Allen Van Gelder. The alternating xpoint of logic programs with negation. In Proceedings of the Eighth ACM SIGACT-SIGMODSIGART Symposium on Principles of Database Systems, Philadelphia, Pennsylvania, pages 1-10. ACM Press, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....was basically developed in [Wen02a] We nally turn to the well founded semantics. In this approach, again, a distinguished three valued model is assigned to each given program P , called the well founded model of P , and introduced in [VGRS91] It is tightly related to the stable model semantics [Gel89, Prz89]. for P such that each A 2 dom(l) satis es one of the following conditions. 5 (WFi) A 2 M and there exists A L 1 ; L n in ground(P ) such that for all i we have L i 2 M and l(A) l(L i ) WFii) A 2 M and for each A A 1 ; A n ; B 1 ; Bm in P (WFiia) There ....

Allen Van Gelder. The alternating xpoint of logic programs with negation. In Proceedings of the Eighth ACM SIGACT-SIGMODSIGART Symposium on Principles of Database Systems, Philadelphia, Pennsylvania, pages 1-10. ACM Press, 1989.

....GL P is in general not monotonic for normal programs P . However it is antitonic, i.e. whenever I J B P then GL P (J) GL P (I) As a consequence, the operator GL P , obtained by applying GL P twice, is monotonic, and hence has a least xed point L P and a greatest xed point G P . In [vG89] it was shown that GL P (L P ) G P , L P = GL P (G P ) and that L P [ B P n G P ) coincides with the well founded model of P . This is called the alternating xed point characterization of the well founded semantics. These two orderings in fact correspond to the knowledge and truth ....

Allen van Gelder. The alternating xpoint of logic programs with negation. In Proceedings of the Eighth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, Philadelphia, Pennsylvania, pages 1-10. ACM Press, 1989.

....P is in general not monotonic for normal programs P . However it is antitonic, i.e. whenever I J B P then GL P (J) GL P (I) As a consequence, the operator GL P , obtained by applying GL P twice, is monotonic and hence has a least xed point L P and a greatest xed point G P . Van Gelder [vG89] has shown that GL P (L P ) G P , L P = GL P (G P ) and that L P [ B P n G P ) coincides with the well founded model of P . This is called the alternating xed point characterization of the well founded semantics. 2.4 Example Consider the program P from Example 2.2. The subprogram Q ....

....logic program P has a total well founded model if and only if there is a total model I of P and a (total) level mapping l such that P satis es (WF) with respect to I and l. As a further example for the application of our proof scheme, we use Theorem 5. 2 in order to prove a result by van Gelder [vG89] which we mentioned in the introduction, concerning the 12 alternating xed point characterization of the well founded semantics. Let us rst introduce some temporary notation, where P is an arbitrary program. L 0 = G 0 = B P L 1 = GL P (G ) for any ordinal G 1 = GL P (L ) for any ....

Allen van Gelder. The alternating xpoint of logic programs with negation. In Proceedings of the Eighth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, Philadelphia, Pennsylvania, pages 1-10. ACM Press, 1989.

....abductive reasoning. In the sequel, our technical exploration will be based on ground programs. 2.2. Regular extension semantics The regular extension semantics, initially formulated in the context of priority logic [40] is based on the notion of alternating xpoint, rst used by Van Gelder [36], and later by a number of authors (cf. 3, 28, 36, 38] to de ne and study the semantics of normal programs and default theories. The interest in this technique is that the regular model semantics for normal programs, and all the semantics equivalent to it, can be expressed by maximal normal ....

....exploration will be based on ground programs. 2.2. Regular extension semantics The regular extension semantics, initially formulated in the context of priority logic [40] is based on the notion of alternating xpoint, rst used by Van Gelder [36] and later by a number of authors (cf. [3, 28, 36, 38]) to de ne and study the semantics of normal programs and default theories. The interest in this technique is that the regular model semantics for normal programs, and all the semantics equivalent to it, can be expressed by maximal normal alternating xpoints. First let us de ne a function FP ....

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A. Van Gelder. The alternating xpoint of logic programs with negation. J. Computer and System Sciences, pages 185-221, 1993.

.... theory, commonly referred to as the Tarski Knaster xpoint theorem, F 2 P possesses a least xpoint, a maximum xpoint, and possibly some others over the domain of a complete lattice (in this case the set of all subsets of default negations) These xpoints have been called alternating xpoints [23]. De nition 2.6 Let P be a general logic program. The well founded model of P is de ned by the least xpoint of F 2 P . The regular models of P are maximal xpoints M of F 2 P satisfying M F P (M ) 2 Since a xpoint E of F P is a maximal xpoint of F 2 P satisfying E F P (E) a stable ....

A. Van Gelder. The alternating xpoint of logic programs with negation. J. Computer and System Sciences, 47(1):185-221, 1993.

....update request set : fentry(mon; 10; 0) entry(mon; 10; 23)g w.r.t. the EDB instance DB 0 of Example 2. 15 results in the new state: DB 0 E = DB 0 [ fentry(mon; 10; 23)g n fentry(mon; 10; 0)g 2 In logic databases the semantics of the IDB atoms often is given by the well founded model [vG89, vGRS91] which subsumes most standard models de ned for restricted program classes, see e.g. AB94] It is denoted WFM( for a union = P IDB [DB of the ( xed) set of IDB rules P IDB and an EDB instance DB. De nition 3.17 [DB Interpretation] For logic databases, we de ne the DB ....

....developed a concept [Che97] that has turned out to be similar to the ULTRA language described in [WFF98b] as well as in Sections 2.2 and 3.3 of this paper. However, Chen s main goal is to de ne a well founded semantics for update programs by tailoring van Gelders s alternating xpoint procedure [vG89] to operate on a structure built over update request sets, whereas in the ULTRA approach, the emphasis lies in the integration of arbitrary basic operations and in a transactional foundation. Up to now we have not considered to permit negation of basic or de nable update atoms. An alternative ....

A. van Gelder. The alternating xpoint of logic programs with negation (extended abstract) . In Proc. 8th ACM Symp. on Principles of Database Systems, pages 1-10, 1989.

.... and Dix [BD94, BD97, BD98b] However, their residual program can grow to exponential size, whereas for function free programs our program remainder is always polynomial in the size of the extensional database (EDB) Our approach is also closely related to the alternating xpoint procedure [VG89, VG93]. However, the alternating xpoint procedure is known to have ineciencies due to redundant recomputations of possible facts. By using conditional facts that can be deleted directly instead of recomputing the complement our approach is guaranteed to need not more work than the alternating ....

....model of a normal program. In this paper we characterize good bottom up methods in terms of elementary program transformations. Essentially, the bottom up algorithms that compute the well founded model of a normal program are: The alternating xpoint approach, introduced by Van Gelder [VG89, VG93] and further developed by Kemp, Stuckey and Srivastava [KSS95] The computation of the residual program, suggested by Bry [Bry89, Bry90] and independently by Dung Kanchanasut [DK89a, DK89b] and extended by Brass and Dix [BD97, BD98c] The alternating xpoint procedure is known to have eciency ....

[Article contains additional citation context not shown here]

Allen Van Gelder. The alternating xpoint of logic programs with negation. Journal of Computer and System Sciences, 47(1):185-221, 1993.

.... and Dix [BD94, BD97, BD98b] However, their residual program can grow to exponential size, whereas for function free programs our program remainder is always polynomial in the size of the extensional database (EDB) Our approach is also closely related to the alternating xpoint procedure [VG89, VG93]. However, the alternating xpoint procedure is known to have ineciencies due to redundant recomputations of possible facts. By using conditional facts that can be deleted directly instead of recomputing the complement our approach is guaranteed to need not more work than the alternating ....

....model of a normal program. In this paper we characterize good bottom up methods in terms of elementary program transformations. Essentially, the bottom up algorithms that compute the well founded model of a normal program are: The alternating xpoint approach, introduced by Van Gelder [VG89, VG93] and further developed by Kemp, Stuckey and Srivastava [KSS95] The computation of the residual program, suggested by Bry [Bry89, Bry90] and independently by Dung Kanchanasut [DK89a, DK89b] and extended by Brass and Dix [BD97, BD98c] The alternating xpoint procedure is known to have eciency ....

[Article contains additional citation context not shown here]

Allen Van Gelder. The alternating xpoint of logic programs with negation. In Proc. of the Eighth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS'89), pages 1-10, 1989.

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