Citation Context

...d the "canonical model" approach. 1.1 Program Completion Semantics The original "program completion" approach, due to Clark [6], and discussed in detail by Shepherdson [37, 38], Ku=-=nen [17], and Lloyd [20]-=-, has been to define a new program, called the completed program (sometimes called the completed database). The completed program is treated simply as a first order formula (see Section 4). Then the n...

Citation Context

...thus faulty. Thus failure to have a perfect model was thought to indicate a flaw in the program rather than in the definition of perfect models. Recent experience has cast doubt on this attitude (see =-=[11] for discu-=-ssion), and spurred the search for further improvements in the definition of the "canonical model." Gelfond and Lifschitz propose an elegant definition of a stable model that is closely rela...

Citation Context

... directions, which may be summarized as the "program completion" approach and the "canonical model" approach. 1.1 Program Completion Semantics The original "program completion=-=" approach, due to Clark [6]-=-, and discussed in detail by Shepherdson [37, 38], Kunen [17], and Lloyd [20], has been to define a new program, called the completed program (sometimes called the completed database). The completed p...

Citation Context

...ms in an unfounded set generalizes negation by failure in Horn clause programs; if H is the Herbrand base and I is the set of atoms that represents the minimum Herbrand model of a Horn clause program =-=[39]-=-, then H \Gamma I , the set of atoms not in I , is unfounded with respect to I . We now formalize the intuition of the preceding discussion. It is immediate that the union of arbitrary unfounded sets ...

Citation Context

... all of its predicates can be assigned a rank such that ffl no predicate depends positively on one of greater rank, and ffl no predicate depends negatively on one of equal or greater rank in any rule =-=[4, 1, 19, 40]-=-. In the context of an IDB and EDB, the EDB, being a set of simple facts, has rank 0. IDB predicates whose defining rules involve no negation also have rank 0. IDB predicates whose only negative depen...

Citation Context

...as come to be known as Datalog), the Herbrand universe is finite and the construction is effective. In this section we show that the data complexity of the well-founded semantics, as defined by Vardi =-=[44]-=-, is polynomial. From this standpoint it is competitive with other methods, such as stratified semantics, whose data complexity has been studied elsewhere [5, 44, 12, 14], and the Fitting model (as re...

Citation Context

...nsequences of the completed program. 5 Stable Models Gelfond introduced an approach to negation through stable models [10], and motivated it by appealing to autoepistemic logic, as developed by Moore =-=[26]-=-. The theory has been further developed by Gelfond and Lifschitz [11], and also by Marek and Truszczynski [24, 23]. In this section we follow the definition of [11], which defines stability without re...

Citation Context

...program is consistent but unintuitive. More importantly, on many natural examples it yields a surprisingly weak reasoning ability. We shall illustrate these claims with examples in Section 7. Fitting =-=[9]-=- and Kunen [17] gave markedly different, more uniform, semantics by interpreting the completed program in a 3-valued constructive logic, elegantly eliminating some difficulties of the Clark program co...

Citation Context

...A.1 in the appendix), the question of whether a "strict" program is possible in the Fitting semantics is open. Nonstrict programs in the Fitting semantics are known to exist, by the work of =-=Immermann [14]-=-, but are quite complicated; details are discussed elsewhere by Van Gelder [42]. As another motivational example, we consider a program that is not locally stratified, as defined in Section 6, yet has...

Citation Context

... treated in [4, 1, 19, 1 By "interpreter," we mean any mechanism for executing the program, including a compiler. 40], and elsewhere. The locally stratified class, defined and studied by Prz=-=ymusinski [31]-=-, is a superset of the class of stratified programs. He defined perfect models, and showed that every locally stratified program has a unique perfect model. These classes are discussed further in Sect...

Citation Context

...l-founded semantics, as defined by Vardi [44], is polynomial. From this standpoint it is competitive with other methods, such as stratified semantics, whose data complexity has been studied elsewhere =-=[5, 44, 12, 14]-=-, and the Fitting model (as remarked below). In this discussion of complexity we restrict attention to function-free programs, so a program's Herbrand universe is just the set of constants appearing i...

Citation Context

...ion" approach and the "canonical model" approach. 1.1 Program Completion Semantics The original "program completion" approach, due to Clark [6], and discussed in detail by She=-=pherdson [37, 38], Kunen [17]-=-, and Lloyd [20], has been to define a new program, called the completed program (sometimes called the completed database). The completed program is treated simply as a first order formula (see Sectio...

Citation Context

...e the preliminary version of this paper was presented at a conference [43], several alternative formulations of negation that appear to be equivalent to the well-founded semantics have been developed =-=[3, 8, 32, 42]. We belie-=-ve that this indicates a robustness of the semantics, and provides evidence that it coincides well with "common sense" and intuition. 2 General Logic Programs and Partial Interpretations In ...

Citation Context

... by Ross and Topor [34], and is essentially the same as the "securable set" in [36]. Unfounded sets are defined in Section 3. Since the preliminary version of this paper was presented at a c=-=onference [43]-=-, several alternative formulations of negation that appear to be equivalent to the well-founded semantics have been developed [3, 8, 32, 42]. We believe that this indicates a robustness of the semanti...

Citation Context

...tes in effect that the predicate is completely defined by the given rules. The formal details, including handling of variables and introduction of axioms for equality, are described in several places =-=[6, 2, 20, 9, 17]-=-. Example 4.1. Recall the last four rules of Example 3.1, whose atoms formed an unfounded set: p(d) / q(a); not q(b): p(d) / q(b); not q(c): q(a) / p(d): q(b) / q(a): The Clark completion combines the...

Citation Context

... models [10], and motivated it by appealing to autoepistemic logic, as developed by Moore [26]. The theory has been further developed by Gelfond and Lifschitz [11], and also by Marek and Truszczynski =-=[24, 23]-=-. In this section we follow the definition of [11], which defines stability without reference to autoepistemic logic. We show that if a program has a total well-founded model, that model is the unique...

Citation Context

...r limit ordinal ff, I ff = [ fi!ff I fi Note that 0 is a limit ordinal, and I 0 = ;. 2. For successor ordinal ff = fl + 1, I fl+1 = WP (I fl ) 3. Finally, define I 1 = [ ff I ff Following Moschovakis =-=[29]-=-, for any literal p in I 1 , we define the stage of p to be the least ordinal ff such that p 2 I ff . We observe that the stage is always a successor ordinal for literals in I 1 . Lemma 3.4. I ff as d...

Citation Context

...le syntax, we require two mutually recursive IDB relations. The interested reader can work out the equivalent rules using a single relation in a language that supports a richer syntax for rule bodies =-=[21, 28]-=-. ffl color (Obj ; Pt ; C) means that the visible color of Obj at Pt is C (looking down from above). ffl dominated(Obj ; Pt ; O1; L1) holds when two objects that are components of the same Obj overlap...

Citation Context

...". However, factoring possibilities in the given program do not always carry over to the completed program, and as:a does not simplify to true in either 3-valued logic [9, 17] or intuitionistic l=-=ogic [7]. Thus cau-=-tion is needed to keep a coherent system. The overly trivial P 2 in Example 5.3 might lead one to believe that a factoring capability can be easily "patched in" by just checking for a negati...

Citation Context

...e present some examples to support our position that well-founded models are natural and intuitive. Example 7.1. This example is abstracted from the "Yale shootout" example due to Hanks and =-=McDermott [13]-=-. The program P is noise(T ) / loaded (T ); shoots(T ): loaded (0). loaded (T ) / succ(S; T ); loaded (S); not shoots(S): shoots(T ) / triggers(T ): triggers(1). succ(0; 1). We regard triggers and suc...

Citation Context

...e the preliminary version of this paper was presented at a conference [43], several alternative formulations of negation that appear to be equivalent to the well-founded semantics have been developed =-=[3, 8, 32, 42]. We belie-=-ve that this indicates a robustness of the semantics, and provides evidence that it coincides well with "common sense" and intuition. 2 General Logic Programs and Partial Interpretations In ...

Citation Context

... idea, the closed world assumption, was introduced in the context of deductive databases by Reiter [33]. The generalized closed world assumption was proposed by Minker to handle disjunctive databases =-=[25]-=- without producing the inconsistency typical of the closed world assumption; it is discussed in Example 3.1. SLDNF is applied to the original program. Clark showed the procedure to be sound in the sen...

Citation Context

... all of its predicates can be assigned a rank such that ffl no predicate depends positively on one of greater rank, and ffl no predicate depends negatively on one of equal or greater rank in any rule =-=[4, 1, 19, 40]-=-. In the context of an IDB and EDB, the EDB, being a set of simple facts, has rank 0. IDB predicates whose defining rules involve no negation also have rank 0. IDB predicates whose only negative depen...

Citation Context

... 95064. resolution plus the negation as failure rule) has been studied extensively. A closely related idea, the closed world assumption, was introduced in the context of deductive databases by Reiter =-=[33]-=-. The generalized closed world assumption was proposed by Minker to handle disjunctive databases [25] without producing the inconsistency typical of the closed world assumption; it is discussed in Exa...

Citation Context

...n's main theorem is that this interpretation characterizes the 3-valued logical consequences of the completed program. 5 Stable Models Gelfond introduced an approach to negation through stable models =-=[10]-=-, and motivated it by appealing to autoepistemic logic, as developed by Moore [26]. The theory has been further developed by Gelfond and Lifschitz [11], and also by Marek and Truszczynski [24, 23]. In...

Citation Context

...iverse, due to their containing what is called unsafe negation. A simple way to remove this behavior is to augment the program, as described in this appendix. We proceed informally here, and refer to =-=[22]-=- for a formal discussion. Definition A.1. Any general logic program P has an associated augmented program that is formed by adding the apparently nonsensical rule: $p($f($c)) / $p($f($c)): where $p, $...

Citation Context

... all of its predicates can be assigned a rank such that ffl no predicate depends positively on one of greater rank, and ffl no predicate depends negatively on one of equal or greater rank in any rule =-=[4, 1, 19, 40]-=-. In the context of an IDB and EDB, the EDB, being a set of simple facts, has rank 0. IDB predicates whose defining rules involve no negation also have rank 0. IDB predicates whose only negative depen...

Citation Context

..."program completion" approach and the "canonical model" approach. 1.1 Program Completion Semantics The original "program completion" approach, due to Clark [6], and discu=-=ssed in detail by Shepherdson [37, 38]-=-, Kunen [17], and Lloyd [20], has been to define a new program, called the completed program (sometimes called the completed database). The completed program is treated simply as a first order formula...

Citation Context

...here, the expressive power of the well-founded semantics has been compared to several forms of fixpoint logic [42]. A corresponding procedural semantics has been reported for some classes of programs =-=[35, 32]-=-. Acknowledgements We wish to thank Jerzy Jaromczyk, Phokion Kolaitis, Vladimir Lifschitz, Wiktor Marek, Rodney Topor, and Moshe Vardi for helpful discussions and comments about this work. We also tha...

Citation Context

... all of its predicates can be assigned a rank such that ffl no predicate depends positively on one of greater rank, and ffl no predicate depends negatively on one of equal or greater rank in any rule =-=[4, 1, 19, 40]-=-. In the context of an IDB and EDB, the EDB, being a set of simple facts, has rank 0. IDB predicates whose defining rules involve no negation also have rank 0. IDB predicates whose only negative depen...

Citation Context

...le syntax, we require two mutually recursive IDB relations. The interested reader can work out the equivalent rules using a single relation in a language that supports a richer syntax for rule bodies =-=[21, 28]-=-. ffl color (Obj ; Pt ; C) means that the visible color of Obj at Pt is C (looking down from above). ffl dominated(Obj ; Pt ; O1; L1) holds when two objects that are components of the same Obj overlap...

Citation Context

...cases, which have a total model. The key idea in our formulation is the concept of an "unfounded set," which is an adaptation of the "closed set" developed for disjunctive database=-=s by Ross and Topor [34], and is e-=-ssentially the same as the "securable set" in [36]. Unfounded sets are defined in Section 3. Since the preliminary version of this paper was presented at a conference [43], several alternati...

Citation Context

...s also destroys the perfect model even though move(a; a) does not occur in the EDB. 4 Recently, Przymusinska and Przmusinski have defined weakly perfect models to handle programs such as this example =-=[30]-=-. The next example was inspired by an informal presentation by K. Morris of Stanford University [27]. It shows how the negation issues addressed by this paper might easily arise in practical settings....

Citation Context

...e the preliminary version of this paper was presented at a conference [43], several alternative formulations of negation that appear to be equivalent to the well-founded semantics have been developed =-=[3, 8, 32, 42]. We belie-=-ve that this indicates a robustness of the semantics, and provides evidence that it coincides well with "common sense" and intuition. 2 General Logic Programs and Partial Interpretations In ...

Citation Context

...alued logical consequence semantics, a "strict" logic program without function symbols cannot define a predicate that is true in the transitive closure, false in its complement, and nowhere =-=undefined [18]. Informal-=-ly, a "strict" program is one in which the dependence of one predicate on another (or itself) is either through an even number of negations or through an odd number, but not both. Because Ku...

Citation Context

..."program completion" approach and the "canonical model" approach. 1.1 Program Completion Semantics The original "program completion" approach, due to Clark [6], and discu=-=ssed in detail by Shepherdson [37, 38]-=-, Kunen [17], and Lloyd [20], has been to define a new program, called the completed program (sometimes called the completed database). The completed program is treated simply as a first order formula...

Citation Context

...e the preliminary version of this paper was presented at a conference [43], several alternative formulations of negation that appear to be equivalent to the well-founded semantics have been developed =-=[3, 8, 32, 42]-=-. We believe that this indicates a robustness of the semantics, and provides evidence that it coincides well with \common sense" and intuition. 2. General Logic Programs and Partial Interpretations In...

Citation Context

... models [10], and motivated it by appealing to autoepistemic logic, as developed by Moore [26]. The theory has been further developed by Gelfond and Lifschitz [11], and also by Marek and Truszczynski =-=[24, 23]-=-. In this section we follow the definition of [11], which defines stability without reference to autoepistemic logic. We show that if a program has a total well-founded model, that model is the unique...

Citation Context

...l-founded semantics, as defined by Vardi [44], is polynomial. From this standpoint it is competitive with other methods, such as stratified semantics, whose data complexity has been studied elsewhere =-=[5, 44, 12, 14]-=-, and the Fitting model (as remarked below). In this discussion of complexity we restrict attention to function-free programs, so a program's Herbrand universe is just the set of constants appearing i...

Citation Context

...e the preliminary version of this paper was presented at a conference [43], several alternative formulations of negation that appear to be equivalent to the well-founded semantics have been developed =-=[3, 8, 32, 42]. We belie-=-ve that this indicates a robustness of the semantics, and provides evidence that it coincides well with "common sense" and intuition. 2 General Logic Programs and Partial Interpretations In ...

Citation Context

...artial) model. this program turns out to be closely related to a game described by Kolaitis, and used to prove that there are queries in fixpoint logic that are not expressible by stratified programs =-=[16]-=-. In this respect, the program can be viewed as describing a game where one wins if the opponent has no moves, as in checkers (draughts). winning(X)/move(X; Y ); not winning(Y ): Some sample move grap...

Citation Context

...dels. Interestly, this program turns out to be closely related to a game described by Kolaitis, and used to prove that there are queries insxpoint logic that are not expressible by stratied programs =-=[16]-=-. In this respect, the program can be viewed as describing a game where one wins if the opponent has no moves, as in checkers (draughts). winning(X) move(X;Y ); not winning(Y ): Some sample move graph...

Citation Context

...ed program. Jaffar, Lassez and Lloyd showed that SLDNF was complete (in the same sense) for Horn programs with non-floundering queries consisting of a conjunction of positive and/or negative literals =-=[15]. SLDNF wa-=-s further investigated for general logic programs by Lloyd [20] (who coined the term SLDNF), Shepherdson [37, 38] (q.v. for further bibliography), and others. This approach is "logically" im...

Citation Context

...ell-founded semantics, as dened by Vardi [44], is polynomial. From this standpoint it is competitive with other methods, such as stratied semantics, whose data complexity has been studied elsewhere =-=[4, 44, 12, 14]-=-, and the Fitting model (as remarked below). In this discussion of complexity we restrict attention to function-free programs, so a program's Herbrand universe is just the set of constants appearing i...

Citation Context

...igning a canonical model to a program is that one now has a standard for correctness of an interpreter 1 on all goals -- it must conform to the canonical model, and succeed or fail appropriately. See =-=[41]-=- for a discussion of how the canonical model approach can benefit application development. Another motivation for concentrating on canonical models is the view, expounded by Reiter [33], that many log...

Citation Context

...tion is the concept of an "unfounded set," which is an adaptation of the "closed set" developed for disjunctive databases by Ross and Topor [35], and is essentially the same as the "securable set" in =-=[36]-=-. Unfounded sets are defined in Section 3. Since the preliminary version of this paper was presented at a conference [43], several alternative formulations of negation that appear to be equivalent to ...

Citation Context

...musinska and Przmusinski have defined weakly perfect models to handle programs such as this example [30]. The next example was inspired by an informal presentation by K. Morris of Stanford University =-=[27]-=-. It shows how the negation issues addressed by this paper might easily arise in practical settings. Example 7.4. We imagine a logic program that might be part of a VLSI CAD system, whose function is ...

Citation Context

...tion is the concept of an "unfounded set," which is an adaptation of the "closed set" developed for disjunctive databases by Ross and Topor [34], and is essentially the same as the=-= "securable set" in [36]-=-. Unfounded sets are defined in Section 3. Since the preliminary version of this paper was presented at a conference [43], several alternative formulations of negation that appear to be equivalent to ...