Table Of ContentFixpoint 3-valued semantics for autoepistemic logic
Marc Denecker Victor Marek and Miros law Truszczyn´ski
Department of Computer Science Computer Science Department
K.U.Leuven University of Kentucky
Celestijnenlaan 200A, B-3001 Heverlee, Belgium Lexington, KY 40506-0046
[email protected] marek|[email protected]
9
9
9 Abstract alsoknowntobeequivalenttoseveralothermodalnon-
1 monotonic reasoning systems.
n The paper presents a constructive 3-valued se- The semantics for autoepistemic logic (?) assigns to
mantics for autoepistemic logic (AEL). We in-
a a modal theory T a collection of its stable expansions.
troduce a derivation operator and define the se-
J This collection may be empty, may consist of exactly
mantics as its least fixpoint. The semantics is 3-
2 valued in the sense that, for some formulas, the one expansion or may consist of several different ex-
1 least fixpoint does not specify whether they are pansions. Intuitively,expansionsaredesignedtomodel
believed or not. Weshowthat complete fixpoints beliefstatesofagentswithperfectintrospectionpowers:
] of the derivation operator correspond to Moore’s foreveryformulaF,eithertheformulaKF (expressing
O
stable expansions. In the case of modal represen- a belief in F) or the formula ¬KF (expressing that F
L tationsoflogicprogramsourleastfixpointseman- is not believed) belongs to an expansion. We will say
. tics expresses well-founded semantics or 3-valued that expansions contain no meta-ignorance.
s
Fitting-Kunen semantics (depending on the em-
c In many applications, the phenomenon of multiple
bedding used). We show that, computationally,
[
expansions is desirable. There are situations where
our semantics is simpler than the semantics pro-
1 posed by Moore (assuming that the polynomial we are not interested in answers to queries concern-
v hierarchy does not collapse). ing a single atom or formula, but in a collection of
3 atoms or formulas that satisfy some constraints. Plan-
0 ninganddiagnosisinartificialintelligence,andarange
Introduction
0 of combinatorial optimization problems, such as com-
1 We describe a 3-valued semantics for modal theories puting hamilton cycles or k-colorings in graphs, are of
0
that approximatesskepticalmodeofreasoninginthe this type. These problems may be solved by means
9
autoepistemic logic introduced in (?; ?). We present of autoepistemic logic precisely due to the fact that
9
resultsdemonstratingthatourapproachis, indeed,ap- multiple expansions are possible. The idea is to rep-
/
s propriate for modeling autoepistemic reasoning. We resent a problem as an autoepistemic theory so that
c
discuss computational properties of our semantics and solutions to the problem are in one-to-one correspon-
:
v connections to logic programming. dence with stable expansions. While conceptually el-
i Autoepistemic logic is among the most extensively egant, this approach has its problems. Determining
X
studied nonmonotonic formal systems. It is closely re- whether expansionsexistis a ΣP-complete problem(?;
2
ar lated to default logic (?). It can handle default rea- ?),andallknownalgorithmsforcomputingexpansions
sonings under a simple and modular translation in the are highly inefficient.
caseofprerequisite-freedefaults(?). Inthecaseofarbi- In a more standard setting of knowledge representa-
trary default theories, a somewhat more complex non- tion, the goal is to model the knowledge about a do-
modular translation provides a one-to-one correspon- main as a theory in some formal system and, then, to
dence between default extensions and stable (autoepis- usesomeinferencemechanismtoresolvequeriesagainst
temic) expansions (?). Further, under the so called thetheoryor,inotherwords,establishwhetherparticu-
Gelfond translation, autoepistemic logic captures the lar formulas are entailedby this theory. Autoepistemic
semantics of stable models for logic programs (?). Un- logic (as well as other nonmonotonic systems) can be
der the Konolige encoding of logic programs as modal used in this mode, too. Namely, under the so called
theories,stableexpansionsgeneralizetheconceptofthe skeptical model, a formula is entailed by a modal the-
supported model semantics (?). Autoepistemic logic is ory,if it belongs to all stable expansions of this theory.
The problem is, again, with the computational com-
Copyright (cid:13)c 2008, American Association for Artificial In-
plexityofdetermining whethera formulabelongstoall
telligence (www.aaai.org). All rights reserved.
Copyright(cid:13)c1998,AmericanAssociationforArtificialIntel- expansions.
ligence (www.aaai.org). All rights reserved. We propose an alternative semantics for autoepis-
temicreasoningthatallowsustoapproximatetheskep- knowledge). The operator attempts to simulate a con-
tical approach described above. Our semantics has the structive process rational agents might use to produce
propertythatifitassignstoaformulathetruthvaluet, an “elementary” improvement on their current set of
then this formula belongs to all stable expansions and, beliefs and disbeliefs.
dually, if it assigns to a formula the truth value f, then We say that (P ,S ) “better approximates” the
1 1
this formuladoesnotbelong toanyexpansion. Ourse- agent’sbeliefsanddisbeliefsentailedbytheagent’sini-
manticsis 3-valuedandsomeformulasareassignedthe tial assumptions than (P,S) if S ⊆ S ⊆ P ⊆ P. The
1 1
truth value u (unknown). While only approximating operator D is monotone with respect to this ordering
T
the skeptical mode of reasoning, it has one important and,thus,ithastheleastfixpoint. Weproposethisfix-
advantage. Its computational complexity is lower (as- point as a constructive approximationto the semantics
sumingthatthepolynomialhierarchydoesnotcollapse of stable expansions.
on some low level). Namely, the problem to determine A fundamental property that makes the above con-
the truth value of a formula is in the class ∆P. structionmeaningfulisthatcompletebeliefpairs(those
2
Clearly, the semantics we propose can be applied withP equalto S)thatarefixpoints ofD areprecisely
wheneverthesituationrequiresthatautoepistemiclogic Moore’sS5-modelscharacterizingexpansions. Thus,by
be usedin the skepticalmode. However,it hasalsoan- the general properties of fixpoints of monotone opera-
other important application. It can be used as a prun- tors over partially ordered sets, the least fixpoint de-
ing mechanisminalgorithmsthatcompute expansions. scribed above indeed approximates the skeptical rea-
Theideaistofirstcomputeour3-valuedinterpretation soning based on expansions. Moreover, as mentioned
for an autoepistemic theory (which is computationally above, the problem of computing the least fixpoint of
simpler than the task of computing anexpansion) and, theoperatorDrequiresonlypolynomiallymanycallsto
in this way, find some formulas which are in all expan- thesatisfiabilitytestingengine,thatis,itisin∆P. An-
2
sions and some that are in none. This restricts the other property substantiating our approach is that un-
search space for expansions and may yield significant der some naturalencodings of logic programsas modal
speedups. theories,oursemanticsyieldsbothwell-foundedseman-
Conceptually, our semantics plays the role similar tics (?) and the 3-valued Fitting-Kunen semantics (?;
to that played by the well-founded semantics in logic ?).
programming. Deciding whether an atom is in all sta-
Autoepistemic logic — preliminaries
ble models is a co-NP-complete problem. However,the
well-founded semantics, which approximates the stable Thelanguageofautoepistemiclogicisthestandardlan-
model semantics can be computed in polynomial time. guage of propositional modal logic over a set of atoms
Furthermore, well-founded semantics is used both as Σ and with a single modal operator K. We will refer
the basis for top-down query answering implementa- to this language as L . The propositional fragment of
K
tions of logic programming (?), and as a search space L will be denoted by L.
K
pruning mechanism by some implementations to com- The notion of a 2-valued interpretation of the alpha-
pute stable model semantics (?). bet Σ is defined as usual: it is a mapping from Σ to
Our 3-valued semantics for autoepistemic logic is {t,f}. ThesetofallinterpretationsofΣ isdenotedA
Σ
based on the notion of a belief pair. These are pairs (or A, if Σ is clear from the context).
(P,S), where P and S are sets of 2-valued interpreta- A possible-world semantics for autoepistemic logic
tionsoftheunderlyingfirst-orderlanguage,andS ⊆P. was introduced by Moore (?) and proven equivalent
The motivation to consider belief pairs comes from with the semantics of stable expansions. A possible-
Moore’s possible-world characterization of stable ex- worldstructureW (overΣ)isasetof2-valuedinterpre-
pansions (?). Moore characterized consistent expan- tations of Σ. Alternatively, it can be seen as a Kripke
sionsin termsofpossible-world structures,that is,non- structure with a total accessibility relation. Given a
empty sets of 2-valued interpretations. A belief pair pair (W,I), where W is a set of interpretations and I
(P,S)canbeviewedasanapproximationtoapossible- is an interpretation (not necessarily from W), one de-
world structure W such that S ⊆ W ⊆ P: interpreta- fines a truth assignment function H inductively as
W,I
tionsnotinP areknownnottobeinW,andthoseinS follows:
are known to be in W. Observe that while expansions (1) For an atom A, we define H (A)=I(A);
W,I
(orthecorrespondingpossible-worldstructures)donot (2) The boolean connectives are handled in the usual
contain meta-ignorance,belief pairs, in general, do. way;
Our semantics is defined in terms of fixpoints of a (3) For every formula F, we define HW,I(KF) = t if
monotone operator defined on the set of belief pairs. for every J ∈ W,HW,J(F) = t, and HW,I(KF) = f,
This operator, D , is determined by an initial theory otherwise.
T
T. Given a belief pair B = (P,S), D establishes that We write (W,I) |= F to denote that H (F) = t.
T W,I
someinterpretationsthatareinP must,infact,belong Further,foramodaltheoryT,wewillwrite(W,I)|=T
to S. In addition, some other interpretations in P are if H (F) = t for any F ∈ T. Finally, for a possible
W,I
eliminated altogether, as inappropriate for describing world structure W we define the theory of W, Th(W),
a possible state of the world (given the agent’s initial by: Th(W)={F:(W,I)|=F, for all I ∈W}.
It is well known that for every formula F, either H (F ∨F ) =max{H (F ),H (F )}
B,I 1 2 B,I 1 B,I 2
KF ∈ Th(W) or ¬KF ∈ Th(W) (H (KF) is the H (F ∧F ) =min{H (F ),H (F )}
W,I B,I 1 2 B,I 1 B,I 2
same for all interpretations I ∈ W). Thus, possible- H (F ⊃F ) =max{H (F ),H (F )−1}
B,I 2 1 B,I 1 B,I 2
world structures have no meta-ignorance and, as such,
The formula K(F) is evaluated as follows:
are suitable for modeling belief sets of agents with per- t if ∀ H (F)=t
J∈P B,J
fect introspection capabilities. It is precisely this prop- H (K(F))= f if ∃ H (F)=f
B,I J∈S B,J
erty that made possible-world structures fundamental ( u otherwise
objects in the study of modal nonmonotonic logics (?;
?). The truth value of a modal atom KF, HB,I(KF),
doesnotdependonthechoiceofI. Consequently,fora
Definition 1 An autoepistemic model of a modal the-
modalatomKF we willwrite H (KF)to denote this,
B
ory T is a possible-world (S5) structure W which sat-
common to all interpretations from A, truth value of
isfies the following fixpoint equation:
KF.
W ={I:(W,I)|=T}. Let us define the meta-knowledge of a belief pair B
as the set of formulas F ∈L such that H (KF)= t
The following theorem, relating stable expansions of K B
or H (KF) = f. The meta-ignorance is formed by
(?) and autoepistemic models, was proved in (?) and B
all other formulas, that is, those formulas F ∈ L for
was discussed in detail in (?). K
which H (KF)=u.
B
Theorem 1 For any two modal theories T and E, E Clearly, a belief pair B = (W,W) naturally corre-
is a consistent stable expansion of T if and only if E = sponds to a possible-world structure W. Such a belief
Th(W) for some nonempty autoepistemic model W of pair is called complete. We will denote it by (W). The
T. following straightforward result indicates that H is
B,I
a generalization of H to the case of belief pairs. It
W,I
A fixpoint 3-valued semantics for
alsostatesthatacompletebeliefpaircontainsnometa-
autoepistemic logic ignorance.
Our semantics for autoepistemic logic is defined in Proposition 1 If B is a complete belief pair (W),
terms of possible-world structures and fixpoint con- then HB,I is 2-valued. Moreover, for every formula F,
ditions. The key difference with the semantics pro- HB,I(F)=HW,I(F).
posed by Moore is that we consider approximations In our approach to autoepistemic reasoning we will
of possible-world structures by pairs of possible-world model the agent who, given an initial theory T,
structures. Recall from the previous section, that A starts with the belief pair ⊥ (with the smallest meta-
denotes the set of all interpretationsof a fixed proposi- knowledge content) and, then, iteratively constructs a
tional language L. sequenceofbeliefpairswithincreasingmeta-knowledge
Definition 2 A belief pair is a pair (P,S) of sets of (decreasing meta-ignorance) until no more improve-
interpretations P and S such that P ⊇ S. When ment is possible. To this end, we will introduce now a
B =(P,S), S(B) denotes S and P(B) denotes P. The partial ordering on the set B of belief pairs. Given two
belief pair (A,∅) is denoted ⊥. The set {(P,S):P,S ∈ belief pairs B1 and B2, we define B1 ≤p B2 if P(B1)⊇
A and P ⊇S} of all belief pairs is denoted by B. P(B2) and S(B1)⊆S(B2). This ordering is consistent
with the ordering defined by the “amount” of meta-
The interpretations in S(B) can be viewed as states
knowledge contained in a belief pair: the ”higher” a
of the world which are known to be possible (belong
beliefpairintheordering≤ ,themoremeta-knowledge
p
to W). They form a lower approximation to W. The
itcontains(andthelessmeta-ignorance). Itisalsocon-
interpretations in P(B) can be viewed as an upper ap-
sistent with the concept of the information ordering of
proximation to W. In other words, interpretations not thetruthvalues: u≤ f,u≤ t,f6≤ tandt6≤ f.
kn kn kn kn
in P(B) are known not to be in W.
Proposition 2 LetB andB bebeliefpairs. IfB ≤
We will extend now the concept of an interpreta- 1 2 1 p
B then for every F ∈L and for every interpretation
tion to the case of belief pairs and consider the ques- 2 K
I, H (F)≤ H (F).
tion of meta-ignorance and meta-knowledge of belief B1,I kn B2,I
pairs. We will show that, being only approximations The ordered set (B,≤ ) is not a lattice. In fact, for
p
to possible-world structures, belief pairs may contain every W ⊆ A, (W) is a maximal element in (B,≤ ).
p
meta-ignorance. We will use three logical values, f, u If W 6= W , then (W ) and (W ) have no least upper
1 2 1 2
and t. In the definition, we will use the truth ordering: bound (l.u.b.) in (B,≤ ). The pair ⊥ = (A,∅) is the
p
f≤ u≤ t and define f−1 =t,t−1 =f,u−1 =u. least element of (B,≤ ).
tr tr p
The ordered set (B,≤ ) is chain-complete. That is,
Definition 3 Let B = (P,S) be a belief pair and let I p
everysetofpairwisecomparableelementshasthel.u.b.
beaninterpretation. ThetruthfunctionH isdefined
B,I A monotone operator defined on a chain-complete or-
inductively:
deredsetwithaleastelementhasaleastfixpoint,which
HB,I(A) =I(A) (A is an atom) is the limit of the iterations of the operator starting at
H (¬F) =H (F)−1 the least element ⊥ (?).
B,I B,I
We will now define a monotone operator on the or- Consistent stratified autoepistemic theories (?) have
dered set (B,≤ ) and will use it to define a step- a unique autoepistemic model (stable expansion). Our
p
wiseprocessofconstructingbeliefpairswithincreasing semantics coincides with the Moore’s semantics on
meta-knowledge. To this end, we will define two sat- stratified theories.
isfaction relations: weak (denoted by |=w) and strong Theorem 3 If T is a consistent stratified autoepis-
(denoted by |=): temic theory, then D ↑ is complete. Hence, it is the
(B,I)|= F if H (F)6=f (i.e. H (F)≥ u) unique autoepistemic model of T.
w B,I B,I tr
(B,I)|=F if HB,I(F)=t Thus, the semantics defined by the least fixpoint of
Definition 4 Given B ∈B, the value of the derivation the operatorD has severalattractivefeatures. It is de-
operator D is defined as follows: fined for every consistent modal theory T. It coincides
T
D (B)=({I | B,I |= T},{I | B,I |=T}). with the semantics of autoepistemic logic on stratified
T w
Thus, P(D (B)) consists of the states which weakly theories and, in the general case, provides an approxi-
T
satisfyT,accordingtoB,whileS(D (B))arethestates mation to all stable expansions (or, in other words, to
T
whichstronglysatisfy T accordingto B. The subscript skeptical autoepistemic reasoning).
T is omitted when T is clear from the context. An effective implementation of D
Example 1 Consider T ={K(p)⊃q}. Then D(⊥)= The approach proposed and discussed in the previous
(A,{pq,pq}). Indeed, H⊥(Kp) = u. Consequently, section does not directly yield itself to fast implemen-
for every I, H⊥,I(Kp ⊃ q) 6= f, that is, (⊥,I) |=w tations. The definition of the operator D refers to all
Kp ⊃ q. For the same reason, H⊥,I(Kp ⊃ q)) = t interpretations of the language L. Thus, computing
if and only if I(q) = t. To compute D2(⊥), ob- D(B)byfollowingthedefinitionisexponential,evenfor
serve that HD(⊥)(Kp) = f. Consequently, for every I, modal theories of a very simple syntactic form. More-
H (Kp⊃q)=t. It follows that D2(⊥)=(A,A). over, representing belief pairs is costly. Each of the
D(⊥),I
It is also easy to see now that (A,A) is the fixpoint of sets P(B) and S(B) may contain exponentially many
D. Notice that the belief pair (A), obtained by iter- elements.
ating D, corresponds to the possible-worldstructure A In this section, we describe a characterization of the
that defines the unique stable expansion of T. operator D that is much more suitable for investiga-
tions of algorithmic issues associated with our seman-
Basic properties of the operator D are gathered in
tics. The strategy is to represent a belief set B as a
the following proposition.
theory Rep(B). Since the theory Rep(B) needs to rep-
Proposition 3 Let T be a propositionally consistent resenttwosetsofvaluations,Rep(B)willbeatheoryin
modal theory. Then, for every belief pair B: thepropositionallanguageextendedbythreeconstants
(1) DT(B) is a belief pair. t, f and u. These constants will always be interpreted
(2) DT is monotone on B. by the logical values t, f and u, respectively. We will
(3) If B is complete, then DT(B) is complete. call such theories 3-FOL theories.
Since (B,≤ ) is a chain-complete ordered set with Let F be a 3-FOL formula. By (F)wk we denote the
p
least element ⊥ and D is monotonic, D has a least fix- formula obtained by substituting t for all positive oc-
point. We will denote it by D ↑. We propose this currences of u and f for all negative occurrences of u.
fixpoint as the semantics of modal theories. This se- Similarly, by (F)str we denote the formula obtainedby
mantics reflects the reasoning process of an agent who substituting t for all negative occurrences of u and f
gradually constructs belief pairs with increasing infor- for all positive occurrences of u. Given a 3-FOL the-
mation content. In the remainder of the paper, we will ory Y, we define (Y)str and (Y)wk by standard setwise
study properties of this semantics and, more generally, extension.
of fixpoints of the operator D. The next three results Clearly, (F)str and (F)wk do not contain u. Conse-
relate fixpoints of D to Moore’s semantics. quently,theycanberegardedasformulasinthepropo-
sitional language extended by two constants t and f
Theorem 2 Let T ⊆ L . A possible-world structure
K
with standard interpretations as truth and falsity, re-
W is an autoepistemic model of T if and only if (W)
spectively. We will call this language 2-FOL. We will
is a fixpoint of D . If D ↑ is complete then it is the
T T
write ⊢ and|= to denote provabilityand entailmentre-
unique autoepistemic model of T.
lationsin2-FOL.Animportantobservationhereisthat
Using Propositions 1 and 2, we can extract a rela- if an interpretation satisfies the formula (F)str then
tionshipbetweenstableexpansionsandfixpoints ofthe it also satisfies (F)wk. That is (F)str ⊃ (F)wk is a
derivation operator DT. tautology of 2-FOL. For a 2-FOL theory U, we define:
Corollary 1 For any pair T,E of modal theories, E is Mod(U)={I : for all F ∈U, I |=F}. It follows that
a consistent stable expansion of T if and only if E = for a 3-FOL theory Y, Mod((Y)str) ⊆ Mod((Y)wk).
Th(S) for some complete fixpoint (S) of D . Thus, (Mod((Y)wk),Mod((Y)str) is a belief pair and
T
If H (F)=t then F belongs to all expansions of T. Y can be viewed as its representation.
DT↑
IfH (F)=fthenF doesnotbelongtoanyexpansion We show now how, similarly to belief pairs, 3-FOL
DT↑
of T. theories can be used to assign truth values to modal
atoms (and, hence, to all modal formulas). Let Y be a (1) Bel(SD (Y))=D (Bel(Y)).
T T
3-FOL theory, and let F be a modal formula. Define (2) If a belief pair B is a fixpoint of D , then T is a
T B
H (K(F)) by induction of depth of formula F as fol- fixpoint of SD .
Y T
lows: (3) If Y is a fixpoint of SD then Bel(Y) is a fixpoint
T
(1) If F is objective, then define: of D .
T
t if (Y)wk ⊢(F)str Observe that Bel({u})=⊥. It follows directly from
HY(K(F))= f if (Y)str 6⊢(F)wk Theorem6(byinduction)thatforeveryordinalnumber
u otherwise. α, Dα(⊥)=Bel(SDα({u})).
T T
Clearly, if SDα(⊥) = SDα+1(⊥) then Dα(⊥) =
(2) If F is not objective, then replace all modal atoms T T T
Dα+1(⊥). Moreover, by Theorems 4, 5 and 6, and by
mKu(Gla)Fin′.FDbefiynHeYH(YK((KG()F).))T=hisHyYie(lKds(Fan′))o.bjectivefor- inTduction, it is easy to show that if DTα(⊥)=DTα+1(⊥)
then SDα+1(⊥)=SDα+2(⊥).
T T
LetT beamodaltheoryandletY bea3-FOLtheory. Consequently, the least fixpoint of DT (its
By the Y-instance of T, T , we mean a 3-FOL theory polynomial-sizerepresentation)canbe computedbyit-
Y
obtained by substituting all modal literals K(F) (not eratingthe operatorSDT. Inthe case whenT isfinite,
appearing under the scope of any other occurrence of the number of iterations is limited by the number of
K)byH (K(F)). Observethatfora finite modalthe- top level (unnested) modal literals in T. Originally,
Y
oryT andafinite3-FOLtheoryY,T canbecomputed they may all be evaluated to u. However,at each step,
Y
by means of polynomially many calls to the proposi- at least one u changes to either t or f and this value
tional provability procedure. is preserved in the subsequent evaluations. Thus, the
LetT be amodaltheory. We willnowdefine acoun- problemofcomputing a polynomialsize representation
terpart to the operator D . Let Y be a 3-FOL theory. of the least fixpoint of the operator D, the correspond-
T
Define SD (Y)=T . ing 3-FOL theory, is in the class ∆P.
T Y 2
The key property of the operator SD is that, for
T
a finite modal theory T and for a finite 3-FOL theory Relationship to Logic Programming
Y,SDT(Y)canbecomputedbymeansofpolynomially Autoepistemic logic is closely relatedto severalseman-
many calls to the propositional provability procedure. tics for logic programs with negation. It is well-known
We will show that SDT can be used to compute DT. that both stable and supported models of logic pro-
Inparticular,wewillshowthattheleastfixpointofDT grams can be described as expansions of appropriate
can be computed by iterating the operator SDT. To translations of programs into modal theories (see, for
this end, for every 3-FOL theory Y, define Bel(Y) = instance, (?)). In this section, we briefly discuss con-
(Mod((T)wk),Mod((T)str)). nections of the semantics defined by the least point of
First,thefollowingtheoremshowsthatthetruthval- theoperatorD tosome3-valuedsemanticsoflogicpro-
ues of modal atoms evaluated according to a 3-FOL grams. The details will be provided in a forthcoming
theoryT andaccordingtothecorrespondingbeliefpair work.
Bel(T) coincide. Given a logic programming clause
Theorem 4 LetY bea3-FOLtheory. Then, forevery r =a←b ,...,b ,not(c ),...,not(c ),
1 k 1 m
modal formula F,
define:
H (K(F))=H (K(F)).
Bel(Y) Y ael (r)=b ∧...∧b ∧¬Kc ∧...∧¬Kc ⊃a
1 1 k 1 m
Next, let us observe that the operator D can be de-
and
scribedintermsofthe operatorBel. LetT be amodal
ael (r)=Kb ∧...∧Kb ∧¬Kc ∧...∧¬Kc ⊃a
theory and let B be a belief pair. By the B-instance of 2 1 k 1 m
T,T ,wemeana3-FOLtheoryobtainedbysubstitut- Embeddings ael () and ael () can be extended to logic
B 1 2
ing all modal literals K(F) (not appearing under the programs P.
scope of any other occurrence of K) by H (K(F)). LetBbeabeliefpair. Definetheprojection,Proj(B),
B
as the 3-valued interpretation I such that I(p) =
Theorem 5 Let T be a modal theory and let B be a
H (K(p)).
belief pair. Then, D (B)=Bel(T ) B
T B It turns out that fixpoints of the operator D
ael1(P)
Thistheoremindicatesthat,givenamodaltheoryT, (D ,respectively)preciselycorrespondto3-valued
ael2(P)
beliefpairsthatareintherangeoftheoperatorDT can stable (supported, respectively) models of P (the pro-
be representedby objects of size polynomialin the size
jectionfunctionProj(·)establishesthecorrespondence).
of T. Namely, every belief pair of the form D (B) can
T Moreover, complete fixpoints of D describe 2-
be represented by a 3-FOL theory T . ael1(P)
B valuedstable(supported,respectively)modelsofP. Fi-
Theorems 4 and 5 imply the main result of this sec-
nally,theleastfixpointofD capturestheFitting-
tion. ael2(P)
Kunen3-valuedsemanticsofaprogramP,andtheleast
Theorem 6 Let T be a modal theory and let Y be a fixpointofD capturesthewell-foundedsemantics
ael1(P)
3-FOL theory. of P.
Acknowledgments
This work was partially supported by the NSF grants
IRI-9400568and IRI-9619233
