## Extending Classical Logic with Inductive Definitions (2000)

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### BibTeX

@INPROCEEDINGS{Denecker00extendingclassical,

author = {Marc Denecker},

title = {Extending Classical Logic with Inductive Definitions},

booktitle = {},

year = {2000},

pages = {703--717},

publisher = {Springer}

}

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### Abstract

The goal of this paper is to extend classical logic with a generalized notion of inductive definition supporting positive and negative induction, to investigate the properties of this logic, its relationships to other logics in the area of non-monotonic reasoning, logic programming and deductive databases, and to show its application for knowledge representation by giving a typology of definitional knowledge.

### Citations

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Citation Context ...in logic programs is interpreted as classical negation but the implication operator gets a stronger meaning than classical implication and is not truth-functional. The formalisms of logic programming =-=[20]-=-, abductive logic programming (ALP) [19], open logic programming [10], when interpreted under well-founded semantics, can be trivially embedded in ID-logic. For example, an abductive logic program und... |

1603 | Foundations of Databases - Abiteboul, Hull, et al. - 1996 |

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876 | J.S.: The well-founded semantics for general logic programs - Gelder, Ross, et al. - 1991 |

856 | Circumscription: A form of nonmonotonic reasoning - McCarthy - 1980 |

734 | Predicate logic as programming language - Kowalski - 1974 |

577 | The frame problem in the situation calculus: a simple solution (sometimes) and a completeness result for goal regression - Reiter - 1991 |

547 | The role of abduction in logic programming - Kakas, Kowalski, et al. - 1998 |

391 | A Kripke-Kleene Semantics for Logic Programs - Fitting - 1985 |

216 | The alternating fixpoint of logic programs with negation
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(Show Context)
Citation Context ...3-valued) model of the classical logic sentences of T . An interpretation I is a model of a denition D, resp. theory T , i it is a total (i.e. 2-valued) justied interpretation of D, resp. T . 7 In [16], an alternative way of dening the well-founded semantics of predicate rules is proposed; it is based on a dierent treatment of positive and negative occurrences of predicates in the body of rules. ... |

185 | An introduction to inductive definitions - Aczel - 1977 |

179 |
Nonmonotonic Logics -- Context-Dependent Reasoning
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(Show Context)
Citation Context ...embeddings show that logic programming and its extensions with disjunction and explicit negation can be interpreted in dierent ways as default theories or autoepistemic theories (for an overview see [=-=21-=-]). ID-logic provides an alternative knowledge theoretic reading for a dierent class of Logic Programming formalisms. While in default or autoepistemic embeddings, the implication symbol in rules is i... |

175 | Acyclic programs - Apt, Bezem - 1991 |

174 | Elementary Induction on Abstract Structures - Moschovakis - 1974 |

138 | Well-founded semantics coincides with three-valued stable-semantics, Fund - Przymusinski - 1990 |

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82 |
Fixed-point extensions of firstorder logic
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Citation Context ...nitions of n-ary concepts has been investigated in mathematical logic; the standard treatments [24, 2] focus on the theoretical expressivity of this principle. One spin-o of thisseld issxpoint logic [=-=18]-=-, subarea of databases [1]. From a knowledge representation point of view, the treatment of (inductive) definitions in this area still imposes strong limitations. Typically, there is no concern with c... |

80 | Elaboration tolerance - McCarthy - 1998 |

60 | The future of knowledge representation - Brachman - 1990 |

43 | The Well-founded Semantics Is the Principle of Inductive De nition - Denecker - 1998 |

37 | Equality and Domain Closure in First-Order Databases - Reiter - 1980 |

34 | Formal theories for transfinite iterations of generalized inductive definitions and some subsystems of analysis - Feferman - 1970 |

34 | A lattice-theoretical theorem and its applications - Tarski - 1955 |

28 | Hypothetical reasoning with well founded semantics
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(Show Context)
Citation Context ...], open logic programming [10], when interpreted under well-founded semantics, can be trivially embedded in ID-logic. For example, an abductive logic program under the extended well-founded semantics =-=[25-=-] consists of a set of abducible predicates, a set of rules dening non-abducible predicates and a set of integrity constraints. Its embedding consists of a set of FOL axioms (the \integrity constraint... |

24 | A Terminological Interpretation of (Abductive) Logic Programming - Denecker - 1995 |

20 |
Logic-Based Knowledge Representation
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(Show Context)
Citation Context ...tation has been acknowledged and denitions are central in the area of description logics [7]. As an outcome of a series of investigations to the semantics (or lack of semantics) of semantic networks [=-=29-=-], Brachman and Levesque [6] observed that expert knowledge contains an important terminological component; this is the knowledge about the dening properties of the concepts and terminology used by ex... |

18 |
An introduction to inductive de
- Aczel
(Show Context)
Citation Context ...tandardly to non-inductive denitions of unary concepts. The more general class of positive inductive denitions of n-ary concepts has been investigated in mathematical logic; the standard treatments [2=-=4, 2-=-] focus on the theoretical expressivity of this principle. One spin-o of thisseld issxpoint logic [18], subarea of databases [1]. From a knowledge representation point of view, the treatment of (induc... |

18 | KRYPTON: Integrating terminology and assertion
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- 1983
(Show Context)
Citation Context ...ed to design a mixed knowledge representation language, consisting of a Tbox to represent denitional knowledge and an Abox to represent other assertional knowledge. Thesrst description logic Krypton [=-=-=-5] and its successors are based on this schema. 2 To assure computational tractability, the class of denitions expressible in Description logics is limited, standardly to non-inductive denitions of un... |

15 | An inductive definition approach to ramifications - Denecker, Theseider, et al. - 1998 |

14 | The alternating fixpoint of logic programs with negation - Gelder - 1989 |

12 | Inductive Definability and the Situation Calculus - Ternovskaia - 1998 |

12 |
Miroslaw Truszczynski. Approximating operators, stable operators, well-founded fixpoints and applications in non-monotonic reasoning
- Denecker, Marek
- 2000
(Show Context)
Citation Context ...s. In general, a model of an inductive denition is a partial (3-valued) interpretation. However, for broad classes of denitions, the well-founded model is known to be total (2-valued). Recently, in [1=-=3]-=- we proposed an algebraic theory ofsxpoints of non-monotone operators which associates with an arbitrary non-monotone operator a class of specialsstable and well-foundedsxpoints. The stable and well-f... |

11 |
Schreye. A strong correspondence between description logics and open logic programming
- Belleghem, Denecker, et al.
- 1997
(Show Context)
Citation Context ...ics, with a Tbox consisting of the denitions and an Abox consisting of classical logic axioms. The correspondence between description logics and a sub-logic of ID-logic were formally investigated in [=-=3-=-4]. Alsosxpoint logic can be embedded in ID-logic, modulo syntactic sugar. The dierence in syntactic sugar is that in ID-logic, the dened concepts are named by global predicate symbols, whereas insxpo... |

11 | Experiments for integration CLP and abduction - Denecker, Nuffelen - 1999 |

8 | A strong correspondence between description logics and open logic programming - Belleghem, Denecker, et al. - 1997 |

8 | Compositionality of normal open logic programs - Verbaeten, Denecker, et al. - 2000 |

8 |
Refining the structure of terminological systems: Terminology = schema + view
- Buchheit, Donini, et al.
- 1994
(Show Context)
Citation Context ...l problem of common sense knowledge representation. Also in A.I., the role of denitions in knowledge representation has been acknowledged and denitions are central in the area of description logics [7=-=]-=-. As an outcome of a series of investigations to the semantics (or lack of semantics) of semantic networks [29], Brachman and Levesque [6] observed that expert knowledge contains an important terminol... |

6 | Definability and commonsense reasoning - Amati, Aiello, et al. - 1997 |

5 | An inductive definition approach to ramifications. Linkoping - Denecker, Theseider-Dupré, et al. - 1998 |

5 | Fixed-point Extensions of First-Order Logic. Annals of Pure and Applied Logic 32:265–280 - Gurevich, Shelah - 1986 |

5 |
Equality and Domain Closure Databases
- Reiter, R
- 1980
(Show Context)
Citation Context ...a case: cause:f (A[y]; s; x)s[y; s; x] Theorem 5.1 A denition consisting of the above rules is well-founded in the collection of all interpretations that satisfy the Unique Names Axioms (UNA) axioms [=-=30-=-] and the second order induction axiom for the situation sort 10 . It follows from theorem 3.1 that the semantics of this inductive denition coincides with its Clark completion. Note that the completi... |

5 |
De Schreye, Compositionality of normal open logic programs
- Verbaeten, Denecker, et al.
(Show Context)
Citation Context ...itions P red 1 :: n C 1 o and P red 2 :: n C 2 o as the denition P red 1 [P red 2 :: n C 1 [ C 2 o . In general, substituting a pair of denitions by their composition is not equivalence preserving. [36] presents an extensive study of when merging denitions is equivalence preserving in the context of a sub-formalism of the logic dened here. One important example is that a denition hierarchy (Deni... |

4 | On the semantics of Stratified Databases - Przymusinski - 1988 |

4 |
An inductive de approach to rami Linkoping
- Denecker, Dupre, et al.
- 1998
(Show Context)
Citation Context ...c2; s) connected(c1; c2; s)sconnected(c1; c3; s) ^ connected(c3; c2; s) Also, similarly as in [33], ramication rules can be added to this theory. Inductive denitions as an approach to Causality. In [1=-=-=-1] we argued that inductive denitions are a suitable formalization of causality. Causality information is an example of constructive information. Eects and forces propagate in a dynamic system through... |

3 | Schreye. Representing Incomplete Knowledge - Denecker, De - 1995 |

3 | Knowledge Representation: an AI Perspecitive - Reichgelt - 1991 |

2 | Nonmonotonic Reasoning: Compiled vs Interpreted Theories. Distributed at the conference of Nonmonotonic Reasoning NMR96 as considerations for the panel discussion - Reiter - 1996 |