Brass,S.,Dix, J. and Przymusinki,T., Super Logic Programs, Technical report, University of Koblenz-Landau, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....non disjunctive logic programs. We are often required to deal with disjunctive information in our everyday life as well as in various artificial intelligence (AI) applications, for example, reasoning by cases, approximate reasoning, legal reasoning, diagnosis, and natural language understanding [7]. To conveniently and properly handle the representation and reasoning of disjunctive information in logic programming, a great deal of efforts have been given to the problem of finding suitable extensions of logic programming. The extension of logic programs by introducing disjunction in the ....

....work and our study also shows that such a generalization is non trivial and interesting. At the same time, our semantic framework can be naturally established for a wider class than that of disjunctive programs, called bi disjunctive logic programs, which is a subclass of super logic programs [7]. The rest of this paper is arranged as follows: Section 2 is devoted to establish the basic argumentation theoretic framework DAS for disjunctive programs. In Section 3 we mainly define three declarative semantics PDH, CDH and WFDH in DAS, give some examples and extend DAS to the class of ....

[Article contains additional citation context not shown here]

Brass,S.,Dix, J. and Przymusinki,T., Super Logic Programs, Technical report, University of Koblenz-Landau, 1996.

....of the form s A . As follows from the theorem, the resulting set of bisequents will be sucient for representing the circumscribed biconsequence relation. Remark. The above procedure is actually equivalent to the construction of generalized Clark s completion of the residual program suggested in [12] for what is called there super logic programs . The correspondence can be easily established once we notice that, for any A, the set of bisequents R A ( fs A g is representable by the following completion formula in the language f ; g: A : fA s1 ; A sn g) where fs 1 ....

S. Brass, J. Dix and T. Przymusinski (1996) Super logic programs. In L.C. Aiello, J. Doyle, and S.C. Shapiro (eds.) Principles of Knowledge Representation and Reasoning: Proc. Fifth Int. Conference (KR'96), Morgan Kaufmann, San Francisco, CA., pages 529-541.

....characterization of non monotonic reasoning (with an incorporated belief revision) The paper is organized as follows: First we describe the language and the basic concepts of AELKB (Sections 2 and 3) Thereafter in Section 4 we de ne DKS. In Section 5 are reviewed known results of [8] and [2] concerning characterizations of AELK and AELB in terms of Kripke structures. The results of this paper are presented in Section 6 (a possible world semantics of AELKB) in Section 7 (insertions into knowledge and belief theories are characterized in terms of DKS and it is outlined how to compute ....

.... behaviour : is a symptom, rather than the essence of nonstandard inference , see [11] 5 Possible World Semantics A characterization of AELK in terms of Kripke structures was given by Moore, see [8] Similarly, Kripke structures were used as a tool of a characterization of AELB in [2]. In this section we summarize the results of [8] and [2] particularly a characterization of SAE of AELK and AELB theories in terms of Kripke structures. Let us restrict the language LA in such a way that we do not use belief atoms (knowledge atoms) of the form B (K ) The language we denote ....

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Brass, S., Dix, J., Przymusinski, T., Super Logic Programs. Technical Report, Universit at Hannover, Institut fur Informatik, 1997

....Results: 1365 model(s) found. 6300 ms. of CPU time used. yes. eclipse 3] mm(demo2) mm minimal model reasoner Results: Query succeeded. 0 countermodel(s) found. 0 ms. of CPU time used. yes. eclipse 4] 3 General Disjunctive Programs For general disjunctive programs we have focused on STATIC ( BDP96] and the D WFS semantics ( BD95a,BD97b,BD95b] and on generalizations of the abductive approach to incorporate negation. We have studied their mutual relationships, implementation methods and relationship to minimal model reasoning. In Subsection 3.1 we present our transformation approach for ....

Stefan Brass, J#rgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529541. San Francisco, CA, Morgan Kaufmann, 1996.

....to the new version, and automatically generate the updated version. Generalize the scope of update rules: Just as interpretation updates were extended to updating by means of arbitrary inference rules [13] logic program updating should also be generalized, possibly using super logic programs [5]. 2 For a generalization of the results presented here for the case of extended (with explicit negation) logic programs, the reader is referred to [9] Reasoning about the past: Since no information is lost during the update process, i.e. after any iteration we can still characterize any ....

S. Brass, J. Dix and T. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shaphiro, editors, Principles of Knowledge Representation and Reasoning, Proc. of the Third Int'l Conf (KR96), pages 529-541. San Francisco, CA, Morgan-Kaufmann, 1996.

....Closed World Assumption (WGCWA) RLM89, RT88] too. As discussed before, minimal model reasoning for positive programs can be extended for more general programs with non monotonic negation. Currently, we are working on extending minimal model reasoning to handle D WFS and STATIC semantics [BD95a, BD95b, BDP96, DS96, Prz95]. We already have experimental Prolog programs to handle D WFS semantics based on both bottom up partial evaluation and a confluent calculus presented in [BD95b] Further, we are also exploring the applications of this fusion of theorem proving and logic programming technologies. One potential ....

Brass, S., Dix, J., Przymusinski, T.: Super logic programs. To appear in proc. of KR '96.

....Information As the example of Nixon s Diamond shows, multiple inheritance can lead to inherently disjunctive information ( Nixon is a pacifist or he is a hawk ) Vice versa, inheritance can be used to model disjunctive statements in Florid. Suppose, a database should express the following (see [4]) Mary visits Europe or Australia. People visiting one of those destinations are happy and those travelling to both are bankrupt afterwards. Since Florid provides no direct operator for disjunction we introduce two auxiliary classes, eFriends and aFriends. By default, members of those classes ....

S. Brass, J. Dix, and T. C. Przymusinski. Super Logic Programs. In 5. International Conference on Principles of Knowledge Representation and Reasoning, 1996.

....the form s A . As follows from the theorem, the resulting set of bisequents will be sufficient for representing the circumscribed biconsequence relation. Remark. The above procedure is actually equivalent to the construction of generalized Clark s completion of the residual program suggested in [12] for what is called there super logic programs . The correspondence can be easily established once we notice that, for any A, the set of bisequents R A (fl) fsAg is representable by the following completion formula in the language f; g: A : fA s 1 ; A sn g) where fs 1 ....

S. Brass, J. Dix and T. Przymusinski (1996) Super logic programs. In L.C. Aiello, J. Doyle, and S.C. Shapiro (eds.) Principles of Knowledge Representation and Reasoning: Proc. Fifth Int. Conference (KR'96), Morgan Kaufmann, San Francisco, CA., pages 529--541.

....) CnAEB ( T [ fBF : T ff j= min Fg and F 2 LAEB ) ffl Otherwise, define T fi = S ff fi T ff . The sequence fT ff g is monotonically increasing and has a unique fixed point T Pi = T = Psi T (T ) for some ordinal . 2 According to the following recent result established in [BDP96], for any finite belief theory only one iteration of the operator Psi T is needed. Theorem 2.2 [BDP96] For every finite belief theory T , the least static expansion T Pi of T is given by: T Pi = Psi T (T ) CnAEB (T [ fBF : T j= min Fg) 2 Observe that the least static autoepistemic ....

..... The sequence fT ff g is monotonically increasing and has a unique fixed point T Pi = T = Psi T (T ) for some ordinal . 2 According to the following recent result established in [BDP96] for any finite belief theory only one iteration of the operator Psi T is needed. Theorem 2. 2 [BDP96] For every finite belief theory T , the least static expansion T Pi of T is given by: T Pi = Psi T (T ) CnAEB (T [ fBF : T j= min Fg) 2 Observe that the least static autoepistemic expansion T Pi of T contains those and only those formulae which are true in all static autoepistemic ....

[Article contains additional citation context not shown here]

Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR'96), Boston, MA, pages 529--541. San Francisco, CA, Morgan Kaufmann, 1996.

.... group of Dix and Furbach [29] in Koblenz (Germany) in an ambitious project, implements the disjunctive stable semantics and other semantics of disjunctive logic programming; in particular, they implement the recently proposed static semantics [67, 68] which is very similar to the D WFS semantics [14, 16, 17] and the well founded circumscription semantics [90] in fact, for disjunctive datalog, static semantics and wellfounded circumscription semantics coincide (cf. 91] The static semantics associates with each program a set of modal formulas as its meaning; it appears to be the best candidate for ....

....with each program a set of modal formulas as its meaning; it appears to be the best candidate for a natural extension of the well founded semantics to disjunctive logic programs. Notice that, since the static semantics coincides on : free disjunctive logic programs with the minimal model semantics [16], it is clear from our results that DATALOG ; programs can express under this semantics every query in Pi p 2 . Moreover, preliminary results based on characterizations in [16] indicate that it can only express such queries, and thus has the same expressive power as DATALOG ; under the ....

[Article contains additional citation context not shown here]

S. Brass, J. Dix, and T. Przymusinski. Super logic programs. In Proc. International Conf. on Principles of Knowledge Representation and Reasoning (KR-96), Vancouver, Canada. pp. 529--540. Morgan Kaufmann, 1996.

....Information As the example of Nixon s Diamond shows, multiple inheritance can lead to inherently disjunctive information ( Nixon is a pacifist or he is a hawk ) Vice versa, inheritance can be used to model disjunctive statements in Florid. Suppose, a database should express the following (see [1]) Mary visits Europe or Australia. People visiting one of those destinations are happy and those travelling to both are bankrupt afterwards. Since Florid provides no direct operator for disjunction we introduce two auxiliary classes, eFriends and aFriends. By default, members of those classes ....

S. Brass, J. Dix, and T. C. Przymusinski. Super Logic Programs. In 5. International Conference on Principles of Knowledge Representation and Reasoning, 1996.

....modal logic KD. General logic programs are translatable into this formalism by treating disjunction, conjunction and as ordinary classical connectives, while not is translated as B: Since not and B are interde nable, we will use in what follows a formulation of AELB based on not (as is done in [10]) In particular, formulas 1 As is shown in [2] this ordering is adequate for representing nonmonotonic models of MBNF. of the form not F are called default atoms , while models of belief theories will be identi ed with their respective sets of propositional and default atoms. AELB determines ....

....a proposition F is true in all minimal models of T . The set of all such formulas minimally entailed by T will be denoted by T min . Finally, a static completion T of a theory T is de ned as a least xed point of the following operator: T (S) Cn not (T [ fnotF j :F 2 S min g) As is stated in [10] (Theorem 2.6 (Main) T turns out to be equal to a result of a single iteration of the operator T on T , that is, T = Cn not (T [ fnotF j :F 2 T min g) The above brief description of the static semantics will be sucient for our purposes. It should be clear, in particular, that the ....

[Article contains additional citation context not shown here]

S. Brass, J. Dix and T. Przymusinski (1996) Super logic programs. In L.C. Aiello, J. Doyle, and S.C. Shapiro (eds.) Principles of Knowledge Representation and Reasoning: Proc. Fifth Int. Conference (KR'96), Morgan Kauman, San Francisco, CA.

....a2, a3, a4, a5] 1365. a1, a5, a4, a3, a2] 1365 model(s) found. 6300 ms. of CPU time used. eclipse 3] mm(demo2) Query succeeded. 0 countermodel(s) found. 0 ms. of CPU time used. eclipse 4] 2 4. General Disjunctive Programs For general disjunctive programs we have focused on STATIC [19] and the D WFS semantics [9, 15, 10] and on generalizations of the abductive approach to incorporate negation. We have studied their mutual relationships, implementation methods and relationship to minimal model reasoning. There are of course more semantics for disjunctive programs around. The ....

....programs around. The first book on this topic was [56] which introduced many different versions most of which have been implemented by D. Seipel 3 . Also the stable semantics generalizes obviously to disjunctive programs ( 53] and there are also frameworks based on autoepistemic logic ([19]) as well as approaches to reduce the disjunctive case to normal programs [38] In addition, there are works where different sorts of disjunction, ranging from exclusive to inclusive are allowed ( 75, 76, 29] The reason why we focused on D WFS is that (1) some of the semantics mentioned above ....

[Article contains additional citation context not shown here]

Stefan Brass, J urgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529-- 541. San Francisco, CA, Morgan Kaufmann, 1996.

....the set of derivable pure disjunctions: we call this set STATIC(P) While arbitrary belief theories do not necessarily have consistent least static expansions, belief programs do. In fact, they can be characterized quite nicely as follows: Theorem 3. 6 (Least Static Expansion in LAELB , [BDP96b]) Every belief program P in AELB has the least static expansion, which is consistent. It is obtained as the least fixed point P Pi of the following monotonic belief closure operator Phi P : Phi P (S) Cn(P [ fB( p 1 ) Delta Delta Delta B( p n ) B( q 1 : qm ) S j= min :p 1 ....

Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96). San Francisco, CA, Morgan Kaufmann, 1996.

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Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529--541. San Francisco, CA, Morgan Kaufmann, 1996.

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Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Kno]wledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529--541. San Francisco, CA, Morgan Kaufmann, 1996.

....This approach allows for the computation of hundreds of thousands of minimal models. In Section 5, as our second main result, we show how this method can be nicely adapted for the computation of D WFS and STATIC. Our results are obtained by using the characterization of least static expansions in [BDP96] and a characterization of D WFS contained in [BD96b] The paper is organized as follows. In the next Section 2 we recall the definition and basic properties of disjunctive logic programs and semantics as well as the modal framework of the autoepistemic logic of beliefs (AELB) In Section 3 1 ....

....the set of derivable pure disjunctions: we call this set STATIC(P) While arbitrary belief theories do not necessarily have consistent least static expansions, belief programs do. In fact, they can be characterized quite nicely as follows: Theorem 11 (Least Static Expansion in LAELB , [BDP96]) Every belief program P in AELB has the least static expansion, which is consistent. It is obtained as the least fixed point P Pi of the following monotonic belief closure operator Phi P : Phi P (S) Cn(P [ fB( p 1 ) Delta Delta Delta B( pn ) B( q 1 : qm ) S j= min :p 1 ....

Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529--541. San Francisco, CA, Morgan Kaufmann, 1996.

....also in the case when the set of premises has a large number of minimal models. Our second main result shows how this method can be nicely adapted for the computation of the D WFS and static semantics. Our results are obtained by using the characterization of least static expansions from [BDP96] and a characterization of D WFS contained in [BD98a] The paper is organized as follows. In the next Section 2 we recall the definition and basic properties of disjunctive logic programs and semantics as well as the modal framework of the autoepistemic logic of beliefs (AEB) In Section 3 we ....

....A 1 : An resp. A 1 : Am constitute the set of derivable pure disjunctions: we call this set STATIC(P) While arbitrary belief theories do not necessarily have consistent least static expansions, belief programs do. In fact, they can be characterized nicely as follows ( BDP96] Theorem 3.7 (Least Static Expansion in LAEB ) Every belief program P in AEB has the least static expansion, which is consistent. It is obtained as the least fixed point P Pi of the following monotonic belief closure operator Psi P , which assigns to a set S the closure Cn(P [ fB( p 1 ) ....

Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529--541. San Francisco, CA, Morgan Kaufmann, 1996.

....evaluation. At the same time none of them is believed to be a bad teacher and there is some evidence supporting 4 The correctness of the examples discussed in this section has been verified by using the interpreter for static semantics developed by Stefan Brass based on the results obtained in [6]. The interpreter is available from ftp: ftp.informatik.uni hannover.de software static static.html via FTP and WWW. each one of them being a good researcher (i.e. neither B:good researcher(P aul) nor B:good researcher(Keith) is true) and therefore they are not subject to negative ....

Stefan Brass, Juergen Dix, and Teodor C. Przymusinski. Super logic programs. In Proceedings of the International Conference on Principles of Knowledge Representation and Reasoning (KR'96), Vancouver, Canada, page (in press). Morgan Kaufmann, 1996.

....Results: 1365 model(s) found. 6300 ms. of CPU time used. yes. eclipse 3] mm(demo2) mm minimal model reasoner Results: Query succeeded. 0 countermodel(s) found. 0 ms. of CPU time used. yes. eclipse 4] 4 General Disjunctive Programs For general disjunctive programs we have focused on STATIC [BDP96] and the D WFS semantics [BD95a, BD99, BD95b] and on generalizations of the abductive approach to incorporate negation. We have studied their mutual relationships, implementation methods and relationship to minimal model reasoning. There are of course more semantics for disjunctive programs ....

....have been implemented by D. Seipel (visit URL:http: www info1.informatik.uni wuerzburg.de database DisLog introduction.html for more information) Also the stable semantics generalizes obviously to disjunctive programs ( GL91] and there are also frameworks based on autoepistemic logic ( BDP96] as well as approaches to reduce the disjunctive case to normal programs [DGM96] In addition, there are works where different sorts of disjunction, ranging from exclusive to inclusive are allowed ( Sak89, SI94, Cha93] The reason why we focused on D WFS is that (1) some of the semantics ....

[Article contains additional citation context not shown here]

Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529--541. San Francisco, CA, Morgan Kaufmann, 1996.

....clearly link it to other well established non monotonic formalisms. Several approaches to the semantics of disjunctive logic programs have been recently proposed and studied in the literature, e.g. LMR92,Ros92,RT88,GL91] Dix92b] BD94,BD97b,BD97a,BD96a,EG93,EGM93,BLM90,Prz91b,Prz95b] and [Prz95a,BDNP97,BDP96]. Since a more thorough discussion of disjunctive programming is beyond the scope of this brief introduction, we refer the reader to those papers, as well as to papers published in this volume, for more details. 6 Implementations In this section we give a rough overview of what semantics have ....

.... et al. NNS91] was used by Dix Muller [DM93,DM92,Mul92] to implement versions of the stationary semantics of Przymusinski ( Prz91c] Brass Dix have implemented both D WFS and DSTABLE for allowed DATALOG programs ( BD95] 12 ) An implementation of static semantics ( Prz95b] is described in [BDP96] 13 . Seipel has implemented in his DisLog system various (modified versions of) semantics of Minker and his group. His system is publicly available at the URL http: sunwww.informatik.uni tuebingen.de:8080 dislog dislog.tar.Z. Finally, there is the DisLoP project undertaken by the Artificial ....

Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529--541. San Francisco, CA, Morgan Kaufmann, 1996.

....of the residual program as a preprocessing phase. In [BD95a] we also proposed an algorithm for doing the remaining work in the computation of stable models (based on a disjunctive extenstion of Clark s completion) Another application is in our prototype implementation of the STATIC semantics [BDP96]: it also consists of the computation of the residual program plus an algorithm for treating the few remaining hard cases. Again, the general algorithm would be too inefficient if applied directly to the original program, but it is good enough for evaluating the few negative body literals ....

Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529--541. San Francisco, CA, Morgan Kaufmann, 1996.

.... of Bell et al. NNS91] was used by Dix Muller to implement versions of the stationary semantics of Przymusinski ( Prz91b] MD93, DM92, Mul92] Brass Dix have implemented both D WFS and DSTABLE for allowed DATALOG programs ( BD95a] 23 ) An implementation of static semantics is described in [BDP96b] 24 . 23 ftp: ftp.informatik.uni hannover.de software index.html 24 ftp: ftp.informatik.uni hannover.de software static static.html 7 WHAT DO WE WANT AND WHAT IS IMPLEMENTED 76 Seipel has implemented in his DisLog system various (modified versions of) semantics of Minker and his group. ....

Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529--541. San Francisco, CA, Morgan Kaufmann, 1996.

....arbitrary belief theories do not necessarily have consistent least static expansions, belief programs do. In fact, they can be characterized quite nicely as follows: Theorem 3. 5 (Least Static Expansion in LAELB ) Every belief program P in AELB has the least static expansion, which is consistent ([5]) It is obtained as the least fixed point P Pi of the following monotonic belief closure operator Phi P : Phi P (S) is defined by Cn(P [ fB( p 1 ) Delta Delta Delta B( pn ) B( q 1 : qm ) S j= min :p 1 Delta Delta Delta :pn ( q 1 : qm )g) The sequence fP ff g ....

Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529--541. San Francisco, CA, Morgan Kaufmann, 1996.

....be iteratively applied to compute the static semantics. Remark 3.1 It is easy to see that if the belief theory is finite then the construction of the least static expansion will stop after countably many steps. However, the following powerful and somewhat surprising result obtained recently in [BDP96] shows that for every finite belief theory T its least static expansion is in fact obtained by a single iteration of the operator Psi T . Theorem 3.6 [BDP96] Let T be any finite belief theory. The least static expansion of T is always obtained by a single iteration of the operator Psi T . In ....

....static expansion will stop after countably many steps. However, the following powerful and somewhat surprising result obtained recently in [BDP96] shows that for every finite belief theory T its least static expansion is in fact obtained by a single iteration of the operator Psi T . Theorem 3. 6 [BDP96] Let T be any finite belief theory. The least static expansion of T is always obtained by a single iteration of the operator Psi T . In other words, the static completion T of T coincides with Psi T (T ) T = Psi T (T ) Psi T (T ) The following theorem significantly extends Theorem 3.4 ....

[Article contains additional citation context not shown here]

Stefan Brass, Juergen Dix, and Teodor C. Przymusinski. Super logic programs. In Proceedings of the International Conference on Principles of Knowledge Representation and Reasoning (KR'96), Vancouver, Canada, page (in press). Morgan Kaufmann, 1996.

....on Principles of Knowledge Representation and Reasoning (KR 96) L. C. Aiello, J. Doyle and S. C. Shapiro (editors) 1996, pp. 529 541. 2 Partially supported by the National Science Foundation grant #IRI 9313061. the foundation for the interesting results and applications obtained in [5]. The second, closely related result establishes a very interesting and somewhat intriguing relationship between static semantics T and Clark s completions comp(T ) of finite belief theories. It shows that the static semantics T of T is obtained by augmenting T with the set Bcomp(T ) fBF : F 2 ....

....is the fourth characterization of static completions presented in this paper. Needless to say, the existence of an equivalent non fixed point definition of static completions significantly simplifies this notion and the underlying theory. It also provides the foundation for the results obtained in [5]. Example 3.9 Consider the belief theory T from Example 2.9. We already noted that T j= min :F ixed and thus B:F ixed 2 T = Cn AEB i T [ fBF : T j= min Fg j : This implies that T contains also Broken. By Proposition 2.3, BBroken belongs to the static completion and so does :B:Broken. From ....

Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529--541. San Francisco, CA, Morgan Kaufmann, 1996.

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Stefan Brass, Jurgen Dix, and Teodor. C. Przymusinski. Super Logic Programs. In L. C. Aiello, J. Doyle, and S. C. Shapiro, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), pages 529--541. San Francisco, CA, Morgan Kaufmann, 1996.

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/98 Stefan Brass, J urgen Dix, Teodor C. Przymusinski. Super Logic Programs.

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BRASS, S., DIX, J., AND PRZYMUSINSKI, T. 1996a. Super logic programs. In Proceedings of the International Conference on Principles of Knowledge Representation and Reasoning (KR-96) (Vancouver, Canada), Morgan-Kaufmann, San Mateo, CA, 529--540.

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S. Brass, J. Dix and T. Przymusinski (1996) Super logic programs. In L.C. Aiello, J. Doyle, and S.C. Shapiro (eds.) Principles of Knowledge Representation and Reasoning: Proc. Fifth Int. Conference (KR'96), Morgan Kaufmann, San Francisco, CA., pages 529--541.

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