Characterizing D-WFS: Con uence and Iterated ... - Semantic Scholar
Characterizing D-WFS: Con uence and Iterated GCWA Stefan Brass1
Abstract. Quite recently Brass/Dix have introduced the se-
mantics D-WFS for general disjunctive logic programs. The interesting feature of this approach is that it is both semantically and proof-theoretically founded. Any program is associated a normalform res(), called the residual program. We show in this paper, that the original calculus, consisting of some simple transformations, has a very strong and appealing property: it is con uent . This means that all the transformations can be applied in any order: if we arrive at an irreducible program (no more transformation is applicable) then this is already the unique normalform. No proper subset of the calculus has this property. We also give an equivalent characterization of D-WFS in terms of iterated minimal model reasoning. This construction is a generalization of a description of the wellfounded semantics: we introduce a very simple and neat construction of a sequence Di that eventually stops and represents the set of derivable disjunctions. Both characterizations open the way for ecient implementations. The rst because the ordering of the transformations does not matter, the second because special methods from Circumscription might be useful.
1 Introduction
The success of PROLOG as one of the major programming languages for Arti cial Intelligence suggests that extensions of Horn-clauses to general clauses with negation and disjunction might become very useful in Knowledge Representation and Reasoning . Indeed, in recent years much work was going on in Logic Programming to extend semantics based on di erent forms of negation-as-failure from non-disjunctive to disjunctive semantics. Many interrelationships of non-monotonic logics (DL, CIRC, AEL) with semantics of logic programs have been discovered during the last years ([1, 9, 12]). Eiter and Gottlob have shown that disjunctive semantics are strictly more expressive than non-disjunctive semantics and therefore may be better suited for many knowledge representation tasks. One of the main problems today is that although we have a quite good understanding of non-disjunctive semantics, our knowledge of disjunctive semantics is quite limited. Besides the main competing approaches of the wellfounded semantics WFS (which is often too weak) and the stable semantics
1 University of Hannover, Inst. f. Informatik, Lange Laube 22, D30159 Hannover, Germany,[email protected] 2 University of Koblenz, Dept. of Computer Science, Rheinau 1,
D-56075 Koblenz, Germany
and
Jurgen Dix2
STABLE (which is sometimes too strong) there exist many extensions of WFS that approximate STABLE. Unfortunately, only STABLE has a straightforward extension to disjunctive programs: not even for WFS does there exist a canonical disjunctive version. We claim that D-WFS is this counterpart of WFS. The novelty of our approach is that it is not exclusively declarative (like Przymusinski's static or stationary approach) nor exclusively procedural (like the approaches of Minker and his group). We introduce a calculus of program transformations. These are declarative since they express precise semantical properties (e.g. partial evaluation ). But they are also procedural because they can be applied to a program and transform it to another one. Our main results are The calculus is con uent . D-WFS can be equivalently described using iterated minimal model reasoning, which relates it to circumscription. We restrict in this paper for simplicity to nite propositional programs, although the original construction of the normalform of a program also holds for the wider class of allowed DATALOG programs. Let us shortly explain this. Such a restriction is not needed for approaches based on models. If we consider the program P (x) P (f (x)) or, equivalently, the in nite propositional program p0 p1 ; p1 p2 ; : : : ; pi pi+1 ; : : : then by looking at all minimal models of these programs, we immediately get :P (t) for all terms t, resp. :pi for all i 2 IN. But no proof-theoretic approach, that is based on ecent program-transformations and associates to any program a normalform in a constructive way (like D-WFS) can \unfold" such an in nite loop and eliminate all these rules. To do this, something like a !-inference rule is needed, which is not a constructive rule. But if the above loop is nite, this can be recognized in a nite number of steps. Indeed, D-WFS will do this. Therefore a completeness result of a constructive procedure can only hold for the propositional case { or we had to modify the construction of the normalform res() and thus departing from constructive methods. Our paper is structured as follows. We rst introduce some notation and terminology in Section 2. In Section 3, we introduce the semantics D-WFS as the weakest semantics satisfying certain abstract conditions. In Section 4 we prove the very strong result that the set of transformations is con uent. Since the proof is rather complicated and uses previous results we only give a detailed sketch. Section 5 contains our
equivalence result. We introduce semantics SEM and SEM0 which do not involve any particular technical machinery. They are based on minimal consequences of positive programs and use a generalized Gelfond-Lifschitz transformation. The construction is very direct and easy, but gives no insights into the properties and general behaviour of these semantics. We show that D-WFS is sound w.r.t. both SEM and SEM0 and also complete w.r.t. SEM0 . Therefore, D-WFS coincides with SEM0 . Finally, we conclude with Section 6.
2 Preliminaries
In this section we present the language L and introduce all necessary formal notions that we need in the sequel. As already stated in the introduction, we consider nite, instantiated programs over some xed nite language L containing the binary connectives _; ^; !, the unary connective : and the falsum ?. We write AtL for the set of all atoms di erent from ?.
De nition 1 (Program , Possibly True Facts P ()) A logic program is a nite set of rules of the form
A1 _ _ Ak
B1 ^ ^ Bm ^ :C1 ^ ^ :Cn ;
where the A =B =C are L-atoms di erent from ?, k 1, m 0, n 0. We allow empty conjunctions: they are considered as abbreviations for :? (the verum). We identify such a rule with the triple consisting of the following sets of atoms A := fA1 ; : : : ; Ak g, B := fB1 ; : : : ; Bm g, C := fC1 ; : : : ; Cn g, and write it as A B ^ :C . This means, in particular, that we assume the Ai (resp. the Bi , resp. the Ci ) to be pairwise disjoint. We write P () for the set of all atoms occuring in rule heads in : these are atoms that are possibly true. By pure disjunctions we mean those consisting solely of positive or of negative literals.
3 Disjunctive Well-Founded Semantics: D-WFS A logic program or a deductive database is used by posing and answering queries about its contents. While there is no need to consider conjunctive queries (since 1 ^ 2 is answered with \yes" i both subqueries are answered \yes"), we need disjunctive queries , since the result of 1 _ 2 cannot be derived from the results of the single queries. Because many semantics for disjunctive logic programs do not derive mixed disjunctions (i.e. containing negative and positive literals at the same time) we consider the following de nition:
De nition 2 (Operator j and Semantics Sj ) By a semantic operator j we mean a binary relation between
logic programs and pure disjunctions which satis es the following three arguably obvious conditions:
1. Right Weakening: If j and 0 3, then j 0 . 2. Nec. True: If A true 2 for a disjunction A, then j A. 3 Nec. False: If A 62 P () for an atom A, then j :A. 3 i. e.
is a subdisjunction of 0 .
Given such an operator j and a logic program , by the semantics Sj () of determined by j we mean the set of all pure disjunctions derivable by j from , i.e., Sj () := f j j g.
Note that both model-based as well as completion-based approaches t well into this framework, because these approaches provide in a natural way a set of derivable disjunctions: we simply take the sceptical view (truth in all intended models or in all models of the completion). In addition to satisfying the general conditions (1)-(3) listed above, we may want a speci c semantic operator j to be invariant under certain \natural" program transformations.
De nition 3 (Invariance of j under a Transf.)
Suppose that a program transformation Trans : 7! Trans() mapping logic programs into logic programs is given. We say that the operator j is invariant under Trans (or that Trans is a j-equivalence transformation) i j () Trans() j for any pure disjunction and any program .
We now describe several such \natural" transformations Trans which will be later used to de ne the D-WFS semantics. By abuse of language (and to facilitate reading) we will simply say \j (or Sj ) satis es Trans \ meaning that \j is invariant under Trans ". We will illustrate all the transformations on the following running example Example 4 (Running Example 0 ) 0 : A _ B C; :C; :D A_C B C_D :E B :C; :D; :E We begin with partial evaluation in the sense of the \unfolding" operation. It is the \Generalized Principle of Partial Evaluation (GPPE)" ([6, 18]):
De nition 5 (GPPE) Semantics Sj satis es GPPE i it is invariant under the following transformation: Replace a rule A B ^ :C where B contains a distinguished atom B by the rules ? ? ? A [ A n fB g B n fB g [ B ^ : C [ C (i = 1; : : : ; n) where A B ^:C (i = 1; : : : ; n) are all rules with B 2 A . Note that we are free to select a speci c positive occurrence of an atom B and then perform the transformation. The new rules are obtained by replacing B by the bodies of all rules r with head literal B and adding the remaining head atoms of r to the head of the new rule. In our example, we have two possibilities to apply GPPE: one to replace C in the rst clause and one to replace B in the second clause. We choose the rst possibility and get the following program 1 : A _ B B; :C; :D A_B _D :C; :D; :E A_C B C_D :E B :C; :D; :E
The next transformation states that tautological clauses like p p do not in uence the semantics of a logic program. This and the following transformation together correspond to the \Equivalence" principle of [8]:
De nition 6 (Elim. of Tautologies, Non-Min. Rules) Semantics Sj allows a) the Elimination of Tautologies, resp. b) the Elimination of Non-Minimal Rules i j is invariant
under the following transformations: a) Delete a rule A B ^ :C with A \ B 6= ;. b) Delete a rule A B ^ :C if there is another rule A0 B0 ^ :C 0 with A0 A, B0 B, and C 0 C . We can apply the rst transformation to get rid of the rst clause (obtaining 2 ) and the second to get rid of the second clause (because it is subsumed by the last one) and get 3 . We can now apply GPPE again to eliminate B and get 4 : 4 : 3 :
A_C :C; :D; :E A_C B C_D :E C _D :E :C; :D; :E B :C; :D; :E B The last two transformatons allow us to do some simple reductions. We want :A to be derivable if A appears in no rule head. Therefore, it should be possible to evaluate the body literal :A to true, i.e. to delete :A from all rule bodies: this is guaranteed by Positive Reduction . Conversely, if the logic program contains A1 _ _ Ak true , at least one of these atoms must be true, so a rule body containing :A1 ^ ^:Ak is surely false, so the entire rule is useless, and it should be possible to delete it: this gives us Negative Reduction .
De nition 7 (Positive and Negative Reduction)
Semantics Sj allows a) Positive, resp. b) Negative Reduction i j is invariant under the following transformations: ? a) Replace a rule A B ^ :C by A B0 ^ : C \ P () . b) Delete a0 rule A B ^ :C if there is A true with A C . In our example, we can apply positive Reduction to obtain 5 and then negative Reduction to obtain 6 5 : A _ C :C; :D 6 : C _ D
C _D B :C; :D We call a semantics j1 weaker than a semantics j2 i for all and \ j1 implies j2 . As shown in [6] there exists the weakest semantics which is invariant under all the natural transformations discussed in this Section.
Theorem 8 (D-WFS, [6])
There exists the weakest semantics Sj which is invariant under all the natural transformations introduced in this section. In other words, there exists the weakest semantics which satis es the properties of GPPE, Elimination of Tautologies and Non-Minimal Rules, Positive and Negative Reduction. Moreover, this semantics is consistent and closed under logical consequences (as a set consisting of pure disjunctions). We call it the Disjunctive Well-Founded Semantics, or, brie y, D-WFS. In our running example, we end up with fC _ Dg =: 6 Redpos Redneg GPPEB Non-Min Taut GPPEC (0 ):
4 The Con uence of our Transformations
Looking again at 0 , it is not clear that a di erent application of our transformations (in a di erent ordering) leads to the same result. Our main theorem shows that it is indeed so. The proof is not easy. We will do it with the help of a particular normalform res() of a program . The de nition of this normalform is somewhat complicated and we refer the reader to [6]. To make the paper selfcontained, we give the de nition in the proof of our main theorem. In fact, since we prove our calculus to be con uent, this construction is obsolete: It only has to be used in the proof of this theorem. Here is the main theorem of [6, 4]: Theorem 9 (Sound., Compl. of D-WFS wrt 7! res())
a) A semantics Sj is invariant under all transformations if and only if j is invariant under the transformation 7! res(). b) jD?WFS ()there is A with A true 2 res() or there is :A 2 and A 62 P (res()): The second part of the last theorem is especially important. It tells us that once the residual program res() has been produced, the semantics D-WFS can be immediately determined. For our running example, we get that 0 is equivalent to 6 and obviously 6 = res(6 ) = fC _ Dg and therefore D-WFS(0 ) = f:A; :B; :E; C _ Dg. Note that we can not conclude that res(0 ) = 6 because we performed our reductions in a di erent order than in the construction of the residual program. Even the residual program does not correspond to the application of our transformations in a particular ordering. Nevertheless the identity res(0 ) = 6 holds. We now formulate our main theorem. The program 6 has the property that it can not be further reduced because none of our transformations is applicable. Let us denote by ! 0 if the program can be transformed into 0 by applying our transformations in some order (we do not require any speci c ordering). We call irreducible, if there is no 0 6= with ! 0 . The following theorem is a strong result
Theorem 10 (Con uence of our Transformations) Our set of transformations is con uent, i.e. if ! 0 and ! 00 and both 0 ; 00 are irreducible, then 0 = 00 . In addition, for any program there is an irreducible 0 with ! 0 . Obviously, such a 0 is exactly the residual program 0 0 0
res() of because res( ) = since is irreducible.
Before sketching a proof of this theorem let us show that a simple and direct proof is not so simple. Consider the following program and its partial evaluation GPPEA ().
GPPEA() : A A; :B A_C :B; :E A_C :E The example shows that if we would rst apply the Elimination of Tautologies to and then GPPE we would get a di erent result than rst applying GPPE and then Taut: GPPEA(Taut()) = fA _ C :E g which is di erent from fA _ C :B; :E g [ fA _ C :E g = Taut(GPPEA()). A A_C
:
A; :B :E
In this particular example, a further application of positive Reduction will do the job: we obtain fA _ C g Redpos GPPEA Taut() = Redpos Taut GPPEA(): In more complicated cases, just applying one particular transformation does not suce. A general set of identities of the form T1 GPPE Taut T2 = Taut GPPE seems to be also very complicated. Proof. (Detailed Sketch) First of all, we have to de ne ? the residual program res() of a program : res() := R! T! (;) . It remains to de ne the two operators R and T . Let us call a conditional fact a rule without positive body literals, i.e. it is of the form A1 _ _ Ak not C1 ^ ^ not Cm , where k 1 and m 0. For a set ? of conditional facts we denote by T (?) the following set n
A0 [
m? [
m [
A n fB g : C0 [ C there are =1 =1 a ground instance A0 B1 ^ ^ Bm ^ :C0 of a rule in o and cond. facts A :C 2 ? with B 2 A ( = 1; : : : ; m) :
We compute the smallest xpoint of T as usual: We start with ?0 := ; and then iterate ? := T (??1 ) until nothing changes. This must happen because our program is nite and propositional. We now de ne the R-operator and denote by R(?) ? fA : C \ P (?) Aj :C 2 ?; and (1) there is no A0 true 2 ? with A0 C , (2) there is no A0 :C 0 2 ? with A0 A, C 0 C where at least one is properg: To prove our theorem, it suces to show that ! 0 implies res() = res(0 ); because it is obvious that an irreducible program irr is already in normalform, i.e. res(irr ) = irr . It also suces to show this result only for our transformations 7! 0 , because then it also holds for any sequence of them. We denote by GPPE(), resp. Taut(), Non-Min(), Redpos=neg () the program obtained from by applying one of the respective transformations. Obviously, the R-operator exactly performs the other transformations. Now the proof of Theorem 12 in [4] already indicates that lfp(T ) is obtained from \in a sense" by applying GPPE and Elimination of Tautologies | but it does not exactly correspond to it. Nevertheless a careful inspection of the proofs of Theorem 12, Lemma 23 and 24 in [4] gives us res() = res(Trans()) for all our transformations, and we are done. We note that simple counterexamples show that the calculus does not remain con uent, if any of the transformations is cancelled.
5 An Equivalent Characterization of D-WFS
We rst present the general idea which leads, in its simplest form, to a semantics SEM . This semantics, however, is too
strong as shown by Example 13 (nevertheless, it satis es all our transformations). This example leads to the modi ed version SEM0 that turns out to be identical to D-WFS.
De nition 11 (=Dis)
Let be a disjunctive logic program over L and let Dis be a set of pure disjunctions over L. Let =Dis be the program obtained from by doing the following reductions for all :C and C1 _ : : : _ Ck
if :C 2 Dis, then remove all ocurrences of :C , if C1 _ : : : _ Ck 2 Dis then remove all rules that contain f:C1 ; : : : ; :Ck g in their bodies. =Dis is obviously a slight generalization of the Gelfond-Lifschitz transformation. While the latter is de ned relative to a set N AtL in such a way that =N is always positive, our =Dis still is a disjunctive program containing possibly negative literals. In fact, the GL-transform can be obtained from our transform by setting =N = =DisN where DisN := N [ f:X : X 2 AtL n N g. The underlying idea of following de nition of a semantics SEM is to use =AtL (i.e. we just delete all rules containing negative literals) for deriving positive disjunctions and to use =:AtL (i.e. we set all negative literals to true) for deriving negative literals:
De nition 12 (SEM )
Let be a disjunctive logic program over L. We de ne a set Dis () of pure disjunctions as follows fA1 _ : : : _ Ak : =AtL j= A1 _ : : : _ A k g [ f:A1 : =:AtL j=min :A1 g The semantics SEM of a program is de ned as the limit of the following growing sets Di of disjunctions (this series eventually gets constant after some nite number of steps). We start with D0 := Dis () and set4
Dn +1 := Dn [ Dis (=Dn ):
Note that although this is a very handy and easy de nition, it does not give us any insights of SEM . In addition, SEM is already very strong and not even closed under logical consequence.
Example 13 (Behaviour of SEM ) : A _ B :C : A _ B B :B B :B C :C We have SEM ( ) = f:Ag and SEM ( ) = f:A; A _ B g. In the rst case, the derivation of :A seems to be very strong (B; C are unde ned). In the second case, we not only have :A but also A _ B without having B . Of course, we could modify
the de nition so that it is closed under logical consequence, but then the derivation of B would be very unituitive. 4 Ilkka Niemela noted that simply setting D := Dis (=Dn ) n+1 does not produce a monotonically growing series Dn . For the program consisting of A _ B :A, B :B; :C and C , we get D0 = f:A; C g, D1 = fA _ B; C g.
The reason for the shortcoming of SEM is the derivation of negative literals :A1 . We therefore modify SEM by weakening this condition.
De nition 14 (SEM0 )
Let be a disjunctive logic program over L. We de ne a set Dis() of pure disjunctions as follows fA1 _ : : : _ Ak : =AtL j= A1 _ : : : _ A k g [ f:A1 : =DisN j=min :A1 for all N AtL s. t.5N j= =AtL g Here, the series Di de ned by Dn+1 := Dis(=Dn ) grows monotonically and eventually gets constant. We de ne the set SEM0 () to be the limit of this series. So instead of deleting all negative literals at a blow (=:AtL ) and looking at all minimal models of =:AtL , we compute =N for any N AtL that is consistent (N j= =AtL and we only derive those :A that are true in all minimal models of such =N . In the Example 13, this prevents us from deriving :A. Therefore SEM0 ( ) = ; and SEM0 ( ) = fA _ B g. The following lemma is the key to establish the soundness of D-WFS with respect to SEM and its completeness with respect to SEM0 . We recall from the last section that ! 0 means that can be transformed into 0 by applying any ( nite) sequence of the transformations introduced in Section 3. Lemma 15 Let ! 0 . Then the following holds: Dis() = Dis (00 ). =Dis() ! =Dis(0 ). The same properties hold for Dis instead of Dis, Proof. It suces to show these results only for our basic transformations 7! 0 (because then it also holds for any sequence of them). In fact, for if Trans is any of Elimination of Tautologies , Elimination of Non-Minimal Rules or Positive Reduction we have Trans()=DisN = Trans(=DisN ) () for all N AtL. Therefore Dis(Trans()) = Dis(). Moreover it is straightforward to show Trans(=Dis()) = Trans()=Dis(Trans()) from which the result follows. Exactly the same reasoning holds for Dis instead of Dis. The condition () does not hold for Negative Reduction Red?. We refer to our example given in the footnote of De nition 14: \A _ B; C :A; :B ". For N0 := ; the left-hand side of () consists of \A _ B " while the right-hand side is \A _ B; C ". But we have Red? ()=AtL = Red? (=AtL) and also Dis(Red?()) = Dis(): For this last result we use the fact that only those N with N j= =AtL are considered. It also holds Red?(=Dis()) = Red?()=Dis(Red? ()) from which the desired result follows. The proof for Dis instead of Dis is literally the same. 5 We need to consider only those N AtL that are consistent with =AtL . For \A _ B; C :A; :B" we do not want to consider N0 := ; because then :C would not be derivable.
Corollary 16 (SEM0 , SEM are sound for 7! res()) ! 0 implies SEM ()=SEM (0 ) and SEM0 ()=SEM0 (0 ). Theorem 17 (D-WFS = SEM0 ) D-WFS is complete wrt SEM0 : D-WFS() = SEM0 () for all .
Proof. Let res() be the residual program from . The last corollary tells us that SEM0 () = SEM0 (res()). It suces to show that A1 _ : : : _ Ak 2 Dis(res())()9 head true 2 res(), with head A1 _ : : : _ Ak , :A 2 Dis(res()) ()A 62 P (res()); because then, res()=D0 = res()=Dis(res()) = res() and therefore the sequence D0 ; D1 ; : : : immediately ends with D0 . The result follows by Theorem 9 b). We show the rst equivalence. Let A1 _ : : : _ Ak follow from res()=AtL. Since res()=AtL consists of only positive disjunctions, there is a subset of fA1 ; : : : ; Ak g occurring as a disjunction in res()=AtL. Because of the de nition of res()=AtL, we are done. The opposite direction is trivial. Let now :A 2 Dis(res()) and suppose A appears in some rule A _ B1 _ : : : _ Bm :C1 ; : : : ; :Cn : We have to derive a contradiction. Let N := AtLnfC1 ; : : : ; Cn g. Obviuosly, N j= res()=AtL, because otherwise there would be a rule A where A is a subset of fC1 ; : : : ; Cn g and res() would not be the residual program (because negative Reduction could be applied). Then res()=DisN consists of exactly those rule heads of rules res() whose bodies are subsets of f:C1 ; : : : ; :Cn g. By Elimination of Non-minimal Rules such a head cannot be a strict subset of fA; B1 ; : : : ; Bm g. Therefore the interpretation J which makes exactly the Bi false and all other atoms true is a model of res()=DisN . So there is also a minimal model of res()=DisN below J which is still smaller. But in this minimal model A is true (because of the disjunction A _ B1 _ : : : _ Bm ). Therefore res()=DisN does not minimally imply :A and we have a contradiction. The opposite direction again is trivial.
6 Conclusions
In this paper we obtained two main results. First the proof of the con uence of a very simple calculus of program transformations. Second an equivalent formulation in terms of minimal model reasoning. Although the very de nition of the semantics D-WFS and the program-transformations have been introduced in [6] the results of this paper are completely new. The con uence of our calculus is not just a simple corollary to the results of [6]. To our knowledge, this is the rst time that a con uent calculus has been de ned as a proof-theoretical attempt for de ning a semantics for logic programs. The con uence of such a calculus is certainly the strongest property one can have. It is worth mentioning that an implementation using the con uence of the calculus can be much more ecient than simply taking a certain ordering of the transformations which is xed in advance. We are currently experimenting and comparing a
xed bottom-up implementation with one based on the con uence of our calculus. The equivalence of our semantics with a construction based on minimal model reasoning is also very interesting, because it opens the way to apply methods from this area for ef cient implementations. In particular, I. Niemela suggested very ecient methods for computing minimal models of positive disjunctive programs. Because of our characterization, such methods can be immediately used to implement D-WFS (note that we only need to consider positive disjunctive programs in De nition 14). Of course, there are still many open questions left for further research. An important property of our approach is its applicability also to semantics stronger than D-WFS. If only our transformation 7! res() is sound for such a semantics (e.g. STATIONARY, STATIC or STABLE) we can use our calculus and already decide many queries without fully computing the semantics. Therefore, putting something on top of res() should make it possible to obtain these stronger semantics { we already did this to obtain and implement STABLE in [4]. Similar ideas to obtain STATIC are currently under investigation. Let us conclude with an overview of the properties of some well-known semantics (see also [5] for interesting characterizations of STABLE):
Theorem 18 ([6])
Properties of Logic-Programming Semantics Semantics Taut. GPPE Red. Nonmin. Clark's comp [7] | WFS [19] GCWA [11] WGCWA [16] | | Positivism [2] | STABLE [10, 13] Strong WFS [17] | | | STATIONARY [14] STATIC [15] D-WFS [6] REG-SEM [20]
ACKNOWLEDGEMENTS
We are grateful to some anonymous referees as well as to Teodor Przymusinski for their useful comments. We are also indebted to Ilkka Niemela for pointing out two weaknesses in former versions of De nitions 14 and 12.
REFERENCES
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[4] Stefan Brass and Jurgen Dix, `A General Approach to Bottom-Up Computation of Disjunctive Semantics', in Nonmonotonic Extensions of Logic Programming, eds., J. Dix, L. Pereira, and T. Przymusinski, LNAI 927, 127{155, Springer, Berlin, (1995). [5] Stefan Brass and Jurgen Dix, `Characterizations of the Stable Semantics by Partial Evaluation', in Logic Programming and Non-Monotonic Reasoning, Proceedings of the Third International Conference, eds., A. Nerode, W. Marek, and M. Truszczynski, LNCS 928, pp. 85{98, Berlin, (June 1995). Springer. [6] Stefan Brass and Jurgen Dix, `Disjunctive Semantics based upon Partial and Bottom-Up Evaluation', in Proceedings of the 12th Int. Conf. on Logic Programming, Tokyo, ed., Leon Sterling, pp. 199{213. MIT Press, (June 1995). [7] K. L. Clark, `Negation as Failure', in Logic and Data-Bases, eds., H. Gallaire and J. Minker, 293{322, Plenum, New York, (1978). [8] Jurgen Dix, `A Classi cation-Theory of Semantics of Normal Logic Programs: II. Weak Properties', Fundamenta Informaticae, XXII(3), 257{288, (1995). [9] Jurgen Dix, `Semantics of Logic Programs: Their Intuitions and Formal Properties. An Overview.', in Logic, Action and Information { Essays on Logic in Philosophy and Arti cial Intelligence, eds., Andre Fuhrmann and Hans Rott, pp. 241{ 327. DeGruyter, (1995). [10] Michael Gelfond and Vladimir Lifschitz, `Classical Negation in Logic Programs and Disjunctive Databases', New Generation Computing, 9, 365{387, (1991). [11] Jack Minker, `On inde nite databases and the closed world assumption', in Proceedings of the 6th Conference on Automated Deduction, New York, pp. 292{308, Berlin, (1982). Springer. [12] Jack Minker, `An Overview of Nonmonotonic Reasoning and Logic Programming', Journal of Logic Programming, Special Issue, 17, (1993). [13] Teodor Przymusinski, `Stable Semantics for Disjunctive Programs', New Generation Computing Journal, 9, 401{424, (1991). [14] Teodor Przymusinski, `Stationary Semantics for Normal and Disjunctive Logic Programs', in DOOD '91, Proceedings of the 2nd International Conference, eds., C. Delobel, M. Kifer, and Y. Masunaga, Berlin, (December 1991). Muenchen, Springer. LNCS 566. [15] Teodor Przymusinski, `Static Semantics For Normal and Disjunctive Logic Programs', Annals of Mathematics and Arti cial Intelligence, Special Issue on Disjunctive Programs, (1995). to appear. [16] Arcot Rajasekar, Jorge Lobo, and Jack Minker, `Weak Generalized Closed World Assumption', Journal of Automated Reasoning, 5, 293{307, (1989). [17] Kenneth A. Ross, `The well-founded semantics for disjunctive logic programs', in Proceedings of the rst International Conference on Deductive and Object Oriented Databases, Kyoto, Japan, pp. 1{22, (1989). [18] Chiaki Sakama and Hirohisa Seki, `Partial Deduction of Disjunctive Logic Programs: A Declarative Approach', in Logic Program Synthesis and Transformation { Meta Programming in Logic, LNCS 883, pp. 170{182, Berlin, (1995). Springer. [19] Allen van Gelder, Kenneth A. Ross, and John S. Schlipf, `The well-founded semantics for general logic programs', Journal of the ACM, 38, 620{650, (1991). [20] Jia-Huai You and Li-Yan Yuan, `A three-valued semantics for deductive databases and logic programs', Journal of Computer and System Sciences, 49(2), 334{361, (1994).
Abstract. Quite recently Brass/Dix have introduced the se-
mantics D-WFS for general disjunctive logic programs. The interesting feature of this approach is that it is both semantically and proof-theoretically founded. Any program is associated a normalform res(), called the residual program. We show in this paper, that the original calculus, consisting of some simple transformations, has a very strong and appealing property: it is con uent . This means that all the transformations can be applied in any order: if we arrive at an irreducible program (no more transformation is applicable) then this is already the unique normalform. No proper subset of the calculus has this property. We also give an equivalent characterization of D-WFS in terms of iterated minimal model reasoning. This construction is a generalization of a description of the wellfounded semantics: we introduce a very simple and neat construction of a sequence Di that eventually stops and represents the set of derivable disjunctions. Both characterizations open the way for ecient implementations. The rst because the ordering of the transformations does not matter, the second because special methods from Circumscription might be useful.
1 Introduction
The success of PROLOG as one of the major programming languages for Arti cial Intelligence suggests that extensions of Horn-clauses to general clauses with negation and disjunction might become very useful in Knowledge Representation and Reasoning . Indeed, in recent years much work was going on in Logic Programming to extend semantics based on di erent forms of negation-as-failure from non-disjunctive to disjunctive semantics. Many interrelationships of non-monotonic logics (DL, CIRC, AEL) with semantics of logic programs have been discovered during the last years ([1, 9, 12]). Eiter and Gottlob have shown that disjunctive semantics are strictly more expressive than non-disjunctive semantics and therefore may be better suited for many knowledge representation tasks. One of the main problems today is that although we have a quite good understanding of non-disjunctive semantics, our knowledge of disjunctive semantics is quite limited. Besides the main competing approaches of the wellfounded semantics WFS (which is often too weak) and the stable semantics
1 University of Hannover, Inst. f. Informatik, Lange Laube 22, D30159 Hannover, Germany,[email protected] 2 University of Koblenz, Dept. of Computer Science, Rheinau 1,
D-56075 Koblenz, Germany
and
Jurgen Dix2
STABLE (which is sometimes too strong) there exist many extensions of WFS that approximate STABLE. Unfortunately, only STABLE has a straightforward extension to disjunctive programs: not even for WFS does there exist a canonical disjunctive version. We claim that D-WFS is this counterpart of WFS. The novelty of our approach is that it is not exclusively declarative (like Przymusinski's static or stationary approach) nor exclusively procedural (like the approaches of Minker and his group). We introduce a calculus of program transformations. These are declarative since they express precise semantical properties (e.g. partial evaluation ). But they are also procedural because they can be applied to a program and transform it to another one. Our main results are The calculus is con uent . D-WFS can be equivalently described using iterated minimal model reasoning, which relates it to circumscription. We restrict in this paper for simplicity to nite propositional programs, although the original construction of the normalform of a program also holds for the wider class of allowed DATALOG programs. Let us shortly explain this. Such a restriction is not needed for approaches based on models. If we consider the program P (x) P (f (x)) or, equivalently, the in nite propositional program p0 p1 ; p1 p2 ; : : : ; pi pi+1 ; : : : then by looking at all minimal models of these programs, we immediately get :P (t) for all terms t, resp. :pi for all i 2 IN. But no proof-theoretic approach, that is based on ecent program-transformations and associates to any program a normalform in a constructive way (like D-WFS) can \unfold" such an in nite loop and eliminate all these rules. To do this, something like a !-inference rule is needed, which is not a constructive rule. But if the above loop is nite, this can be recognized in a nite number of steps. Indeed, D-WFS will do this. Therefore a completeness result of a constructive procedure can only hold for the propositional case { or we had to modify the construction of the normalform res() and thus departing from constructive methods. Our paper is structured as follows. We rst introduce some notation and terminology in Section 2. In Section 3, we introduce the semantics D-WFS as the weakest semantics satisfying certain abstract conditions. In Section 4 we prove the very strong result that the set of transformations is con uent. Since the proof is rather complicated and uses previous results we only give a detailed sketch. Section 5 contains our
equivalence result. We introduce semantics SEM and SEM0 which do not involve any particular technical machinery. They are based on minimal consequences of positive programs and use a generalized Gelfond-Lifschitz transformation. The construction is very direct and easy, but gives no insights into the properties and general behaviour of these semantics. We show that D-WFS is sound w.r.t. both SEM and SEM0 and also complete w.r.t. SEM0 . Therefore, D-WFS coincides with SEM0 . Finally, we conclude with Section 6.
2 Preliminaries
In this section we present the language L and introduce all necessary formal notions that we need in the sequel. As already stated in the introduction, we consider nite, instantiated programs over some xed nite language L containing the binary connectives _; ^; !, the unary connective : and the falsum ?. We write AtL for the set of all atoms di erent from ?.
De nition 1 (Program , Possibly True Facts P ()) A logic program is a nite set of rules of the form
A1 _ _ Ak
B1 ^ ^ Bm ^ :C1 ^ ^ :Cn ;
where the A =B =C are L-atoms di erent from ?, k 1, m 0, n 0. We allow empty conjunctions: they are considered as abbreviations for :? (the verum). We identify such a rule with the triple consisting of the following sets of atoms A := fA1 ; : : : ; Ak g, B := fB1 ; : : : ; Bm g, C := fC1 ; : : : ; Cn g, and write it as A B ^ :C . This means, in particular, that we assume the Ai (resp. the Bi , resp. the Ci ) to be pairwise disjoint. We write P () for the set of all atoms occuring in rule heads in : these are atoms that are possibly true. By pure disjunctions we mean those consisting solely of positive or of negative literals.
3 Disjunctive Well-Founded Semantics: D-WFS A logic program or a deductive database is used by posing and answering queries about its contents. While there is no need to consider conjunctive queries (since 1 ^ 2 is answered with \yes" i both subqueries are answered \yes"), we need disjunctive queries , since the result of 1 _ 2 cannot be derived from the results of the single queries. Because many semantics for disjunctive logic programs do not derive mixed disjunctions (i.e. containing negative and positive literals at the same time) we consider the following de nition:
De nition 2 (Operator j and Semantics Sj ) By a semantic operator j we mean a binary relation between
logic programs and pure disjunctions which satis es the following three arguably obvious conditions:
1. Right Weakening: If j and 0 3, then j 0 . 2. Nec. True: If A true 2 for a disjunction A, then j A. 3 Nec. False: If A 62 P () for an atom A, then j :A. 3 i. e.
is a subdisjunction of 0 .
Given such an operator j and a logic program , by the semantics Sj () of determined by j we mean the set of all pure disjunctions derivable by j from , i.e., Sj () := f j j g.
Note that both model-based as well as completion-based approaches t well into this framework, because these approaches provide in a natural way a set of derivable disjunctions: we simply take the sceptical view (truth in all intended models or in all models of the completion). In addition to satisfying the general conditions (1)-(3) listed above, we may want a speci c semantic operator j to be invariant under certain \natural" program transformations.
De nition 3 (Invariance of j under a Transf.)
Suppose that a program transformation Trans : 7! Trans() mapping logic programs into logic programs is given. We say that the operator j is invariant under Trans (or that Trans is a j-equivalence transformation) i j () Trans() j for any pure disjunction and any program .
We now describe several such \natural" transformations Trans which will be later used to de ne the D-WFS semantics. By abuse of language (and to facilitate reading) we will simply say \j (or Sj ) satis es Trans \ meaning that \j is invariant under Trans ". We will illustrate all the transformations on the following running example Example 4 (Running Example 0 ) 0 : A _ B C; :C; :D A_C B C_D :E B :C; :D; :E We begin with partial evaluation in the sense of the \unfolding" operation. It is the \Generalized Principle of Partial Evaluation (GPPE)" ([6, 18]):
De nition 5 (GPPE) Semantics Sj satis es GPPE i it is invariant under the following transformation: Replace a rule A B ^ :C where B contains a distinguished atom B by the rules ? ? ? A [ A n fB g B n fB g [ B ^ : C [ C (i = 1; : : : ; n) where A B ^:C (i = 1; : : : ; n) are all rules with B 2 A . Note that we are free to select a speci c positive occurrence of an atom B and then perform the transformation. The new rules are obtained by replacing B by the bodies of all rules r with head literal B and adding the remaining head atoms of r to the head of the new rule. In our example, we have two possibilities to apply GPPE: one to replace C in the rst clause and one to replace B in the second clause. We choose the rst possibility and get the following program 1 : A _ B B; :C; :D A_B _D :C; :D; :E A_C B C_D :E B :C; :D; :E
The next transformation states that tautological clauses like p p do not in uence the semantics of a logic program. This and the following transformation together correspond to the \Equivalence" principle of [8]:
De nition 6 (Elim. of Tautologies, Non-Min. Rules) Semantics Sj allows a) the Elimination of Tautologies, resp. b) the Elimination of Non-Minimal Rules i j is invariant
under the following transformations: a) Delete a rule A B ^ :C with A \ B 6= ;. b) Delete a rule A B ^ :C if there is another rule A0 B0 ^ :C 0 with A0 A, B0 B, and C 0 C . We can apply the rst transformation to get rid of the rst clause (obtaining 2 ) and the second to get rid of the second clause (because it is subsumed by the last one) and get 3 . We can now apply GPPE again to eliminate B and get 4 : 4 : 3 :
A_C :C; :D; :E A_C B C_D :E C _D :E :C; :D; :E B :C; :D; :E B The last two transformatons allow us to do some simple reductions. We want :A to be derivable if A appears in no rule head. Therefore, it should be possible to evaluate the body literal :A to true, i.e. to delete :A from all rule bodies: this is guaranteed by Positive Reduction . Conversely, if the logic program contains A1 _ _ Ak true , at least one of these atoms must be true, so a rule body containing :A1 ^ ^:Ak is surely false, so the entire rule is useless, and it should be possible to delete it: this gives us Negative Reduction .
De nition 7 (Positive and Negative Reduction)
Semantics Sj allows a) Positive, resp. b) Negative Reduction i j is invariant under the following transformations: ? a) Replace a rule A B ^ :C by A B0 ^ : C \ P () . b) Delete a0 rule A B ^ :C if there is A true with A C . In our example, we can apply positive Reduction to obtain 5 and then negative Reduction to obtain 6 5 : A _ C :C; :D 6 : C _ D
C _D B :C; :D We call a semantics j1 weaker than a semantics j2 i for all and \ j1 implies j2 . As shown in [6] there exists the weakest semantics which is invariant under all the natural transformations discussed in this Section.
Theorem 8 (D-WFS, [6])
There exists the weakest semantics Sj which is invariant under all the natural transformations introduced in this section. In other words, there exists the weakest semantics which satis es the properties of GPPE, Elimination of Tautologies and Non-Minimal Rules, Positive and Negative Reduction. Moreover, this semantics is consistent and closed under logical consequences (as a set consisting of pure disjunctions). We call it the Disjunctive Well-Founded Semantics, or, brie y, D-WFS. In our running example, we end up with fC _ Dg =: 6 Redpos Redneg GPPEB Non-Min Taut GPPEC (0 ):
4 The Con uence of our Transformations
Looking again at 0 , it is not clear that a di erent application of our transformations (in a di erent ordering) leads to the same result. Our main theorem shows that it is indeed so. The proof is not easy. We will do it with the help of a particular normalform res() of a program . The de nition of this normalform is somewhat complicated and we refer the reader to [6]. To make the paper selfcontained, we give the de nition in the proof of our main theorem. In fact, since we prove our calculus to be con uent, this construction is obsolete: It only has to be used in the proof of this theorem. Here is the main theorem of [6, 4]: Theorem 9 (Sound., Compl. of D-WFS wrt 7! res())
a) A semantics Sj is invariant under all transformations if and only if j is invariant under the transformation 7! res(). b) jD?WFS ()there is A with A true 2 res() or there is :A 2 and A 62 P (res()): The second part of the last theorem is especially important. It tells us that once the residual program res() has been produced, the semantics D-WFS can be immediately determined. For our running example, we get that 0 is equivalent to 6 and obviously 6 = res(6 ) = fC _ Dg and therefore D-WFS(0 ) = f:A; :B; :E; C _ Dg. Note that we can not conclude that res(0 ) = 6 because we performed our reductions in a di erent order than in the construction of the residual program. Even the residual program does not correspond to the application of our transformations in a particular ordering. Nevertheless the identity res(0 ) = 6 holds. We now formulate our main theorem. The program 6 has the property that it can not be further reduced because none of our transformations is applicable. Let us denote by ! 0 if the program can be transformed into 0 by applying our transformations in some order (we do not require any speci c ordering). We call irreducible, if there is no 0 6= with ! 0 . The following theorem is a strong result
Theorem 10 (Con uence of our Transformations) Our set of transformations is con uent, i.e. if ! 0 and ! 00 and both 0 ; 00 are irreducible, then 0 = 00 . In addition, for any program there is an irreducible 0 with ! 0 . Obviously, such a 0 is exactly the residual program 0 0 0
res() of because res( ) = since is irreducible.
Before sketching a proof of this theorem let us show that a simple and direct proof is not so simple. Consider the following program and its partial evaluation GPPEA ().
GPPEA() : A A; :B A_C :B; :E A_C :E The example shows that if we would rst apply the Elimination of Tautologies to and then GPPE we would get a di erent result than rst applying GPPE and then Taut: GPPEA(Taut()) = fA _ C :E g which is di erent from fA _ C :B; :E g [ fA _ C :E g = Taut(GPPEA()). A A_C
:
A; :B :E
In this particular example, a further application of positive Reduction will do the job: we obtain fA _ C g Redpos GPPEA Taut() = Redpos Taut GPPEA(): In more complicated cases, just applying one particular transformation does not suce. A general set of identities of the form T1 GPPE Taut T2 = Taut GPPE seems to be also very complicated. Proof. (Detailed Sketch) First of all, we have to de ne ? the residual program res() of a program : res() := R! T! (;) . It remains to de ne the two operators R and T . Let us call a conditional fact a rule without positive body literals, i.e. it is of the form A1 _ _ Ak not C1 ^ ^ not Cm , where k 1 and m 0. For a set ? of conditional facts we denote by T (?) the following set n
A0 [
m? [
m [
A n fB g : C0 [ C there are =1 =1 a ground instance A0 B1 ^ ^ Bm ^ :C0 of a rule in o and cond. facts A :C 2 ? with B 2 A ( = 1; : : : ; m) :
We compute the smallest xpoint of T as usual: We start with ?0 := ; and then iterate ? := T (??1 ) until nothing changes. This must happen because our program is nite and propositional. We now de ne the R-operator and denote by R(?) ? fA : C \ P (?) Aj :C 2 ?; and (1) there is no A0 true 2 ? with A0 C , (2) there is no A0 :C 0 2 ? with A0 A, C 0 C where at least one is properg: To prove our theorem, it suces to show that ! 0 implies res() = res(0 ); because it is obvious that an irreducible program irr is already in normalform, i.e. res(irr ) = irr . It also suces to show this result only for our transformations 7! 0 , because then it also holds for any sequence of them. We denote by GPPE(), resp. Taut(), Non-Min(), Redpos=neg () the program obtained from by applying one of the respective transformations. Obviously, the R-operator exactly performs the other transformations. Now the proof of Theorem 12 in [4] already indicates that lfp(T ) is obtained from \in a sense" by applying GPPE and Elimination of Tautologies | but it does not exactly correspond to it. Nevertheless a careful inspection of the proofs of Theorem 12, Lemma 23 and 24 in [4] gives us res() = res(Trans()) for all our transformations, and we are done. We note that simple counterexamples show that the calculus does not remain con uent, if any of the transformations is cancelled.
5 An Equivalent Characterization of D-WFS
We rst present the general idea which leads, in its simplest form, to a semantics SEM . This semantics, however, is too
strong as shown by Example 13 (nevertheless, it satis es all our transformations). This example leads to the modi ed version SEM0 that turns out to be identical to D-WFS.
De nition 11 (=Dis)
Let be a disjunctive logic program over L and let Dis be a set of pure disjunctions over L. Let =Dis be the program obtained from by doing the following reductions for all :C and C1 _ : : : _ Ck
if :C 2 Dis, then remove all ocurrences of :C , if C1 _ : : : _ Ck 2 Dis then remove all rules that contain f:C1 ; : : : ; :Ck g in their bodies. =Dis is obviously a slight generalization of the Gelfond-Lifschitz transformation. While the latter is de ned relative to a set N AtL in such a way that =N is always positive, our =Dis still is a disjunctive program containing possibly negative literals. In fact, the GL-transform can be obtained from our transform by setting =N = =DisN where DisN := N [ f:X : X 2 AtL n N g. The underlying idea of following de nition of a semantics SEM is to use =AtL (i.e. we just delete all rules containing negative literals) for deriving positive disjunctions and to use =:AtL (i.e. we set all negative literals to true) for deriving negative literals:
De nition 12 (SEM )
Let be a disjunctive logic program over L. We de ne a set Dis () of pure disjunctions as follows fA1 _ : : : _ Ak : =AtL j= A1 _ : : : _ A k g [ f:A1 : =:AtL j=min :A1 g The semantics SEM of a program is de ned as the limit of the following growing sets Di of disjunctions (this series eventually gets constant after some nite number of steps). We start with D0 := Dis () and set4
Dn +1 := Dn [ Dis (=Dn ):
Note that although this is a very handy and easy de nition, it does not give us any insights of SEM . In addition, SEM is already very strong and not even closed under logical consequence.
Example 13 (Behaviour of SEM ) : A _ B :C : A _ B B :B B :B C :C We have SEM ( ) = f:Ag and SEM ( ) = f:A; A _ B g. In the rst case, the derivation of :A seems to be very strong (B; C are unde ned). In the second case, we not only have :A but also A _ B without having B . Of course, we could modify
the de nition so that it is closed under logical consequence, but then the derivation of B would be very unituitive. 4 Ilkka Niemela noted that simply setting D := Dis (=Dn ) n+1 does not produce a monotonically growing series Dn . For the program consisting of A _ B :A, B :B; :C and C , we get D0 = f:A; C g, D1 = fA _ B; C g.
The reason for the shortcoming of SEM is the derivation of negative literals :A1 . We therefore modify SEM by weakening this condition.
De nition 14 (SEM0 )
Let be a disjunctive logic program over L. We de ne a set Dis() of pure disjunctions as follows fA1 _ : : : _ Ak : =AtL j= A1 _ : : : _ A k g [ f:A1 : =DisN j=min :A1 for all N AtL s. t.5N j= =AtL g Here, the series Di de ned by Dn+1 := Dis(=Dn ) grows monotonically and eventually gets constant. We de ne the set SEM0 () to be the limit of this series. So instead of deleting all negative literals at a blow (=:AtL ) and looking at all minimal models of =:AtL , we compute =N for any N AtL that is consistent (N j= =AtL and we only derive those :A that are true in all minimal models of such =N . In the Example 13, this prevents us from deriving :A. Therefore SEM0 ( ) = ; and SEM0 ( ) = fA _ B g. The following lemma is the key to establish the soundness of D-WFS with respect to SEM and its completeness with respect to SEM0 . We recall from the last section that ! 0 means that can be transformed into 0 by applying any ( nite) sequence of the transformations introduced in Section 3. Lemma 15 Let ! 0 . Then the following holds: Dis() = Dis (00 ). =Dis() ! =Dis(0 ). The same properties hold for Dis instead of Dis, Proof. It suces to show these results only for our basic transformations 7! 0 (because then it also holds for any sequence of them). In fact, for if Trans is any of Elimination of Tautologies , Elimination of Non-Minimal Rules or Positive Reduction we have Trans()=DisN = Trans(=DisN ) () for all N AtL. Therefore Dis(Trans()) = Dis(). Moreover it is straightforward to show Trans(=Dis()) = Trans()=Dis(Trans()) from which the result follows. Exactly the same reasoning holds for Dis instead of Dis. The condition () does not hold for Negative Reduction Red?. We refer to our example given in the footnote of De nition 14: \A _ B; C :A; :B ". For N0 := ; the left-hand side of () consists of \A _ B " while the right-hand side is \A _ B; C ". But we have Red? ()=AtL = Red? (=AtL) and also Dis(Red?()) = Dis(): For this last result we use the fact that only those N with N j= =AtL are considered. It also holds Red?(=Dis()) = Red?()=Dis(Red? ()) from which the desired result follows. The proof for Dis instead of Dis is literally the same. 5 We need to consider only those N AtL that are consistent with =AtL . For \A _ B; C :A; :B" we do not want to consider N0 := ; because then :C would not be derivable.
Corollary 16 (SEM0 , SEM are sound for 7! res()) ! 0 implies SEM ()=SEM (0 ) and SEM0 ()=SEM0 (0 ). Theorem 17 (D-WFS = SEM0 ) D-WFS is complete wrt SEM0 : D-WFS() = SEM0 () for all .
Proof. Let res() be the residual program from . The last corollary tells us that SEM0 () = SEM0 (res()). It suces to show that A1 _ : : : _ Ak 2 Dis(res())()9 head true 2 res(), with head A1 _ : : : _ Ak , :A 2 Dis(res()) ()A 62 P (res()); because then, res()=D0 = res()=Dis(res()) = res() and therefore the sequence D0 ; D1 ; : : : immediately ends with D0 . The result follows by Theorem 9 b). We show the rst equivalence. Let A1 _ : : : _ Ak follow from res()=AtL. Since res()=AtL consists of only positive disjunctions, there is a subset of fA1 ; : : : ; Ak g occurring as a disjunction in res()=AtL. Because of the de nition of res()=AtL, we are done. The opposite direction is trivial. Let now :A 2 Dis(res()) and suppose A appears in some rule A _ B1 _ : : : _ Bm :C1 ; : : : ; :Cn : We have to derive a contradiction. Let N := AtLnfC1 ; : : : ; Cn g. Obviuosly, N j= res()=AtL, because otherwise there would be a rule A where A is a subset of fC1 ; : : : ; Cn g and res() would not be the residual program (because negative Reduction could be applied). Then res()=DisN consists of exactly those rule heads of rules res() whose bodies are subsets of f:C1 ; : : : ; :Cn g. By Elimination of Non-minimal Rules such a head cannot be a strict subset of fA; B1 ; : : : ; Bm g. Therefore the interpretation J which makes exactly the Bi false and all other atoms true is a model of res()=DisN . So there is also a minimal model of res()=DisN below J which is still smaller. But in this minimal model A is true (because of the disjunction A _ B1 _ : : : _ Bm ). Therefore res()=DisN does not minimally imply :A and we have a contradiction. The opposite direction again is trivial.
6 Conclusions
In this paper we obtained two main results. First the proof of the con uence of a very simple calculus of program transformations. Second an equivalent formulation in terms of minimal model reasoning. Although the very de nition of the semantics D-WFS and the program-transformations have been introduced in [6] the results of this paper are completely new. The con uence of our calculus is not just a simple corollary to the results of [6]. To our knowledge, this is the rst time that a con uent calculus has been de ned as a proof-theoretical attempt for de ning a semantics for logic programs. The con uence of such a calculus is certainly the strongest property one can have. It is worth mentioning that an implementation using the con uence of the calculus can be much more ecient than simply taking a certain ordering of the transformations which is xed in advance. We are currently experimenting and comparing a
xed bottom-up implementation with one based on the con uence of our calculus. The equivalence of our semantics with a construction based on minimal model reasoning is also very interesting, because it opens the way to apply methods from this area for ef cient implementations. In particular, I. Niemela suggested very ecient methods for computing minimal models of positive disjunctive programs. Because of our characterization, such methods can be immediately used to implement D-WFS (note that we only need to consider positive disjunctive programs in De nition 14). Of course, there are still many open questions left for further research. An important property of our approach is its applicability also to semantics stronger than D-WFS. If only our transformation 7! res() is sound for such a semantics (e.g. STATIONARY, STATIC or STABLE) we can use our calculus and already decide many queries without fully computing the semantics. Therefore, putting something on top of res() should make it possible to obtain these stronger semantics { we already did this to obtain and implement STABLE in [4]. Similar ideas to obtain STATIC are currently under investigation. Let us conclude with an overview of the properties of some well-known semantics (see also [5] for interesting characterizations of STABLE):
Theorem 18 ([6])
Properties of Logic-Programming Semantics Semantics Taut. GPPE Red. Nonmin. Clark's comp [7] | WFS [19] GCWA [11] WGCWA [16] | | Positivism [2] | STABLE [10, 13] Strong WFS [17] | | | STATIONARY [14] STATIC [15] D-WFS [6] REG-SEM [20]
ACKNOWLEDGEMENTS
We are grateful to some anonymous referees as well as to Teodor Przymusinski for their useful comments. We are also indebted to Ilkka Niemela for pointing out two weaknesses in former versions of De nitions 14 and 12.
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