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## The Well-Founded Semantics for General Logic Programs (1991)

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Venue: | Journal of the ACM |

Citations: | 997 - 15 self |

### Citations

2080 |
Foundations of Logic Programming
- Lloyd
- 1984
(Show Context)
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... |

1845 | V.: The stable model semantics for logic programming
- Gelfond, Lifschitz
- 1988
(Show Context)
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... |

1069 |
Negation as failure.
- Clark
- 1978
(Show Context)
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... |

808 | Predicate logic as a programming language
- Kowalski
- 1974
(Show Context)
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 ... |

682 |
Towards a theory of declarative knowledge
- Apt, Blair, et al.
(Show Context)
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... |

586 |
Complexity of relational query languages
- Vardi
- 1982
(Show Context)
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... |

498 |
Semantical considerations on nonmonotonic logic.
- Moore
- 1985
(Show Context)
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... |

446 | A Kripke-Kleene semantics for logic programs
- Fitting
- 1985
(Show Context)
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... |

316 | Relational Queries Computable in Polynomial Time
- Immerman
- 1982
(Show Context)
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... |

279 |
On the declarative semantics of deductive databases and logic programs
- Przymusinski
(Show Context)
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... |

262 | Structure and Complexity of Relational Queries
- Chandra, Harel
- 1980
(Show Context)
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... |

253 |
Negation in logic programming
- Kunen
- 1987
(Show Context)
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... |

244 | The alternating fixpoint of logic programs with negation.
- Gelder
- 1989
(Show Context)
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 ... |

231 |
Unfounded sets and well-founded semantics for general logic programs.
- Gelder, Ross, et al.
- 1988
(Show Context)
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... |

209 |
Contributions to the theory of logic programming.
- Apt, Emden
- 1982
(Show Context)
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... |

207 |
Autoepistemic Logic
- Marek, Truszczy'nski
- 1991
(Show Context)
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... |

195 |
Elementary Induction on Abstract Structures
- Moschovakis
- 1974
(Show Context)
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... |

194 |
Making prolog more expressive.
- Lloyd, Topor
- 1984
(Show Context)
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... |

165 |
1977] Elements of Intuitionism
- Dummett
(Show Context)
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... |

156 |
Default reasoning, non-monotonic logics, and the frame problem.
- Hanks, McDermott
- 1986
(Show Context)
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... |

156 | Every logic program has a natural stratification and an iterated least fixed point model.
- Przymusinski
- 1989
(Show Context)
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 ... |

154 |
On indefinite databases and the closed world assumption.
- Minker
- 1982
(Show Context)
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... |

139 |
Negation as failure using tight derivations for general logic programs.
- Gelder
- 1988
(Show Context)
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... |

123 |
On Closed World Databasesâ€ť.
- Reiter
- 1978
(Show Context)
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... |

120 |
On stratified autoepistemic theories.
- Gelfond
- 1987
(Show Context)
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... |

83 |
Equivalences of logic programs
- Maher
- 1988
(Show Context)
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, $... |

79 |
clause queries and generalizations
- Horn
- 1985
(Show Context)
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... |

76 |
Negation in logic programming.
- Shepherdson
- 1987
(Show Context)
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... |

60 | A procedural semantics for well-founded negation in logic programs
- Ross
- 1989
(Show Context)
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... |

58 |
On the Declarative Semantics of Logic Programs with Negation
- Lifschitz
- 1988
(Show Context)
Citation Context |

49 |
Design overview of the NAIL! system
- Morris, Ullman, et al.
- 1986
(Show Context)
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... |

48 |
Inferring Negative Information from Disjunctive Databases, .I. Automated Reasoning
- Ross, Topor
- 1988
(Show Context)
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... |

47 |
Weakly Perfect Model Semantics for Logic Programs
- Przymusinska, Przymusinski
- 1988
(Show Context)
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.... |

45 |
Logic programming as constructivism: a formalization and its application to databases.
- Bry
- 1989
(Show Context)
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 ... |

27 |
Some Remarks on the Completed Database
- Kunen
- 1988
(Show Context)
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... |

20 |
Negation as Failure
- Shepherdson
- 1985
(Show Context)
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... |

16 |
The alternating of logic programs with negation
- Gelder
- 1992
(Show Context)
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... |

13 | Stable theories in autoepistemic logic
- Marek
- 1989
(Show Context)
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... |

11 |
Fixed-point extensions of first order logic
- Gurevich, Shelah
- 1986
(Show Context)
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... |

10 |
A natural semantics for logic programs with negation.
- Dung, Kanchanasut
- 1989
(Show Context)
Citation Context |

10 |
The Expressive Power of Stratified Programs.
- Kolaitis
- 1991
(Show Context)
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... |

4 |
The expressive power of strati programs
- Kolaitis
- 1991
(Show Context)
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... |

3 |
Completeness of the negation-as-failure rule
- Jaar, Lassez, et al.
- 1983
(Show Context)
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... |

3 |
Fixed-point extensions of order logic
- Gurevich, Shelah
- 1986
(Show Context)
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... |

2 |
Modeling simultaneous events with default reasoning and tight derivations
- Gelder
- 1990
(Show Context)
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... |

2 |
Negation by securable failure in logic programming. (manuscript
- Schlipf
- 1987
(Show Context)
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 ... |

1 |
Talk at Workshop XP8.3i
- Morris
- 1987
(Show Context)
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 ... |

1 |
Negation by securable failure in logic programming
- Schlipf
- 1987
(Show Context)
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 ... |