## Default reasoning over domains and concept hierarchies (2004)

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Venue: | In Proceedings of the 27th German conference on Artificial Intelligence (KI-04 |

Citations: | 8 - 4 self |

### BibTeX

@INPROCEEDINGS{Hitzler04defaultreasoning,

author = {Pascal Hitzler},

title = {Default reasoning over domains and concept hierarchies},

booktitle = {In Proceedings of the 27th German conference on Artificial Intelligence (KI-04},

year = {2004},

pages = {351--365},

publisher = {Springer}

}

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### Abstract

Abstract. W.C. Rounds and G.-Q. Zhang have proposed to study a form of disjunctive logic programming generalized to algebraic domains [1]. This system allows reasoning with information which is hierarchically structured and forms a (suitable) domain. We extend this framework to include reasoning with default negation, giving rise to a new nonmonotonic reasoning framework on hierarchical knowledge which encompasses answer set programming with extended disjunctive logic programs. We also show that the hierarchically structured knowledge on which programming in this paradigm can be done, arises very naturally from formal concept analysis. Together, we obtain a default reasoning paradigm for conceptual knowledge which is in accordance with mainstream developments in nonmonotonic reasoning. 1

### Citations

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Citation Context ...inimally closed by rules in P # (and thus is the only answer set of P # ), then P does not have any models. In particular, Theorem 1 shows that the minimal model semantics for definite logic programs =-=[16]-=- can be recovered using the original approach from [1] without default negation. Likewise, the same holds for the stable model semantics for normal logic programs [17], which are non-disjunctive ones ... |

1492 | The stable model semantics for logic programming
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Citation Context ...tics for definite logic programs [16] can be recovered using the original approach from [1] without default negation. Likewise, the same holds for the stable model semantics for normal logic programs =-=[17]-=-, which are non-disjunctive ones without negation . 5 Formal Concept Analysis Formal concept analysis is a powerful lattice-based approach to symbolic data analysis. It was devised in the 1980s [18] a... |

1423 |
A Logic for Default Reasoning
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Citation Context ...ation. We are motivated by the gain in expressiveness through the use of negation in artificial intelligence paradigms related to nonmonotonic reasoning. This approach, using ideas from default logic =-=[3]-=-, treats negation under the intuition that the negation of an item 2 Pascal Hitzler shall be believed if there is no reason to believe the item itself. This perspective on negation has recently led to... |

847 | Classical negation in logic programs and disjunctive databases
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- 1991
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Citation Context ...ls [4, 5]. We will indeed see that the extension of the approach by Rounds and Zhang by default negation is a natural generalization of answer set programming with extended disjunctive logic programs =-=[6]-=-. On the other hand, building on the work reported in [7], we establish a strong connection between the clausal logic on algebraic domains mentioned above, and fundamental notions from formal concept ... |

811 | Formal Concept Analysis: Mathematical Foundations - GANTER, WILLE - 1999 |

456 | Domain Theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ... in the form of answer set programming, on the other. The relationships thus worked out serve a threefold purpose, namely (1) to obtain a sound domain-theoretic perspective on answer set programming, =-=(2)-=- to provide a formal link between domain logics and formal concept analysis for the purpose of cross-transfer of methods and results, and (3) to devise a reasoning paradigm which encompasses two forme... |

410 |
Restructuring lattice theory: An approach based on hierarchies of concepts
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- 1981
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Citation Context ...s [17], which are non-disjunctive ones without negation . 5 Formal Concept Analysis Formal concept analysis is a powerful lattice-based approach to symbolic data analysis. It was devised in the 1980s =-=[18]-=- and was originally inspired by ideas from philosophy, more precisely by Port Royal Logic, which describes a conceptsas consisting of a set of objects (the extent of the concept) and a set of attribut... |

310 | Extending and Implementing the Stable Model Semantics
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- 2002
(Show Context)
Citation Context ...tive on negation has recently led to the development of applications in the form of nonmonotonic reasoning systems known as answer set programming, the two most popular probably being dlv and smodels =-=[4, 5]-=-. We will indeed see that the extension of the approach by Rounds and Zhang by default negation is a natural generalization of answer set programming with extended disjunctive logic programs [6]. On t... |

259 |
Domains for denotational semantics
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(Show Context)
Citation Context ...eferees on an earlier version of this paper. 2 Clausal Logic and Logic Programming in Algebraic Domains The study of domain theory from a logical perspective has a long tradition, and originates from =-=[9]-=-, where a logical characterization (more precisely, a categorical equivalence) of bounded complete algebraic cpo's (with Scott continuous functions as morphisms) was given. Rounds and Zhang [1] have r... |

102 | A Deductive System for Nonmonotonic Reasoning
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(Show Context)
Citation Context ...tive on negation has recently led to the development of applications in the form of nonmonotonic reasoning systems known as answer set programming, the two most popular probably being dlv and smodels =-=[4, 5]-=-. We will indeed see that the extension of the approach by Rounds and Zhang by default negation is a natural generalization of answer set programming with extended disjunctive logic programs [6]. On t... |

88 |
Description logics of minimal knowledge and negation as failure
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(Show Context)
Citation Context ...efault reasoning on concept hierarchies has also been studied before, for example in the form of default reasoning in semantic networks, e.g. [31], and as nonmonotonic reasoning with ontologies, e.g. =-=[32, 33]-=-. Since ontology creation is a currently evolving area of application for formal concept analysis, we expect that our paradigm will also be useful for similar purposes. Another related paradigm is log... |

65 | Disjunctive logic programs with inheritance, in
- Buccafurri, Faber, et al.
- 1999
(Show Context)
Citation Context ...urrently evolving area of application for formal concept analysis, we expect that our paradigm will also be useful for similar purposes. Another related paradigm is logic programming with inheritance =-=[34]-=-, where the underlying order structures are is-a hierarchies, which do not have a similarly rich logical structure as the logic RZ or Galois subhierarchies of formal concept lattices. 7 Conclusions an... |

49 |
T ω as a universal domain
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- 1978
(Show Context)
Citation Context ...# t. This induces a pointwise ordering on the space T V of all interpretations (or partial truth assignments). The partially ordered set T V is a coherent algebraic cpo 2 and has been studied e.g. in =-=[11]-=- in a domain-theoretic context, and in [12] in a logic programming context. Compact elements in T V are those interpretations which map all but a finite number of propositional variables to u. We deno... |

46 |
Local compactness and continuous lattices
- Hofmann, Mislove
(Show Context)
Citation Context ...on of all non-empty Scott-compact saturated subsets of D, ordered by reverse inclusion --- and the latter is isomorphic to the Smyth powerdomain of D. This result rests on the Hofmann-Mislove theorem =-=[10]-=-. It is also shown that a theory is logically closed if and only if it is an ideal, 1 and also that a clause is a logical consequence of a theory T if and only if it is a logical consequence of a fini... |

31 |
A Kripke-Kleene-semantics for general logic programs
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Citation Context ...he space T V of all interpretations (or partial truth assignments). The partially ordered set T V is a coherent algebraic cpo 2 and has been studied e.g. in [11] in a domain-theoretic context, and in =-=[12]-=- in a logic programming context. Compact elements in T V are those interpretations which map all but a finite number of propositional variables to u. We denote compact elements by strings such as pqr,... |

30 |
Embedding defaults into terminological representation systems, J. Automated Reasoning 14
- Baader, Hollunder
- 1995
(Show Context)
Citation Context ...efault reasoning on concept hierarchies has also been studied before, for example in the form of default reasoning in semantic networks, e.g. [31], and as nonmonotonic reasoning with ontologies, e.g. =-=[32, 33]-=-. Since ontology creation is a currently evolving area of application for formal concept analysis, we expect that our paradigm will also be useful for similar purposes. Another related paradigm is log... |

27 | Generalized metrics and uniquely determined logic programs
- Hitzler, Seda
(Show Context)
Citation Context ...Relationships between domain theory and nonmonotonic reasoning have hardly been studied in the literature, except from series of papers by Rounds and Zhang, e.g. [1,26,27], and Hitzler and Seda, e.g. =-=[28,29,30]-=-. This is remarkable since domain theory has become a respected paradigm in the theory of computing with widespread applications. We believe that this relationship deserves much more attention in orde... |

22 |
Default Reasoning in Semantic Networks: A Formalization of Recognition and Inheritance
- Shastri
- 1989
(Show Context)
Citation Context ...c reasoning and other artificial intelligence paradigms. Default reasoning on concept hierarchies has also been studied before, for example in the form of default reasoning in semantic networks, e.g. =-=[31]-=-, and as nonmonotonic reasoning with ontologies, e.g. [32, 33]. Since ontology creation is a currently evolving area of application for formal concept analysis, we expect that our paradigm will also b... |

17 | Clausal logic and logic programming in algebraic domains
- Rounds, Zhang
(Show Context)
Citation Context ... Department of Computer Science, Dresden University of Technology Abstract. W.C. Rounds and G.-Q. Zhang have proposed to study a form of disjunctive logic programming generalized to algebraic domains =-=[1]-=-. This system allows reasoning with information which is hierarchically structured and forms a (suitable) domain. We extend this framework to include reasoning with default negation, giving rise to a ... |

17 |
R.: Contextual attribute logic
- Ganter, Wille
- 1999
(Show Context)
Citation Context ...onotonic aspects which are di#cult to deal with e#ciently and e#ectively. 6 Related Work Logical aspects of formal concept analysis have certainly received ample attention in the literature, see e.g. =-=[19, 20]-=-. In particular, the contextual attribute logic due to Ganter and Wille [19] is closely related to our results in Section 5, and for the finite case this was spelled out in [7]. The study of relations... |

17 | Topology and iterates in computational logic
- Seda, Hitzler
- 1997
(Show Context)
Citation Context ...Relationships between domain theory and nonmonotonic reasoning have hardly been studied in the literature, except from series of papers by Rounds and Zhang, e.g. [1,26,27], and Hitzler and Seda, e.g. =-=[28,29,30]-=-. This is remarkable since domain theory has become a respected paradigm in the theory of computing with widespread applications. We believe that this relationship deserves much more attention in orde... |

15 |
spaces, concept lattices, and domains
- Chu
- 1922
(Show Context)
Citation Context ...ts in Section 5, and for the finite case this was spelled out in [7]. The study of relationships between formal concept analysis and domain theory has only recently received attention. Zhang and Shen =-=[21, 22]-=- approach the issue from the perspective of Chu spaces and Scott information systems. A category-theoretical setting was developed from these investigations in [23]. The work just mentioned has a di#e... |

12 | Approximable concepts, Chu spaces and information systems. Theory and Application of Categories
- Zhang, Shen
- 2006
(Show Context)
Citation Context ...ts in Section 5, and for the finite case this was spelled out in [7]. The study of relationships between formal concept analysis and domain theory has only recently received attention. Zhang and Shen =-=[21, 22]-=- approach the issue from the perspective of Chu spaces and Scott information systems. A category-theoretical setting was developed from these investigations in [23]. The work just mentioned has a di#e... |

10 |
Bridges between classical and nonmonotonic logic
- Makinson
- 2003
(Show Context)
Citation Context ...that this imprecise reasoning mechanism (amongst other things) allows us to deal with the world e#ectively. Formally, jumping to conclusions can be studied by investigating supraclassical logics, see =-=[15]-=-, where supraclassicality means, roughly speaking, that under such a logic more conclusions can be drawn from a set of axioms (or knowledge base) than could be drawn using classical (e.g. propositiona... |

8 | Some issues concerning fixed points in computational logic: Quasi-metrics, multivalued mappings and the Knaster-Tarski theorem
- Hitzler, Seda
- 1999
(Show Context)
Citation Context ...Relationships between domain theory and nonmonotonic reasoning have hardly been studied in the literature, except from series of papers by Rounds and Zhang, e.g. [1,26,27], and Hitzler and Seda, e.g. =-=[28,29,30]-=-. This is remarkable since domain theory has become a respected paradigm in the theory of computing with widespread applications. We believe that this relationship deserves much more attention in orde... |

7 |
W.C.: Semantics of logic programs and representation of Smyth powerdomains
- Zhang, Rounds
- 2001
(Show Context)
Citation Context ...be worked out how it relates to ours. Relationships between domain theory and nonmonotonic reasoning have hardly been studied in the literature, except from series of papers by Rounds and Zhang, e.g. =-=[1, 26, 27]-=-, and Hitzler and Seda, e.g. [28--30]. This is remarkable since domain theory has become a respected paradigm in the theory of computing with widespread applications. We believe that this relationship... |

6 | A cartesian closed category of approximable concept structures
- Hitzler, Zhang
(Show Context)
Citation Context ...eived attention. Zhang and Shen [21, 22] approach the issue from the perspective of Chu spaces and Scott information systems. A category-theoretical setting was developed from these investigations in =-=[23]-=-. The work just mentioned has a di#erent focus than our result in Section 5 and [7], but develops along similar basic intuitions and is mainly compatible with ours. Its flavour is more category-theore... |

5 | M.: Formal concept analysis and resolution in algebraic domains
- Hitzler, Wendt
(Show Context)
Citation Context ...pproach by Rounds and Zhang by default negation is a natural generalization of answer set programming with extended disjunctive logic programs [6]. On the other hand, building on the work reported in =-=[7]-=-, we establish a strong connection between the clausal logic on algebraic domains mentioned above, and fundamental notions from formal concept analysis [8]. More precisely, we will see that in certain... |

4 | A logical approach to datadriven classification
- Osswald, Petersen
- 2003
(Show Context)
Citation Context ...lar basic intuitions and is mainly compatible with ours. Its flavour is more category-theoretical and targets categorial constructions which may be used for ontology engineering. Osswald and Petersen =-=[24, 25]-=- study an approach to encoding knowledge in order structures which is inspired from linguistics. They obtain a framework which is more flexible than formal concept analysis, and appears to be compatib... |

4 |
Reasoning with power defaults (preliminary report
- Zhang, Rounds
- 1997
(Show Context)
Citation Context ...be worked out how it relates to ours. Relationships between domain theory and nonmonotonic reasoning have hardly been studied in the literature, except from series of papers by Rounds and Zhang, e.g. =-=[1, 26, 27]-=-, and Hitzler and Seda, e.g. [28--30]. This is remarkable since domain theory has become a respected paradigm in the theory of computing with widespread applications. We believe that this relationship... |

3 |
A resolution theorem for algebraic domains
- HITZLER
- 2003
(Show Context)
Citation Context ...normal form, e.g. {pqr, pq, r} translates to (p # q # r) # (p # q) # r. In [1], it was shown that the logic RZ is compact. A proof theory for it was also given. An alternative version was reported in =-=[13, 14]-=-. The logic RZ provides a framework for reasoning with disjunctive information on a lattice which encodes background knowledge. Indeed it was shown [7] that it relates closely to formal concept analys... |

2 |
Boolean judgement logic
- Wille
- 2001
(Show Context)
Citation Context ...onotonic aspects which are di#cult to deal with e#ciently and e#ectively. 6 Related Work Logical aspects of formal concept analysis have certainly received ample attention in the literature, see e.g. =-=[19, 20]-=-. In particular, the contextual attribute logic due to Ganter and Wille [19] is closely related to our results in Section 5, and for the finite case this was spelled out in [7]. The study of relations... |

1 | A generalized resolution theorem
- Hitzler
(Show Context)
Citation Context ...normal form, e.g. {pqr, pq, r} translates to (p # q # r) # (p # q) # r. In [1], it was shown that the logic RZ is compact. A proof theory for it was also given. An alternative version was reported in =-=[13, 14]-=-. The logic RZ provides a framework for reasoning with disjunctive information on a lattice which encodes background knowledge. Indeed it was shown [7] that it relates closely to formal concept analys... |

1 |
conditionals, and defaults
- Osswald
(Show Context)
Citation Context ...lar basic intuitions and is mainly compatible with ours. Its flavour is more category-theoretical and targets categorial constructions which may be used for ontology engineering. Osswald and Petersen =-=[24, 25]-=- study an approach to encoding knowledge in order structures which is inspired from linguistics. They obtain a framework which is more flexible than formal concept analysis, and appears to be compatib... |