W. Marek and M. Truszczyński. Nonmonotonic logics; contextdependent reasoning. SpringerVerlag, Berlin, 1993.
 [PP94] H. Przymusińska and T. Przymusiński. Stationary default extensions. Fundamenta Informaticae, 21(12):67–87, 1994.
 [Rei80] R. Reiter. A logic for default reasoning. Artificial Intelligence, 13(12):81–132, 1980.
 [Sch95] J. Schlipf. The expressive powers of the logic programming semantics. Journal of the Computer Systems and Science, 51(1):64–86, 1995.
 [Tar55] A. Tarski. Latticetheoretic fixpoint theorem and its applications. Pacific Journal of Mathematics, 5:285–309, 1955.
 [Tru91] M. Truszczyński. Modal interpretations of default logic. In Proceedings of IJCAI91, pages 393–398, San Mateo, CA, 1991. Morgan Kaufmann.
 [vEK76] M.H. van Emden and R.A. Kowalski. The semantics of predicate logic as a programming language. Journal of the ACM, 23(4):733–742, 1976.
 [VRS91] A. Van Gelder, K.A. Ross, and J.S. Schlipf. The wellfounded semantics for general logic programs. Journal of the ACM, 38(3):620–650, 1991.