## Logic programming revisited: logic programs as inductive definitions (2001)

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Venue: | ACM Transactions on Computational Logic |

Citations: | 34 - 21 self |

### BibTeX

@ARTICLE{Denecker01logicprogramming,

author = {Marc Denecker and Maurice Bruynooghe and Victor Marek},

title = {Logic programming revisited: logic programs as inductive definitions},

journal = {ACM Transactions on Computational Logic},

year = {2001},

volume = {2},

pages = {2001}

}

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### Abstract

Logic programming has been introduced as programming in the Horn clause subset of first order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iff-definitions. A second approach was to identify a unique canonical, preferred or intended model among the models of the program and to appeal to common sense to validate the choice of such model. Another line of research developed the view of logic programming as a non-monotonic reasoning formalism strongly related to Default Logic and Auto-epistemic Logic. These competing approaches have resulted in some confusion about the declarative meaning of logic programming. This paper investigates the problem and proposes an alternative epistemological foundation for the canonical model approach, which is not based on common sense but on a solid mathematical information principle. The thesis is developed that logic programming can be understood as a natural and general logic of inductive definitions. In particular, logic programs with negation represent non-monotone inductive definitions. It is argued that this thesis results in an alternative justification of the well-founded model as the unique intended model of the logic program. In addition, it equips logic programs with an easy to comprehend meaning

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Citation Context ...on of the query. Techniques for analysis of termination of SLD [De Schreye and Decorte 1994] and of SLG [Verbaeten et al. 2001] exist. Floundering can be analyzed by means of abstract interpretation [=-=Cousot and Cousot 1977-=-], more specifically by groundness analysis [Marriott and Søndergaard 1993]. Of course, due to the undecidability results, these techniques cannot be complete. 5.3 Inductive definitions in the absence... |

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Citation Context ...bove rule to the default: q : :r p In this view, Logic Programming is seen as a restricted form of default logic or auto-epistemic logic. This approach resulted in stable semantics of logic programs [=-=Gelfond and Lifschitz 1-=-988] and was the foundation for Answer Set Programming [Niemela 1999; Marek and Truszczynski 1999]. ACM Transactions on Computational Logic, Vol. TBD, No. TBD, TBD TBD. 4 Marc Denecker et al. It is e... |

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Citation Context ...and can dene each predicate of the program. In recent years, several Logic Programming extensions capable of representing incomplete knowledge have been proposed. One is Abductive Logic Programming [K=-=akas et al. 1992-=-; Denecker 1995], an extension of logic programming by means of abductive reasoning. An abductive logic program is a triple hA; P; ICi consisting of a set A of abducible (or open) predicates, a logic ... |

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Citation Context ...te, while the \membership-in-all" problem is co-NPcomplete [Marek and Truszczynski 1991]. With the introduction of tabling [Tamaki and Sato 1986] and the development of the SLG resolution procedu=-=re [Chen and Warren 1996-=-], more powerful top down proof procedures became available. It is proven in [Chen and Warren 1996] that SLG is sound and search space complete with respect to the well-founded partial model and is po... |

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Citation Context ...ach was extended to larger classes of programs. It started with work on the perfect model semantics for stratied programs [Apt et al. 1988; Van Gelder 1988], which was extended to locally stratied [Pr=-=zymusinski 19-=-88] and weakly stratied [Przymusinska and Przymusinski 1990] programs. This direction culminated in the well-founded semantics which denes a unique (possibly 3-valued) model for all normal programs [V... |

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Citation Context ...rm of default logic or autoepistemic logic. This approach resulted in stable semantics of logic programs [Gelfond and Lifschitz 1988] and was the foundation for answer set programming [Niemel"a 1999; =-=Marek and Truszczy'nski 1999-=-]. ACM Transactions on Computational Logic, Vol. TBD, No. TBD, TBD TBD.s4 \DeltasMarc Denecker et al. It is easy to see that the above approaches are based on different commonsense interpretations of ... |

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Citation Context ...ce problem for stable models of Datalog programs is NP-complete. Similarly, "membership-in-some" problem for such programs is also NP-complete, while the "membership-in-all" problem is coNP-complete [=-=Marek and Truszczy'nski 1991-=-]. With the introduction of tabling [Tamaki and Sato 1986] and the development of the SLG resolution procedure [Chen and Warren 1996], more powerful top down proof procedures became available. It is p... |

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Citation Context ...ained through some iterated application of this operator until a fixpoint is reached. Consequently the study of inductive definitions is closely related to the study of operators and their fixpoints [=-=Aczel 1977-=-]. ACM Transactions on Computational Logic, Vol. TBD, No. TBD, TBD TBD.s10 · Marc Denecker et al. Originally, mathematical logicians focused on monotone inductive definitions. When an operator Γ is mo... |

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(Show Context)
Citation Context ...in terms of absence of domain elements bsin X . It is certainly much easier tosnd applications of iterated induction than of in ationary induction. The applications of in ationary induction, e.g. in [=-=Moschovakis 197-=-4], tend to be for dening highly abstract concepts in set theory. Although in ationary induction is expressive [Moschovakis 1974; Kolaitis and Papadimitriou 1991; Gurevich and Shelah 1986], it turns o... |

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Citation Context ...n) predicates, a logic program P dening the set of non-abducible predicates and a set IC of classical logic constraints. Another new paradigm is Answer Set Programming [Marek and Truszczynski 1999; Li=-=fschitz 19-=-99; Niemela 1999]. This approach is based on the stable model semantics [Gelfond and Lifschitz 1988] and is fundamentally dierent from the view we have developed so far. Under the stable model semanti... |

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Citation Context ...on of the query. Techniques for analysis of termination of SLD [De Schreye and Decorte 1994] and of SLG [Verbaeten et al. 2001] exist. Floundering can be analyzed by means of abstract interpretation [=-=Cousot and Cousot 1-=-977], more specically by groundness analysis [Marriott and Sndergaard 1993]. Of course, due to the undecidability results, these techniques cannot be complete. 5.3 Inductive denitions in the absence o... |

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Citation Context ... with work on the perfect model semantics for stratied programs [Apt et al. 1988; Van Gelder 1988], which was extended to locally stratied [Przymusinski 1988] and weakly stratied [Przymusinska and Prz=-=ymusinski 199-=-0] programs. This direction culminated in the well-founded semantics which denes a unique (possibly 3-valued) model for all normal programs [Van Gelder et al. 1991]. A third major direction was motiva... |

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Citation Context ... transformations has been conducted by Brass and Dix [1999] who showed that several classes of semantics can be characterized in these terms. Another familiar example is the vanilla meta-interpreter [=-=Bowen and Kowalski 1982-=-] which consists of the following rules: demo(true). demo((P,Q)):-demo(P),demo(Q). demo(P):-atomic(P),clause(P:-Q),demo(Q). demo(not P):- not demo(P). This program induces a transformation of a normal... |

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Citation Context ... then not only should compute answers to a query but also check the consistency of the program, that is the fact that the well-founded model is total. The latter is in general an undecidable problem [=-=Schlipf 1995-=-a] 14 . Even for programs for which it is feasible to prove that they are total, the cost of doing so could be prohibitive 15 . Moreover, in some complex applications, partial models simply cannot be ... |

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Citation Context ... induction, e.g. in [Moschovakis 1974], tend to be for dening highly abstract concepts in set theory. Although in ationary induction is expressive [Moschovakis 1974; Kolaitis and Papadimitriou 1991; G=-=urevich and Shelah 198-=-6], it turns out to be very dicult to use it to encode even simple concepts. This is illustrated by Van Gelder [1993] with a discussion of the denition of the complement of the transitive closure of a... |

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Citation Context ...e applications of inflationary induction, e.g., in Moschovakis [1974], tend to be for defining highly abstract concepts in set theory. Although inflationary induction is expressive [Moschovakis 1974; =-=Kolaitis and Papadimitriou 1991-=-; Gurevich and Shelah 1986], it turns out to be very difficult to use it to encode even simple concepts. This is illustrated by Van Gelder [1993] with a discussion of the definition of the complement ... |

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Citation Context ...tegy followed by SLD-proof procedures as explained in the seminal Kowalski [1974] paper. Meanwhile, the group of Colmerauer developed a programming language along the same lines and called it Prolog [=-=Colmerauer et al. 1973-=-] as abbreviation for PROgrammation en LOGique. Many researchers were attracted by the new paradigm: application programmers by the ability to program at a, until then, unprecedented level of abstract... |

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(Show Context)
Citation Context ...th work on the perfect model semantics for stratified programs [Apt et al. 1988; Van Gelder 1988], which was extended to locally stratified [Przymusinski 1988] and weakly stratified [Przymusinska and =-=Przymusinski 1990-=-] programs. This direction culminated in the well-founded semantics which defines a unique (possibly three-valued) model for all normal programs [Van Gelder et al. 1991]. A third major direction was m... |

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55 | Semantics of (disjunctive) logic programs based on partial evaluation
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(Show Context)
Citation Context ...ogram is its perfect model. Second, many transformations of the type illustrated in the previous section which may transform stratied into unstratied programs, preserve the wellfounded model | see [Br=-=ass and Dix 199-=-9]. The third argument is based on the analogy between the well-founded semantics and the semantic principle used in IID and stratied logic programming. Przymusinski [1989a] showed that each logic pro... |

55 | Precise and efficient groundness analysis for logic programs - Marriott, Søndergaard - 1993 |

54 | The Family of Stable Models
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(Show Context)
Citation Context ...8 Marc Denecker et al. Until recently, nosxpoint theory for general (monotone or non-monotone) operators was known that was modeling the principle of iterated induction. Building on Fitting's work [F=-=itting 1993-=-] on semantics of Logic Programming in bilattices, [Denecker et al. 2000] developed Approximation Theory, an algebraicsxpoint theory for general (monotone and non-monotone) operators in a lattice. Thi... |

52 | Stable semantics for logic programs and default reasoning - Marek, Truszczynski - 1999 |

49 |
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(Show Context)
Citation Context ...ions, 3-valued well-founded semantics offers an answer to an old critique on classical logic, namely that it collapses totally in the case of inconsistency. 5.2 Computational aspects As discussed in [=-=Apt and Blair 1990-=-], the perfect model and hence the well-founded model is not recursively enumerable for all programs and no effective proof procedure is possible for the general case. However computation is effective... |

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43 | The Well-founded Semantics Is the Principle of Inductive De nition
- Denecker
- 1998
(Show Context)
Citation Context ...la. The problem is that this encoding is extremely tedious and blurs the simple intuitions behind this work. For more details on the encoding of iterated inductive denitions, we refer the reader to [D=-=enecker 1-=-998]. In ationary inductive denitions and Iterated Inductive denitions are not equivalent and are based on very dierent intuitions. At present, there is no standard well-motivated treatment of non-mon... |

43 |
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(Show Context)
Citation Context ... then not only should compute answers to a query but also check the consistency of the program, that is the fact that the well-founded model is total. The latter is in general an undecidable problem [=-=Schlipf 1995-=-a] 14 . Even for programs for which it is feasible to prove that they are total, the cost of doing so could be prohibitive 15 . Moreover, in some complex applications, partial models simply cannot be ... |