Static Semantics For Normal and Disjunctive Logic ... - Semantic Scholar

Static Semantics For Normal and Disjunctive Logic Programs Teodor C. Przymusinski Department of Computer Science University of California Riverside, CA 92521 ([email protected])

Dedicated to Jack Minker

Abstract

In this paper, we propose a new semantic framework for disjunctive logic programming by introducing static expansions of disjunctive programs. The class of static expansions extends both the classes of stable, well-founded and stationary models of normal programs and the class of minimal models of positive disjunctive programs. Any static expansion of a program P provides the corresponding semantics for P consisting of the set of all sentences logically implied by the expansion. We show that among all static expansions of a disjunctive program P there is always the least static expansion which we call the static completion P of P . The static completion P can be de ned as the least xed point of a natural minimal model operator and can be constructed by means of a simple iterative procedure. The semantics de ned by the static completion P is called the static semantics of P . It coincides with the set of sentences that are true in all static expansions of P . For normal programs, it coincides with the well-founded semantics. The class of static expansions represents a semantic framework which diers signi cantly from the other semantics proposed recently for disjunctive programs and databases. It is also de ned for a much broader class of programs. Keywords: Normal and Disjunctive Logic Programs and Deductive Databases, Semantics of Logic Programs, Non-Monotonic Reasoning, Belief Theories.

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

1 Introduction During the last couple of years a signi cant body of knowledge has been accumulated providing us with a better understanding of semantic issues in logic programming and the theory of deductive databases. In particular, the class of perfect models [ABW88, VG89, Prz88] was shown to provide a suitable semantics for (locally) strati ed logic programs. Subsequently, two closely related [PP90] extensions of the class of perfect models to normal, non-disjunctive , logic programs were introduced and extensively investigated. One of them is the class of well-founded models [VGRS90] and the other is the class of stable models [GL88]. In [Prz90, Prz91c], another extension of the class of perfect models, namely the class of partial stable models , later renamed stationary models [Prz91b], was introduced1 , for arbitrary normal programs. Stationary models include both stable and well-founded models. Moreover, every normal program has the least stationary model which coincides with its well-founded model. The problem of extending these approaches and de ning a suitable semantics for the class of disjunctive logic programs and deductive databases turned out to be a dicult one, as evidenced by a large number of papers [Ros89, BLM90, BLM89, Prz91c, GL90, Prz91b, Dix91] and the recent book [LMR92] devoted to this issue. It is well known that disjunctive programs are signi cantly more expressive than normal programs [Got92]. In this paper, we propose a new semantic framework for disjunctive logic programs obtained by de ning, for every disjunctive program P , the class of static expansions Pb of P . Static expansions Pb are rst-order extensions of P and thus provide the meaning or semantics for P consisting of the set of sentences logically derivable from Pb . We show that among all static expansions Pb of a disjunctive program P there is always the least static expansion P called the static completion of P . The semantics de ned by the static completion P is called the static semantics of P . Consequently, the static semantics always coincides with the set of sentences that are true in all static expansions of P . The static completion P is shown to be the least xed point of a natural monotonic minimal model operator and it can be iteratively constructed by beginning with P 0 = P , i.e., with the program P itself, and then, at every successor step, adding to the previous iteration P n those sentences which are true in all minimal models of P n. The construction continues until no new sentences can be added, i.e., until P n+1 = P n , at which point the static completion, P , is obtained. We prove that for normal programs P there is a one-to-one correspondence between static expansions Pb of P and stationary (or partial stable) models M of P . In particular, the least stationary model (i.e., the well-founded model) MP of a normal program corresponds to its least static expansion (i.e., the static completion) P . This implies that, for normal programs, the well-founded, stationary and static semantics all coincide . Consequently, static expansions provide a new and natural characterization of stationary, and, in particular, stable and well-founded, models of normal programs and extend these important classes of models to disjunctive programs. For positive disjunctive programs, static semantics coincides with the minimal model semantics . More precisely, every positive disjunctive program P has precisely one static expansion P which describes the minimal model semantics of P . However, when restricted 1 Partial

stable models were also called 3-valued stable models .

1

to strati ed disjunctive programs, the static semantics is in general weaker than the perfect model semantics [Prz88]. This is the consequence of the fact that the static semantics derives a minimal , in some sense, set of conclusions that can be inferred from a disjunctive program. If a given application area requires us to infer more facts, we can achieve it by explicitly adding additional axioms which then become an intrinsic part of the program and thus can be more easily revised and modi ed. As a result, we are less likely to infer undesirable conclusions . The class of static expansions represents a semantic framework which diers signi cantly from the other semantic approaches proposed recently for disjunctive programs and is also de ned for a much broader class of programs. In particular, static expansions dier signi cantly from stationary expansions introduced earlier in [Prz91b, Prz91d]. We view the de nition of static expansions as a \rational reconstruction" of the de nition of stationary expansions in the class of disjunctive programs which coincides with the old de nition in the class of normal programs. Static expansions are closely related to Moore's stable autoepistemic expansions. In [Prz94b, Prz94a] we introduce the Autoepistemic Logic of Knowledge and Beliefs , AELB . Static autoepistemic expansions in AELB generalize both stable and static expansions (see Section 7.3). The paper is organized as follows:  In Section 2 we extend the propositional language in order to be able to formally express the non-monotonic (commonsense) negation not C which appears in premises of program clauses.  In Section 3 we select the non-monotonic formalism providing the desired meaning for the non-monotonic negation not C , de ne static expansions and static semantics of logic programs and prove their basic properties.  In Section 4 (respectively, Section 5) we give examples and establish further properties of static expansions and static semantics in normal (respectively, disjunctive) logic programs.  In Section 6 we show how our approach can be easily generalized to the class of extended logic programs with the so called \classical" (or strong ) negation.  In Section 7 we discuss other natural extensions and modi cations of the proposed semantic framework.  Section 8 contains concluding remarks and a discussion of the relationship between static semantics and other semantics proposed for normal and disjunctive logic programs.

2 Propositional Language with Beliefs A disjunctive logic program (or a disjunctive deductive database) P is a set of informal clauses of the form A1 _ : : : _ Al B1 ^ : : : ^ Bm ^ not C1 ^ : : : ^ not Cn (1) 2

where l  1; m; n  0 and Ai , Bi and Ci 's are atomic formulae. If l = 1, for all clauses, then the program is called normal or non-disjunctive . As usual, we assume (see [PP90]) that the program P has been already instantiated and thus all of its clauses (possibly in nitely many) are propositional. This assumption allows us to restrict our considerations to a xed objective propositional language L: In particular, if the original (uninstantiated) program is nite and function-free then the resulting objective language L is also nite. Clauses (1) are informal because the negation symbol not C does not denote the classical negation :C of C but rather a non-monotonic (commonsense) negation . Various meanings can be associated with not C leading, in general, to dierent semantics. We are interested in the intended meaning that can be roughly described by:

not C



C is believed to be false

 :C

is believed to be true:

Before providing a suitable formalization of the non-monotonic negation not C one must rst be able to express it in the language. After all, propositional logic includes only one connective for negation, namely \:", and this connective represents the classical , rather than non-monotonic, negation. In order to achieve this goal, we rst augment the propositional language L with a modal belief operator B . Accordingly, the language considered in this paper is a propositional modal language, LB , with standard connectives (_, ^, , :), the propositional letter ? (denoting false ) and a modal operator B, called the belief operator. The atomic formulae of the form BF , where F is an arbitrary formula of LB , are called belief atoms. The formulae of LB in which B does not occur are called objective and the set of all such formulae is denoted by L.

De nition 2.1 (Belief Theory) By an autoepistemic theory of beliefs, or just a belief

theory, we mean an arbitrary theory in the language LB , i.e., a (possibly in nite) set of arbitrary clauses of the form:

B1 ^ : : : ^ B k ^ B G 1 ^ : : : ^ B G l  A1 _ : : : _ A m _ B F 1 _ : : : _ B F n where k; l; m; n  0, Ai 's and Bi 's are objective atoms and Fi 's and Gi 's are arbitrary

formulae of LB . Equivalently, a belief theory consists of a set of arbitrary clauses of the form:

B1 ^ ::: ^ Bk ^ BG1 ^ ::: ^ BGl ^ :BF1 ^ ::: ^ :BFn  A1 _ ::: _ Am which say that if the Bi 's are true, the Gi 's are believed, and the Fi 's are not believed then one of the Ai 's is true. 2 We assume the following two simple axiom schemata and one inference rule describing the arguably obvious properties of belief atoms:

(D) Consistency Axiom:

:B?

(2)

(K) Normality Axiom: For any formulae F and G: B(F  G) 

3

(BF  BG)

(3)

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

(4)

The rst axiom states that tautologically false formulae are not believed. The second axiom states that if we believe that a formula F implies a formula G and if we believe that F is true then we believe that G is true as well. The necessitation inference rule states that if a formula F has been proven to be true then F is believed to be true.

Remark 2.1 For readers familiar with modal logics it should be clear by now that we

are, in eect, considering here a normal modal logic with a modality B which satis es the consistency axiom (D) [MT94]. The axiom (K) is called \normal" because all normal modal logics satisfy it [MT94]. It is easy to see that, in the presence of the axiom (K), the consistency axiom (D) is equivalent to the axiom: BF  :B:F: (5) stating that if we believe in a formula F then we do not believe in :F . The assumption of the necessitation inference rule (N) is, strictly speaking, super uous because all static expansions (de ned below) automatically satisfy this rule anyway. It is included here for compatibility with the related work presented in [APP94a, APP94b] and also because all modal logics assume it. 2 In Section 7.2 we consider additional axioms that, if necessary, can be used to enlarge the set of facts logically derivable from belief theories. 2

De nition 2.2 (Formulae Derivable from a Belief Theory) We say that a formula

F is derivable from a given belief theory T if it belongs to the smallest set, Cn(T ), of formulae of the language LB which contains the theory T and all the (substitution instances

of) the axioms (K) and (D), and is closed under the necessitation rule (N). We denote this by T ` F . Consequently: Cn (T ) = fF : T ` F g: We call a belief theory T consistent if the theory Cn (T ) is consistent. Thus T is consistent if T 6` ?. 2

2.1 Logic Programs as Belief Theories

There are several dierent ways in which the informal disjunctive logic programs P given by clauses (1) can be naturally translated into formal belief theories. In this paper, we are primarily interested in the following translation.

De nition 2.3 (Translating Logic Programs into Belief Theories) Let P be any disjunctive logic program consisting of the informal clauses:

A1 _ : : : _ A l

B1 ^ : : : ^ Bm ^ not C1 ^ : : : ^ not Cn 4

(6)

where l  1; m; n  0 and Ai , Bi and Ci 's are objective atoms. The translation of P into the belief theory T (P ) is given by the set of the corresponding clauses:

B1 ^ : : : ^ Bm ^ B:C1 ^ : : : ^ B:Cn  A1 _ : : : _ Al (7) in the language LB obtained by replacing the non-monotonic negation not F by the formula B:F and by replacing the implication symbol ! by the standard material implication . 2 The translation, T (P ), corresponds therefore to: not F

def

 B:F 

F is believed to be false

 :F

is believed to be true (8) and is patterned after the translation introduced earlier in [Prz91a]. The transformation de ned above allows us to formally talk about two dierent types of negation:  Classical, monotonic negation :F ,  Non-monotonic, commonsense negation not F de ned by B:F . It is important to point out that in the above translation T (P ) the negation symbol \:" represents the true classical negation and not the so called \classical" or strong negation in the sense of Gelfond and Lifschitz [GL90]. Similarly, the implication symbol  also represents the classical material implication and thus we can freely move literals from one side of the implication to the other. In Section 6 we show how one can easily extend this class of theories to also allow the use of strong negation. In Section 7.4 we will also discuss another translation of logic programs into belief theories obtained by replacing the non-monotonic negation notF by the formula :BF thus resulting in the intended meaning for notF given by \F is not believed to be true" (see [Gel87]). For normal programs, both translations lead to the same (stationary or partial stable) semantics. However, for disjunctive programs, they produce signi cantly dierent results.

2.2 Armative Belief Theories

The de nitions introduced in the previous section lead us to the introduction of an important class of belief theories which is broader than then the class of all (translations of) disjunctive logic programs and yet, as we will show later, has some very natural properties which allow us to view them as generalized disjunctive logic programs.

De nition 2.4 (De nite Belief Theories) By an armative belief theory we mean a theory in the language LB consisting of a (possibly in nite) set of clauses of the form:

B1 ^ : : : ^ B m ^ B G 1 ^ : : : ^ B G k  A1 _ : : : _ A l _ B F 1 _ : : : _ B F n

(9)

or, equivalently:

B1 ^ : : : ^ Bm ^ BG1 ^ : : : ^ BGk ^ :BF1 ^ : : : ^ :BFn  A1 _ : : : _ Al; (10) where , l > 0, k; m; n  0, Ai 's and Bi 's are objective atoms and Fi 's and Gi's are arbitrary formulae. 2 5

In other words, armative belief theories are precisely those belief theories that satisfy the condition that all of their clauses contain at least one objective atom in their heads. Clearly all translations of disjunctive logic programs are special cases of armative belief theories. Armative belief theories can also be viewed as formal counterparts of more general disjunctive logic programs :

A1 _ : : : _ Al

B1 ^ : : : ^ Bm ^ not C1 ^ : : : ^ not Ck ^ not D1 ^ : : : ^ not Dn

(11)

allowing two types of non-monotonic negation, namely, not F , de ned by B:F , and, not F , de ned by :BF . Armative belief theories represent a very broad class of theories. For example, they allow clauses of the form: A ^ B(C _ D)  C _ BD or, equivalently: A ^ B(C _ D) ^ :BD  C which say that if A holds, C _ D is believed and D is not believed then C is true.

3 Static Expansions and Static Completions We rst choose a speci c non-monotonic formalism on which to base our beliefs BF . In other words, we state precisely what it means to say that \F is believed" . We select a form of Minker's Generalized Closed World Assumption GCWA (see [Min82, GPP89]) or McCarthy's Circumscription [McC80] which say that a formula F is believed to be true if and only if F is true in all minimal models of the theory, i.e., if and only if F is minimally entailed . In other words, the intended meaning of belief atoms BF will be given by the principle of predicate minimization: BF 

F is believed to be true



F is true in all minimal models.

Accordingly, beliefs considered in this paper can be called minimal beliefs . The intended meaning of the non-monotonic negation not F is therefore given by:

not F

def

 B:F 

F is false in all minimal models.

(12)

We now give a formal de nition of minimal models and minimal entailment. Throughout the paper we represent models as (consistent) sets of literals . An atom A is true in a model M if and only if A belongs to M . An atom A is false in a model M if and only if :A belongs to M . A model M is total if for every atom A either A or :A belongs to M . Otherwise, the model is called partial . Unless stated otherwise, all models are assumed to be total models. A (total) model M is smaller than a (total) model N if it contains less positive literals (atoms). When describing models we usually list only those of their members that are relevant to our considerations, typically those whose predicate symbols appear in the program that we are currently discussing. 6

De nition 3.1 (Minimal Models) By a minimal model of a belief theory T we mean a model M of T with the property that there is no smaller model N of T which coincides with M on belief atoms BF . If a formula F is true in all minimal models of T then we write:

T j=min F and say that F is minimally entailed by T .

2

For readers familiar with circumscription , this means that we are considering predicate circumscription CIRC (T ; L) of the theory T in which atoms from the objective language L are minimized while the belief atoms BF are xed2 . In other words, minimal models are obtained by assigning arbitrary truth values to belief atoms BF and then minimizing objective atoms.

Remark 3.1 The main reason for using minimal model entailment as a basis for beliefs is

the fact that it plays a fundamental role in the semantics of disjunctive logic programs and deductive databases and seems particularly well-suited to represent disjunctive information in these application domains. However, various other non-monotonic formalisms can be used to de ne the meaning of beliefs. In Section 7 we discuss some possible alternatives. The reason that we minimize only objective atoms is that the objective atoms A represent objective-level knowledge which, according to the closed world assumption, has to be minimized in order to arrive at minimal beliefs BA. On the other hand, belief atoms BF intuitively describe meta-level knowledge, namely, a plausible rational belief scenario , which is therefore not subject to minimization. 2 We now de ne static expansions and static completions of an arbitrary belief theory T . The de nition is based on the idea of building an extension Tb of T obtained by augmenting T with precisely those belief atoms BF that satisfy the condition that F is true in all minimal models of Tb. Consequently, the de nition of static expansions enforces the intended meaning of belief atoms BF discussed above.

De nition 3.2 (Static Expansions) A static expansion of a belief theory T is any belief theory Tb which satis es the following xed point condition:

Tb = Cn ( T [

f BF

: Tb j=min F g );

(13)

2

where F ranges over all formulae of LB .

Although the de nition of static expansions is similar to the de nition of Moore's stable autoepistemic expansions [Moo85], it uses minimal model entailment Tb j=min F instead of the logical implication Tb j= F and it does not specify when a belief atom BF is false in the expansion. As a result, properties of static expansions are signi cantly dierent from the properties of stable autoepistemic expansions. In particular, it turns out that every belief theory T has the least (in the sense of inclusion) static expansion T which is called the static completion of T . Static completion T has an iterative de nition as the least xed 2 The

author is grateful to L.Yuan for pointing out the need to use circumscription that minimizes only objective rather than all propositional atoms [YYar].

7

point of a monotonic belief closure operator T de ned below. Although static completions may, in general, be inconsistent theories, we will show that all armative belief theories always have consistent static completions. These properties of static expansions sharply contrast with the properties of stable autoepistemic expansions which do not admit natural least xed point de nitions and, in general, do not have least elements.

De nition 3.3 (Belief Closure Operator) For any belief theory T de ne the belief clo-

sure operator T by the formula:

T (S ) = Cn( T [ fBF : S j=min F g ); where S is an arbitrary belief theory and F ranges over all formulae of LB .

(14)

2

We begin with the following easy observation.

Proposition 3.1 A theory Tb is a static expansion of the belief theory T if and only if Tb is

a xed point of the operator T , i.e., if Tb = T (Tb).

Tb

Proof. Theory Tb is a xed point of the operator T if Tb = T (Tb) = Cn( T [ fBF : j=min F g ) which is equivalent to Tb being a static expansion of T . 2

Consequently, in order to show that every belief theory has the least static expansion we need to prove that the operator T has the least xed point. We rst establish the (restricted) monotonicity of the operator T .

Proposition 3.2 (Monotonicity of the Belief Closure Operator) The operator T

is monotonic. More precisely, suppose that the theories V 0 and V 00 are extensions of T obtained by adding some belief atoms BF to T and let T 0 = Cn(V 0) and T 00 = Cn (V 00). If T 0  T 00 then T (T 0 )  T (T 00) . Proof. Suppose that V 0 = T [ f BFs : s 2 S 0g, V 00 = T [ f BFs : s 2 S 00g and let T 0 = Cn (V 0) and T 00 = Cn (V 00 ). We have to show that T (T 0)  T (T 00). Since Cn(V 0 )  Cn (V 00) we can clearly assume that S 0  S 00. It suces to show that if T 0 j=min F then T 00 j=min F . Suppose that T 0 j=min F and let M be an arbitrary minimal model of T 00. Since 0 V  V 00 and V 0 and V 00 dier only on the set of belief atoms and since minimal models do not minimize belief atoms, M is also a minimal model of V 0 and thus also a minimal model of T 0. We conclude that M j= F and therefore T 00 j=min F . 2

Using Propositions 3.1 and 3.2 we can easily adapt the proof of the well-known Theorem of Tarski, ensuring the existence of least xed points of monotonic operators, to obtain:

Theorem 3.1 (Least Static Expansion) Every belief theory T has the least static expansion, namely, the least xed point T of the monotonic belief closure operator T .

8

The least static expansion T of a belief theory T can be constructed as follows. Let T 0 = Cn(T ) and suppose that T  has already been de ned for any ordinal number  < . If =  + 1 is a successor ordinal then de ne3: T +1 = T (T ) = Cn ( T [ fBF : T  j=min F g ); where F ranges over all formulae in LB . Else, if is a limit ordinal then de ne: [ T = T : <

The sequence fT g is monotonically increasing and thus has a unique xed point T = T  = T (T ), for some ordinal . Moreover, if the objective language L is nite then the xed point T is reached after nitely many steps . Proof. From Proposition 3.2 it easily follows that the sequence fT g is monotonically increasing and thus has a unique xed point T = T  = T (T ), for some ordinal . This xed point must therefore be the least xed point of the operator T . From Proposition 3.1 we infer that any xed point of the operator T is a static expansion of T . If the objective language L is nite then clearly the construction has to stop after nitely many steps. 2

De nition 3.4 (Static Completion) The least static expansion T of a belief theory T is

2

called the static completion of T .

Since the static completion of a belief theory T is obtained by augmenting P with the least possible number of belief atoms BF , it can be viewed as the most skeptical among static expansions (see [HTT87]). Like Clark's predicate completion comp(P ) of a program P , the static completion T of a belief theory T describes the corresponding semantics of T :

De nition 3.5 (Static Semantics) By the static semantics of a belief theory T we mean the set of all formulae that belong to the static completion T of T .

2

More generally, any class K of static expansions of a belief theory T naturally de nes the corresponding semantics consisting of those sentences that belong to all expansions from the class K. Observe that since the static completion of a belief theory T is the least (in the sense of inclusion) static expansion of T , it contains those and only those formulae which belong to all static expansions of T . One of the important strengths of static semantics is the fact that it can be constructed by means of the iterative minimal model procedure given in Theorem 3.1. As a result, static semantics not only has an elegant xed point characterization but it can simply be viewed as the iterated minimal model semantics . The last fact is important from the procedural point of view. Namely, once a suitable procedure is devised to compute the minimal model semantics, it can then be iteratively applied to compute the static semantics. The following theorem signi cantly extends Theorem 3.1 and provides a complete characterization of all static expansions of a belief theory. T

3 Since

the sequence fT  g is monotonically increasing we can equivalently de ne T = Cn ( T  [ fBF : j=min F g ).

9

Theorem 3.2 (Characterization Theorem) A theory Tb is a static expansion of a belief

theory T if and only if Tb is the static completion T 0 of a theory T 0 = T [ fBFs : s 2 S g, obtained by adding some belief atoms BFs to T , so that the following condition is satis ed:

Tb j=min Fs , for every s 2 S: 2

Proof. (=)) Suppose that Tb is a static expansion of T . Then:

Tb = Cn( T

[ f BF

: Tb j=min F g ):

Tb = Cn( T 0

[ f BF

: Tb j=min F g ):

Let T 0 = T [ f BF : Tb j=min F g. It suces to show that Tb is the least static expansion T 0 of T 0 . Clearly, Tb is a static expansion of T 0 because: Since Tb = Cn (T 0) it is also the least static expansion of T 0. ((=) Suppose now that Tb is the static completion T [ f BFs : s 2 S g of a belief theory T 0 = T [ f BFs : s 2 S g which satis es the condition that Tb j=min Fs , for every s 2 S . Since Tb is the (least) static expansion of the belief theory T [ f BFs : s 2 S g we have:

Tb = Cn ( T [ f BFs : s 2 S g

Since Tb j=min Fs , for every s 2 S we obtain:

Tb = Cn ( T

[ f BF

which shows that Tb is a static expansion of T .

[ f BF

: Tb j=min F g ):

: Tb j=min F g );

2

According to the above theorem, in order to nd a static expansion Tb of a belief theory T one needs to:  



Select a set f BFs : s 2 S g of beliefs. Construct the static completion Tb = T [ f BFs : s 2 S g of the augmented belief theory T 0 = T [ f BFs : s 2 S g (e.g., using the iterative xed point de nition from Theorem 3.1). Show that Tb j=min Fs , for every s 2 S .

It is the rst part, namely, the selection (or guessing) of the set of beliefs fBFs : s 2 S g, that is most dicult. However, one can always choose the empty set to obtain the least static expansion, or, equivalently, the static completion T of the belief theory T . Theorem 3.1 also immediately implies:

Corollary 3.3 (Greatest Static Expansion) Every belief theory T has the greatest static expansion which is an inconsistent theory.

10

Proof. Let Tb = T [ f B(A ^ :A)g, where A is an arbitrary atom. It is easy to see that Tb is inconsistent. Indeed, since any theory logically implies A _ :A we infer that B(A _ :A) belongs to Tb. Consequently, by the Consistency Axiom, :B(A ^ :A) also belongs to Tb which shows that Tb is inconsistent. Since Tb is inconsistent, Tb j= A ^ :A, which, by Theorem 3.1, implies that Tb is a stationary expansion of T . 2

It is time to introduce the rst example. Example 3.1 Consider the following normal logic program P :

Fly

not Ab;

where Ab stands for \Abnormal" , and translate it into the (armative) belief theory T = T (P ) given by: B:Ab  Fly . We will now compute the static completion of T . Let T 0 = Cn (T ) and T 1 = T (T 0 ) = Cn (T [ fBF : T 0 j=min F g): First observe that T 0 j=min :Ab. Indeed, in order to nd minimal models of T 0 we need to assign an arbitrary truth value to the belief atom B:Ab and then minimize the objective atoms Ab and Fly: We easily see that T 0 has the following two (classes of) minimal models (truth values of the remaining belief atoms are irrelevant and are therefore skipped): M1 = fFly; :Ab; B:Abg M2 = f:Fly; :Ab; :B:Abg: Since the atom Ab is false in all minimal models of T 0 , while Fly is false only in some of them, we have: T 0 j=min :Ab but T 0 6j=min :Fly and T 0 6j=min Fly: We obtain therefore: The next iteration is:

fB:Abg T 1:

T 2 = T (T 1) = Cn (T [ fBF : T 1 j=min F g)

and since T 1 j= Fly we immediately obtain:

fB:Ab; BFly g T 2 :

Moreover, one easily veri es that no other belief atom of the form BL, where L is an objective literal, belongs to T 2 and that T 2 is a xed point of the operator T , i.e., T 2 = T (T 2 ). Consequently, T = T 2 = T [ fB:Ab; BFly g is the static completion of T . It is also easy to check, using Theorem 3.2, that T does not have any other (consistent) static expansions. For example, to show that Tb = T [ fBAbg is not a static expansion of T it suces to note that Tb 6j=min Ab. Observe that there is a natural one-to-one correspondence, between the unique (consistent) static expansion T = T [ fB:Ab; BFly g of T = T (P ) and the unique perfect (well-founded , or stable ) model M = f:Ab; Fly g of the program P in which Fly is true and Ab is false. 2 11

Remark 3.2 In the previous example, and in all the other examples considered in this section, we were only interested in establishing which belief atoms of the form BL, where L is an objective literal, belong to the static completion T . This is because the belief theories T considered in this Section are translations of normal logic programs. Observe, however, that the static completion T = T [ fB:Ab; B Flyg of the belief theory T discussed in the previous example also contains many other belief atoms, such as B(Fly _ :Fly ) and B(Fly _ :Ab). Since these belief atoms do not appear in the theory T itself, they do not aect our considerations in any way. 2 Example 3.2 If we add :Fly to the previous theory T we obtain the (non-armative) belief theory T 0 : B:Ab  :Fly

Fly

whose static completion consists of: T 0 = T 0 [ fB:Ab; BFlyg: and is therefore inconsistent because it implies both Fly and :Fly . In fact the theory appears to describe contradictory information. 2 The last example shows that the static completion T of a consistent belief theory T may be inconsistent. It follows from Theorem 3.1 and from Corollary 3.3 that a belief theory T either has a consistent least static expansion (i.e., static completion) T or it does not have any consistent static expansions at all, in which case its least and greatest static expansions coincide. The following theorem shows that static completions of armative belief theories de ned in (9) are always consistent. Theorem 3.4 (Consistency of Static Completions) Static completion T of any armative belief theory T is always consistent. In particular, static completion T of any (translation of a) normal or disjunctive logic program is always consistent. 2 Proof. We will prove by induction that T  , as de ned in Theorem 3.1, is consistent, for every . To see that T 0 = Cn (T ) is consistent it suces to take an interpretation of LB in which all belief atoms are false and all objective atoms are true. Suppose that we already proved that T  is consistent, for any  < . If is a limit ordinal then, by the Compactness Theorem, T must also be consistent as a union of an increasing sequence of consistent theories. If on the other hand =  + 1 then: T = Cn (T [ fBF : T  j=min F g): Since T  is consistent, the class of formulae F minimally entailed by T  also constitutes a consistent theory. De ne an interpretation M so that all the objective atoms and all the belief atoms BF such that T  j=min F are true in M while all the remaining belief atoms are false. Clearly M is a model of T [ fBF : T  j=min F g and since it also clearly satis es axioms (2) and (3) it is a model of T . We conclude that T is consistent, which completes the inductive step. 2 We are now ready to discuss more examples and prove additional properties of static expansions and static semantics of normal and disjunctive logic programs. 12

4 Static Expansions of Normal Logic Programs In this Section we study static expansions of (armative) belief theories T obtained as translations T (P ) of normal, non-disjunctive logic programs P . We show that there is a one-to-one correspondence between stationary (or partial stable) models of a program P and consistent static expansions of its translation T = T (P ) into belief theory.

Example 4.1 Consider the following program P : Bird Ab Fly

not Bird not Ab;

and its translation into the (armative) belief theory T = T (P ):

Bird B:Bird B:Ab

 

Ab Fly:

In order to compute its static completion T we let T 0 = Cn (T ) and de ne:

T 1 = T (T 0) = Cn (T Since T 0 j= Bird we have

[ fBF

: T 0 j=min F g):

fBBirdg  T 1 :

By the Consistency Axiom (2), T 1 j= :B:Bird and therefore T 1 j=min :Ab. Since:

T 2 = T (T 1) = Cn (T

[ fBF

: T 1 j=min F g);

we immediately conclude that: fBBird; B:Abg  T 2 :

Similarly, since T 2 j= Fly we obtain: fBBird; B:Ab; BFly g  T 3 :

Moreover, it is easy to see that no other belief atom of the form BL, where L is an objective literal, belongs to T 3 and that T 3 = T (T 3) is a xed point of T and therefore T = T 3 = T [ fBBird; B:Ab; BFlyg is the static completion of T . Using Theorem 3.2 one easily veri es that T does not have any other (consistent) static expansions. There is an obvious one-to-one correspondence between the unique static expansion T = T [ fBBird; B:Ab; BFly g of T = T (P ) and the unique perfect (well-founded , or stable ) model M = fBird; :Ab; Fly g of the program P in which Bird and Fly are true and Ab is false. 2

13

Example 4.2 Consider now the following normal program P : A B Q Q

not B not A A B and its translation into (armative) belief theory T = T (P ): B:B  A B:A  B A  Q B  Q: Let T 0 = Cn(T ) and de ne:

T 1 = T (T 0) = Cn(T

[ fBF

: T 0 j=min F g):

The theory T 0 has minimal models in which both B:A and B:B are true (respectively, false) and thus all of A; B and Q are true (respectively, false). Consequently, T 0 6j=min A, T 0 6j=min :A, etc. and thus T 1 does not contain any belief atoms of the form BL, where L is an objective literal. This leads us to the conclusion that T 1 = T (T 1) is a xed point of T and thus the static completion T of T . However, in addition to the least static expansion Tb0 = T = T [ f g, this (translated) program T = T (P ) has two more static expansions, namely:

Tb1 = T [ fB:B; BA; BQg Tb2 = T [ fB:A; BB; BQg:

Indeed, according to Theorem 3.2, to show that Tb1 is a static expansion of T it suces to show that T [ fB:B; BA; BQg j=min :B ^ A ^ Q. Clearly, T [ fB:B; BA; BQg j= A ^ Q. Moreover, by the Consistency Axiom (2) Tb1 j= :B:A which implies that Tb1 j=min :B . The proof that Tb2 is a static expansion of T is completely symmetric. It is easy to see that T does not have any other (consistent) static expansions. Observe that again there exists a one-to-one correspondence between stationary (or partial stable) models Mi of the program P [Prz90, Prz91c] and static expansions Tbi of T = T (P ). Indeed, P has three stationary models, namely: M0 = fg M1 = f:B; A; Qg M2 = f:A; B; Qg which precisely correspond to static expansions Tb0 , Tb1 and Tb2, respectively. Models M1 and M2 are (total) stable but the least (well-founded) model M0 is a partial model which corresponds to the least static expansion (static completion) Tb0 = T of T . 2

The above example underlines the fact that our approach uses a strong interpretation of non-monotonic negation, namely, one that requires us to establish that F is false in 14

all minimal models of Tb, rather than just in one of them, in order to conclude not F , or, equivalently, B:F . In the preceding example we could not prove that A is false in all minimal models of T or that B is false in all minimal models of T , and thus we were unable to conclude that either B:A or B:B belongs to T .

Example 4.3 Finally, consider the following normal program due to Van Gelder [VGRS90]: A B Q Q or, after translation:

A B Q Q

not B not A not Q not A;    

B:B B:A B:Q B:A:

This program has precisely three (consistent) static expansions, namely:

Pb0 = P Pb1 = P [ fB:B g = P [ fBA; B:B g Pb2 = P [ fB:Ag = P [ fB:A; BB; BQg:

The proof that Pbi 's are static expansions is identical as in the previous example4 . Again, there exists a one-to-one correspondence between stationary models Mi of P [Prz90, Prz91c] and static expansions of Pbi of P . Indeed, P has three stationary models, namely:

M0 = fg M1 = fA; :Bg M2 = f:A; B; Qg

which precisely correspond to static expansions Pb0 , Pb1 and Pb2 , respectively. Model M2 is (total) stable but both models M1 and M2 are partial. As before, the well-founded model M0 corresponds to the static completion P . 2 It turns out that the one-to-one correspondence between stationary models of a logic program P and consistent static expansions of its translation T (P ) holds for all normal programs.

b

4 To

b

verify that P1 does not contain BQ observe that P1 = P [ fBA; B:B g has precisely two types of minimal models, namely: M1 = fA; :B; Q; BA; B:B; B:Qg M2 = fA; :B; :Q; BA; B:B; :B:Qg: which implies that: P1 6j=min Q and P1 6j=min :Q and therefore P1 6j= BQ and P1 6j= B:Q :

b

b

b

15

b

Theorem 4.1 Let P be an arbitrary normal program and let L denote any (objective) literal. There is a one-to-one correspondence between stationary (or partial stable) models M of the program P and consistent static expansions Tb of its translation T = T (P ). Namely, if M is a stationary model of P , then Tb = T [ fBL : L 2 M g

is a consistent static expansion of T . Conversely, if Tb is a consistent static expansion then

M = fL : BL 2 Tbg

is a stationary model of P . In other words, the correspondence can be described by:

L 2 M  BL 2 Tb; i.e., a literal L belongs to the model M if and only if L is believed to be true in Tb.

Proof. We rst recall the characterization of stationary models [Prz90] of normal programs given in [Prz91c]. It uses the Gelfond Lifschitz quotient operator MP [GL88] which assigns to any normal logic program P and any interpretation M a positive normal logic program MP obtained by deleting all clauses containing negative premises not A such that A is true in M and removing all negative premises not A from the remaining clauses. Characterization of stationary models: [Prz91c] Suppose that P is a normal logic program and M is an arbitrary partial interpretation. De ne M + (respectively, M ? ) to be the total interpretation obtained by making all unde ned atoms in M true (respectively, false). An interpretation M is a stationary model of P if and only if M = MPos [ MNeg , where MPos (respectively, MNeg ) is the set of positive (respectively, negative) literals in the least model of the program MP+ (respectively, MP? ). We now prove Theorem 4.1: ( =)) Suppose that M = MPos [ MNeg is a stationary model of P and let:

Tb = T [ fBL : L 2 M g:

We will show that Tb is a static expansion of T = T (P ). By Theorem 3.2, it suces to show that Tb j=min L, for every L 2 M . Observe that if A 2 MPos then BA 2 Tb and therefore, by the Consistency Axiom (2), :B:A 2 Tb. Similarly, if :A 2 MNeg then B:A 2 Tb and therefore :BA 2 Tb . This means that a negative premise of the form B:A in a program clause is always true in any model N of Tb if :A 2 M and always false in any model N of Tb if A 2 M . Let N be any minimal model of Tb. Since Tb contains, as a subtheory, the translation of the program MP+ , all the positive atoms in the least model of MP+ must belong to N . Consequently, MPos  N . Similarly, since the set of all satis able clauses in Tb is contained in the translation of the program MP? , all the negative literals in the least model of MP? must also belong to N . Consequently, MNeg  N . This shows that Tb j=min L, for every L 2 M and thus completes the proof of the rst part. 16

( (=) The proof in the opposite direction is completely analogous.

2

Static expansions provide therefore a new characterization of stationary models of normal programs. It has been shown in [Prz90, Prz91c] that every normal logic program P has at least one stationary model and that among all of its stationary models there is always the smallest one which coincides with the well-founded model of P . In view of the above Theorem, we immediately conclude: Corollary 4.2 For every normal program P the well-founded model MP of P corresponds to the static completion T of T = T (P ). Consequently, for normal programs, the static, the stationary and the well-founded semantics all coincide. 2 Stationary (or partial stable) models of a normal program P include as a proper subset the class of all (total) stable models [Prz90]. From Theorem 4.1 we immediately infer: Corollary 4.3 Let P be an arbitrary normal program. There is a one-to-one correspondence between stable models M of P and those consistent static expansions Tb of T = T (P ) that satisfy the condition: BA 2 Tb or B:A 2 Tb, for any (objective) atom A . 2 In other words, stable models correspond to those consistent static expansions that completely decide the status of all belief atoms.

5 Static Expansions of Disjunctive Logic Programs We now consider examples of belief theories T = T (P ) obtained as translations of disjunctive logic programs P . According to Theorem 3.4, every armative belief theory T and thus, in particular, every translation T = T (P ) of a disjunctive logic program P , has a consistent least static expansion T . Moreover, the class of consistent static expansions of (translations of) disjunctive programs extends, via the correspondence described in Theorem 4.1, the class of stationary models of normal logic programs. Consequently, static semantics of such programs extends the well-founded semantics of normal programs. As we will see below, static semantics also extends the minimal model semantics of positive disjunctive programs. Distributivity of Conjunctive Beliefs: We rst recall that the axiom (N) implies that beliefs are distributive with respect to conjunctions [MT94], namely, that for any formulae F and G: B(F ^ G)  BF ^ BG : (15) In other words, the conjunction F ^ G of formulae F and G is believed if and only if both F and G are believed. Example 5.1 Let P be the positive disjunctive logic program consisting of a simple disjunction:

A_B :

Its translation T = T (P ) coincides therefore with P itself. Let T 0 = Cn (T ). Since T 0 j=min A _ B and T 0 j=min :A _ :B: 17

we obtain:

fB(A _ B ); B(:A _ :B )g  T 1

= T (T 0 ): It is easy to check that T 1 is a xed point of T and therefore T = T 1 is the static completion of T . It describes the minimal model semantics of the program P , namely, the semantics consisting of formulae A _ B and :A _ :B . The theory T = P does not have any other (consistent) static expansions. 2 Remark 5.1 In the previous example, and in all the other examples considered in this section, we are only interested in establishing which belief atoms of the form BL, where L is a disjunction of positive (respectively, negative) objective literals, belong to the static completion T . This is because the belief theories T considered in this Section are translations of disjunctive logic programs. We recall again, however, that static completions T of belief theories T also contain many other belief atoms, such as B(A _ :A), where A is an arbitrary atom. 2 The next result extends the above observation to all positive disjunctive programs by showing that the static semantics of positive disjunctive programs always corresponds to the minimal model semantics of P . Theorem 5.1 Let P be a positive disjunctive logic program. The translation T = T (P ) of P into belief theory has precisely one consistent static expansion (or static completion) T which naturally corresponds to the minimal model semantics of T , namely: T = T [ fBF : T j=min F g : Proof. Since T = P does not contain any belief premises its static completion T given by Theorem 3.1 is obtained after just one iteration, i.e., T = Cn(T [ fBF : T 0 j=min F g) = T [ fBF : T j=min F g: Clearly, this is the only stationary expansion of T . 2 We now consider examples of non-positive disjunctive programs. Example 5.2 Let P be the disjunctive program consisting of clauses: Write Paper 1 _ Write Paper 2 Write Paper 1 ^ Write Paper 2 Get Crazy Get Fired not Write Paper 1 ^ not Write Paper 2 and let T = T (P ) be its translation into the (armative) belief theory: Write Paper 1 _ Write Paper 2  Get Crazy Write Paper 1 ^ Write Paper 2 B:Write Paper 1 ^ B:Write Paper 2  Get Fired: Let T 0 = Cn(T ) and assume obvious abbreviations. Clearly, in all minimal models of T 0 the disjunctions WP 1 _ WP 2 and :WP 1 _ :WP 2 hold true. Therefore: T 0 j=min WP 1 _ WP 2 ; T 0 j=min :WP 1 _ :WP 2 ; T 0 j=min :GC 18

and, consequently: fB(WP 1 _ WP 2); B(:WP 1 _ :WP 2); B:GC g  T 1

= T (T 0):

Now, by the Consistency Axiom (2), it follows that T 1 j= :B(:WP 1 ^ :WP 2) and T 1 j= :B(WP 1 ^ WP 2). By the Distributivity of Conjunctive Beliefs (15), T 1 j= :B:WP 1 _ :B:WP 2. Consequently: T 1 j=min :GF and therefore: fB(WP 1 _ WP 2); B(:WP 1 _ :WP 2); B:GC; B:GF g  T 2

= T (T 1):

It is easy to see that T 2 = T (T 2) = T 3 is a xed point and therefore:

T = T [ fB(WP 1 _ WP 2); B(:WP 1 _ :WP 2); B:GC; B:GF g: The resulting semantics corresponds to the perfect model semantics [Prz88] of the disjunctive program P . Based on the known fact that one of the two papers will be written and the belief that not both will be written, it establishes the beliefs that the author will neither get red nor get crazy. One easily veri es that T does not have any other (consistent) static expansions. 2

Example 5.3 Consider now the following strati ed disjunctive logic program P describing

the state of mind of a person planning a trip to either Australia or Europe:

Goto Australia _ Goto Europe Goto Both Save Money Cancel Reservation Cancel Reservation

Goto Australia ^ Goto Europe not Goto Both not Goto Australia not Goto Europe and its translation into the (armative) belief theory T = T (P ): Goto Australia _ Goto Europe Goto Australia ^ Goto Europe B:Goto Both B:Goto Australia B:Goto Europe

   

Goto Both Save Money Cancel Reservation Cancel Reservation:

Let T 0 = Cn(T ) and assume obvious abbreviations. Clearly, in all minimal models of T 0 the disjunctions GA _ GE and :GA _ :GE hold true. Therefore:

T 0 j=min GA _ GE; T 0 j=min :GA _ :GE and T 0 j=min :GB and, consequently: fB(GA _ GE ); B(:GA _ :GE ); B:GB g  T 1

19

= T (T 0):

Now T 1 j=min SM and thus fB(GA _ GE ); B(:GA _ :GE ); B:GB; BSM g  T 2 = T (T 1): It is easy to see that there is a minimal model of T 2 in which B:Goto Australia is true and thus also Cancel Reservation is true. But there is also is a minimal model of T 2 in which both B:Goto Australia and B:Goto Europe are false and thus also Cancel Reservation is false. Consequently: T 2 6j=min CR and T 2 6j=min :CR . This leads to the conclusion that T 3 = T (T 2 ) = T 2 is a xed point and therefore the static completion T of T is given by: T = T [ fB(GA _ GE ); B(:GA _ :GE ); B:GB; BSM g: It establishes that the individual is expected to travel either to Australia or to Europe but (s)he is not expected to do both trips and thus will save money. One easily veri es that T does not have any other (consistent) static expansions. 2 It is important to stress that Cancel Reservation is not a logical consequence of the static semantics T of the previously considered (translated) program T = T (P ). This is a consequence of the fact that the static completion T does not infer5 B:Goto Australia _ B:Goto Europe even though it derives B(:Goto Australia _ :Goto Europe). This re ects the notion that from the fact that a disjunction F _ G is believed to be true, one does not necessarily want to conclude that either F is believed or that G is believed. In this particular case, we do not want to cancel our reservations to either Australia or to Europe until we nd out precisely which one of them we will actually not visit. In other words, we usually do not want to assume that the belief operator B is distributive with respect to disjunctions. This feature of static expansions represents one of several major dierences between static expansions and stationary expansions introduced earlier in [Prz91b]. However, one could ensure distributivity of beliefs w.r.t. disjunctions by assuming the Disjunctive Belief Axiom: B(F _ G)  BF _ BG: (16) Another consequence of the fact that the belief operator is not distributive w.r.t. disjunctions is the fact that the static semantics, when restricted to strati ed disjunctive programs, is in general weaker than the perfect model semantics [Prz88]. In this sense the static semantics is similar to the semantics introduced earlier by Ross [Ros89]6. We view this feature as an important strength of static semantics. The semantics derives a minimal , in some sense, set of conclusions that can be inferred from a disjunctive program. If a given application area requires us to infer more facts, we can often achieve it by explicitly adding additional axioms which then become an intrinsic part of the description of the program and thus can be more easily revised and modi ed. As a result, we are less likely to infer undesirable conclusions. For example, in the previous example, the perfect model semantics implies that we should cancel our reservation before we even nd out where we are going, which seems clearly undesirable. 5 However, by the Consistency Axiom (2), T implies the weaker formula: :B(6Goto Australia ^ Goto Europe):BGoto Australia _ :BGoto Europe.

The exact relationship between the two semantics requires further investigation.

20

Example 5.4 Finally, let us consider the following non-strati ed disjunctive program P rst introduced in [Prz91c]):

Work _ Tired _ Sleep Work Sleep Tired Unhappy Paid; and its translation T = T (P ): Work _ Tired _ Sleep B:Tired B:Work B:Sleep Work ^ B:Paid Paid:

not Tired not Work not Sleep Work ^ not Paid

   

Work Sleep Tired Unhappy

Intuitively, it says that the person involved is either asleep or tired or working but it is not exactly clear which. Moreover, (s)he is unhappy when (s)he works and is not paid for it. However, (s)he has been paid and therefore has no reason to be unhappy. This (translated) program T has precisely one (consistent) static expansion, namely:

T = T [ fB(Work _ Tired _ Sleep); B(:Work _ :Tired _ :Sleep); BPaid; B:Unhappyg; which agrees with its intended meaning. Recall that the program does not have any (partial or total) disjunctive stable models [Prz91c, GL90], and therefore its disjunctive (partial or total) stable semantics is unde ned. 2 Observe that the method used in all the discussed examples to construct static completions of logic programs closely resembles the method used to de ne perfect models of strati ed normal programs (see [ABW88, VG89, Prz88]). Its signi cance, however, lies in the fact that it works for all normal and disjunctive programs. In fact, it works for a much broader class of armative belief theories. More work is needed, however, to determine effective ways of computing static completions of disjunctive programs. The main stumbling block is the problem of computing the minimal model semantics.

6 Logic Programs with Strong (or \Classical") Negation As we already know, the negation operator not F used in logic programs does not represent the classical negation :F , but rather a non-monotonic, commonsense negation. Gelfond and Lifschitz pointed out [GL90] that in logic programming it is often useful to have a dierent negation operator, more closely resembling the classical negation. They introduced such an operator, which they called \classical negation", and developed a suitable semantics for extended logic programs with \classical negation" based on the stable model semantics. As pointed out by several researchers [Prz90, PA92, AP92], the form of negation proposed by 21

Gelfond and Lifschitz does not represent real classical negation :F but rather its weaker form, denoted here by F , which does not require the law of excluded middle F _ F . Following Pereira et.al. [AP92] we call it strong negation. In our belief theories, which can be viewed as generalized logic programs, we already allow the full use of real classical negation :F but so far we do not have any counterpart of the strong negation F . In order to illustrate the dierence between the strong negation F and the real classical negation :F observe that the following belief theory T :

B :B

A A

 

is a translation T (P ) of the (extended) logic program P given by:

A A

B :B:

A A

B

Clearly, T implies A because the law of excluded middle B _ :B always holds in classical logic. On the other hand, according to the approach proposed by Gelfond-Lifschitz, the extended logic program P 0 : B:

which uses strong negation B instead of the classical negation :B implies not A and therefore does not imply A. It is easy, however, to extend our language to allow the use of strong negation and thus obtain three types of negation, namely:  Classical, monotonic negation :F ;  Non-monotonic, commonsense negation not F , de ned by B:F ;  Strong negation A. In order to use strong negation it suces to add to the original objective language L new objective atoms \A", with the intended meaning that A is the \strong negation of A", and assume the strong negation axioms : A ^ A  ? , or, equivalently, A  :A ; (17) for any objective atom A in the original language L. Observe that, as opposed to the real classical negation :, the law of excluded middle A _ A is not assumed. As pointed out by Bob Kowalski7 , if the atom A describes the property of being \good" then the atom A may describe the property of being \bad". The strong negation axiom states that things cannot be both \good" and \bad". We do not assume, however, that things must always be either \good" or \bad". The resulting framework provides a strict generalization of the Gelfond-Lifschitz approach. More precisely, one easily proves the following one-to- one correspondence between static expansions and stable models of extended normal logic programs with strong negation. This result strictly generalizes Corollary 4.3. 7 Lecture

1992.

presented during the Workshop in Honor of Professor Jack Minker, Washington, DC, October

22

Corollary 6.1 Let P be an arbitrary extended normal program with strong negation and let T = T (P ) be its translation into belief theory. There is a one-to-one correspondence between stable models M of P (in the sense of [GL90]) and consistent static expansions Tb of T that satisfy the condition that:

b

BA 2 T

or

b

B:A 2 T ,

for any (objective) atom A

i.e., those expansions that completely decide the belief status of all literals.

2

This approach also eliminates some problems (pointed out earlier by Pereira et.al. [PAA92]), in the de nition of stationary (or partial stable) models of extended programs with strong negation originally given in [Prz90].

Example 6.1 [PAA92] Consider the extended logic program P : A

A B

not B not A

and its translation T = T (P ) into belief theory: A B:B  B:A 

A B;

where A denotes strong negation and thus represents a new objective atom. Let T 0 = Cn(T ). Since T 0 j= A, from the strong negation axioms (17) it follows that T 0 j= :A and therefore: Thus T 1 j=min B and therefore:

fBA; B:Ag  T 1 :

fBA; B:A; BB g  T 2:

One easily veri es that T = T 2 is the static completion of T . We believe therefore in B and in not A which seems intuitively correct. The approach previously proposed in [Prz90] was not powerful enough to derive B and not A. It is conjectured by Jose Alferes that the form of negation obtained in this way coincides with the strong negation introduced in [AP92]. 2

7 Other Extensions The proposed formalism is quite exible and allows various other extensions and modi cations. In this section we discuss some of them.

23

7.1 Using a Dierent Non-Monotonic Formalism

In our approach we used the minimal model semantics T j=min F or the Generalized Closed World Assumption GCWA [Min82] to de ne the meaning of our beliefs BF . In other words, F is believed if F is true in all minimal models of the expansion. As illustrated by the following example, by using the weak minimal model semantics T j=wmin F or the Weak Generalized Closed World Assumption WGCWA [RLM89, RT88] instead and thus requiring that F is believed if F is true in all weakly minimal models of T , one can ensure that disjunctions are treated inclusively rather than exclusively.

Example 7.1 Consider the following (translated) positive program T : A_B A^B



C:

Let T 0 = Cn(T ). Clearly, in all weakly minimal models of T 0 the disjunction A _ B holds. On the other hand, while the disjunction :A _ :B is true in all minimal models of T 0 it is not true in all weakly minimal models of T 0. Therefore:

T 0 j=wmin A _ B and T 0 j=min :A _ :B and yet T 0 6j=wmin :A _ :B: As a result: fB(A _ B ); B(:A _ :B ); B:C g  T 1

= T (T 0 ) but only

fB(A _ B )g  Tw1

= wT (T 0);

where by the index \w" we indicate the fact that we are using WGCWA instead of GCWA in the de nition of static expansions and static completion. It is easy to see that T 1 = T (T 1) is a xed point and therefore:

T = T [ fB(A _ B); B(:A _ :B); B:C g: Similarly, Tw1 = wT (Tw1 ) is a xed point and therefore:

T w = T [ fB(A _ B)g:

We conclude that under GCWA we can derive that both A _ B and :A _ :B as well as :C are believed, whereas WGCWA only allows us to believe A _ B . 2 Both GCWA and WGCWA are very natural non-monotonic formalisms which seem to closely correspond to the intuitive meaning of negation in logic programs and deductive databases. However, they also share an important feature which in some applications domains may be viewed as a drawback, namely the fact that they both minimize only positive literals (atoms) thus leading to immediate asymmetry between positive and negative literals. If this feature of GCWA and WGCWA is undesirable, one can use almost any other non-monotonic formalism, naturally leading to a dierent notion of belief and thus to a dierent semantics. In particular, one can use a suitable form of predicate or formula circumscription which minimizes those and only those predicates (formulae) whose minimization is desired. 24

7.2 Adding More Axioms

One can add other axioms or inference rules in order to tailor the formalism of static expansions to the needs of speci c application domains. For example, as we mentioned before, by assuming the Disjunctive Belief Axiom (16): B(F _ G)  BF _ BG one can ensure that the belief operator is distributive with respect to disjunctions. In some application domains, e.g., in disjunctive deductive databases, such an assumption may be very natural. It would ensure, for example, that from the program P :

A_B C C whose translation T = T (P ) is:

not A not B

A_B B:A B:B

 

C C

one can deduce that C is true. Indeed, since T 0 j=min A _ B and T 0 j=min :A _ :B we have T j= B(A _ B ) and T j= B(:A _ :B ). In view of the above axiom, we then infer that T j= B:A _ B:B and therefore T j= C and T j= BC . Observe that the de nition of static expansions and completions Tb carefully distinguishes between these formulae F which are known to be true in the expansion Tb (i.e., those for which Tb j= F ), and those formulae F which are only believed (i.e., those for which Tb j= BF ). This important distinction not only increases the expressiveness of the language but is in fact quite crucial for many forms of reasoning. However, if we wanted to use the closed world assumption to ensure that a formula F is always true whenever it is believed to be true, we could achieve this goal by assuming that all theories are closed under the derivation rule : BF

F ;

which says that if F is believed then F is true. In [Prz94b] we discuss a dierent way of expressing the closed world assumption by means of an axiom rather than an inference rule. The fact that the de nition of static expansions can be relatively easily modi ed to accommodate various domain-speci c requirements underlines the exibility of the proposed framework.

7.3 Combining Static and Stable Expansions

One can combine static and stable expansions using the following extended de nition of static expansions which results in an even more expressive semantic framework:

De nition 7.1 A theory Tb is called a static autoepistemic expansion of a theory T if it satis es the following xed-point equation:

Tb = Cn(T [ fBF : Tb j=min F g [ fLF : Tb j= F g 25

[ f:LF

: Tb 6j= F g): 2

Here by LF we denote the autoepistemic belief operator as de ned in Moore's autoepistemic logic AEL [Moo85]. Observe that the second part of the de nition is identical to the de nition of stable autoepistemic expansions while the rst part coincides with the de nition of static expansions. Naturally, we assume now that the extended language LB;L allows both operators BF and LF . The resulting non-monotonic framework strictly extends several non-monotonic formalisms, including circumscription, autoepistemic logic, various semantics proposed for logic programs and deductive databases (stable semantics, well-founded semantics and stationary semantics) as well as Gelfond's epistemic speci cations. It is studied in detail in [Prz94b, Prz94a].

7.4 Choosing a Dierent Translation of Non-Monotonic Negation

One can translate the non-monotonic negation not F by using a slightly weaker formula def

not F

def

 :BF

;

instead of not F  B:F , thus giving it the intended meaning of \F is not believed". More precisely, let P be any disjunctive logic program consisting of the informal clauses: A1 _ : : : _ Al B1 ^ : : : ^ Bm ^ not C1 ^ : : : ^ not Cn (18) where l  1; m; n  0 and Ai , Bi and Ci 's are objective atoms. The translation of P into the belief theory T (P ) is de ned as the set of the corresponding clauses: B1 ^ : : : ^ Bm ^ :BC1 ^ : : : ^ :BCn  A1 _ : : : _ Al (19) in the language LB obtained by replacing the non-monotonic negation notF by the formula :BF and by replacing the implication symbol ! by the standard material implication . This translation T (P ) is therefore analogous to the one originally proposed by Gelfond for autoepistemic logic [Gel87]. It turns out that for normal programs P the translation T(P ) results in exactly identical semantics as the translation T (P ). However, for disjunctive programs the translation T (P ) in general results in a stronger semantics than the translation T (P ). Example 7.2 Consider again the strati ed disjunctive logic program P discussed in Example 5.3:

Goto Australia _ Goto Europe Goto Both Goto Australia ^ Goto Europe not Goto Both Save Money not Goto Australia Cancel Reservation Cancel Reservation not Goto Europe: This time, however, use its translation T = T(P ) into the belief theory: Goto Australia _ Goto Europe Goto Australia ^ Goto Europe  Goto Both  Save Money :BGoto Both  Cancel Reservation :BGoto Australia  Cancel Reservation: :BGoto Europe 26

Let T 0 = Cn(T ) and assume obvious abbreviations. Clearly, in all minimal models of T 0 the disjunctions GA _ GE and :GA _ :GE hold true. Therefore:

T 0 j=min GA _ GE; T 0 j=min :GA _ :GE and T 0 j=min :GB and, consequently: fB(GA _ GE ); B(:GA _ :GE ); B:GB g  T 1

= T (T 0):

By the consistency axiom, T 1 j= :BGB and thus T 1 j= SM . Similarly, T 1 j= :B(GE ^ GA), or, equivalently, T 1 j= (:BGE _ :BGA) and thus T 1 j= CR. We obtain therefore: fB(GA _ GE ); B(:GA _ :GE ); B:GB; BSM; BCRg  T 2

= T (T 1):

Again, T 3 = T (T 2 ) = T 2 is a xed point and therefore the static completion T of T . The static completion now also derives the fact that the individual is supposed to cancel his reservation. One easily veri es that T does not have any other (consistent) static expansions. 2

8 Concluding Remarks Belief theories represent a broad class of theories that include (translations of) all disjunctive logic programs. Static expansions provide a semantic framework for belief theories T which is based on the idea of building an extension Tb of T in which a formula F is believed to be true (i.e., BF 2 Tb) if F is true in all minimal models of Tb. When restricted to (translations of) logic programs it leads to the interpretation of non-monotonic negation not F as B:F , i.e., F is false in all minimal models. Although similar to the notion of stable autoepistemic expansions [Moo85], static expansions use minimal model entailment instead of logical implication and do not explicitly specify when BF is supposed to be false. Static expansions extend to disjunctive programs the class of stationary (or partial stable) models of normal, non-disjunctive programs and thus generalize both the stable and wellfounded models. They also extend the minimal model semantics of positive disjunctive programs. Static expansions dier signi cantly from the class of stationary expansions introduced earlier in [Prz91b, Prz91d]. The de nition of stationary expansions was based on the idea of building a completion E of a logic program P in which non-monotonic negation not F holds if F is false in all minimal models of E and :not F holds if F is true in all minimal models of E , because it assumed that :not F and not :F are equivalent. As a result, it blurred the essential distinction between :not F and not :F leading to some unintuitive results. It also assumed distributivity of :not with respect to disjunctions thus making it impossible to dierentiate between \disjunction of beliefs" and \beliefs in disjunctions", as illustrated in Example 5.3. Moreover, it used the so called \disjunctive inference rule" which is no longer assumed in static expansions. Although, for normal programs, the static, the stationary and the well-founded semantics all coincide (see Theorem 4.1), the notions of a static and a stationary expansions dier considerably in the class of disjunctive programs, and, consequently, the static and 27

the stationary semantics of disjunctive programs are signi cantly dierent. Static expansions are also de ned for a much broader class of theories and they constitute a special case of a more general non-monotonic framework introduced in [Prz94b] which generalizes several formalizations of non-monotonic reasoning. We view static expansions as a \rational reconstruction" of stationary expansions in the class of disjunctive programs and their generalizations which coincides with the old de nition in the class of normal programs. The proposed approach also diers signi cantly from the other major semantics proposed recently for disjunctive logic programs and databases (see [LMR92]). In particular:

Generalized well-founded semantics, introduced by Minker et. al. [BLM90, BLM89], extends the minimal model semantics of disjunctive programs but it does not extend the well-founded or the stationary or stable models of normal programs. Extended well-founded semantics, introduced by Ross [Ros89], extends the well-founded semantics of normal programs and the minimal model semantics of disjunctive programs. It does not, however, extend the stationary or stable models of normal programs. Disjunctive (partial) stable semantics, introduced in [Prz91c, GL90], extends both classes of stable and stationary models of normal programs as well as the minimal model semantics of disjunctive programs but it is de ned only for a fairly restricted subclass of the class of disjunctive programs.

Similar comments apply to the semantics introduced recently by Dix [Dix92]. As it is the case with most semantics for disjunctive programs, the static semantics is not cumulative (see [Dix91]).

Acknowledgments The author is grateful to Jose Alferes, Roland Bol, Juergen Dix, Michael Gelfond, Vladimir Lifschitz, Luis Pereira, Halina Przymusinska, Mirek Truszczynski and an anonymous referee for their helpful comments. The author is especially grateful to Grigorij Schwartz for his extensive remarks.

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