## A Cartesian Closed Category of Approximable Concept Structures (2004)

### Cached

### Download Links

- [www.wv.inf.tu-dresden.de]
- [www.wv.inf.tu-dresden.de]
- [newton.case.edu]
- [newton.cwru.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | Proceedings of the International Conference On Conceptual Structures |

Citations: | 6 - 4 self |

### BibTeX

@INPROCEEDINGS{Hitzler04acartesian,

author = {Pascal Hitzler and Guo-qiang Zhang},

title = {A Cartesian Closed Category of Approximable Concept Structures},

booktitle = {Proceedings of the International Conference On Conceptual Structures},

year = {2004},

pages = {170--185},

publisher = {Springer}

}

### OpenURL

### Abstract

Infinite contexts and their corresponding lattices are of theoretical and practical interest since they may o#er connections with and insights from other mathematical structures which are normally not restricted to the finite cases. In this paper we establish a systematic connection between formal concept analysis and domain theory as a categorical equivalence, enriching the link between the two areas as outlined in [25].

### Citations

819 |
Formal concept analysis: Mathematical foundations
- Ganter, Wille
- 1999
(Show Context)
Citation Context ...ork was carried out while the author was visiting the second author at the Department of EECS, Case Western Reserve University 2 . ⋆⋆ Corresponding author.s1 Introduction Formal concept analyais (=-=FCA [1]) is-=- a powerful lattice-based tool for symbolic data analysis. In essence, it is based on the extraction of a lattice — called formal concept lattice — from a binary relation called formal context con... |

561 | Lattice Theory - Birkhoff - 1967 |

412 |
Restructuring lattice theory: an approach based on hierarchies of concepts
- Wille
- 1982
(Show Context)
Citation Context ...or FCA, a proof of which can be found in [1]. Recall that a complete lattice is a partial order in which all (possibly infinite) suprema (a.k.a. join) and infima (a.k.a. meet) exist. Theorem 1 (Wille =-=[24]).-=- The set of all (intent-) concepts of a formal context P, ordered under subset inclusion, is a complete lattice, called the concept lattice of P. Furthermore, every complete lattice (D, ≤) is isomor... |

312 | T.: Extending and Implementing the Stable Model Semantics
- Simons, Niemelä, et al.
- 2002
(Show Context)
Citation Context ...atter work has also led to the proposal of a non-monotonic reasoning paradigm on (possibly infinite) formal contexts [9], which is in the spirit of the recent evolving answer set programming paradigm =-=[6, 20]. -=-Also worth noting are potential connections between our work and that of Lamarche [11] and Plotkin [14]. The work reported here can be viewed as part of a unique research program [8, 25–27,19] explo... |

261 |
Domains for denotational semantics
- Scott
- 1982
(Show Context)
Citation Context ...e extracted from a context by letting X ⊢ a just when a ∈ α(ω(X)). This has been observed by many researchers and investigated at a more systematic level in [25] by relating it to information sy=-=stems [18]. -=-The relation ⊢ corresponds to an association rule in data mining and an instance of functional dependence in databases. If we build theories by taking attributes as atomic propositional formulas and... |

209 |
Information Flow: The Logic of Distributed Systems
- Barwise, Seligman
- 1997
(Show Context)
Citation Context ...ts at the same time. A rich collection of constructions including product and function space is immediately possible. In contrast, work in related areas such as Chu spaces [2, 15] or information flow =-=[3] h-=-ave traditionally focused on formal contexts alone; our work achieves an integration of concept lattices in the same framework. ⋆ The first author was supported by a Postdoc-fellowship of the German... |

174 | Formal Ontology, Conceptual Analysis and Knowledge Representation. International Journal of Human and Computer
- Guarino
- 1995
(Show Context)
Citation Context ...lied as a formal method for automated web-menu design, where the top layers of the concept lattice naturally provide a menu hierarchy for the navigation of a website. In ontological engineering (e.g. =-=[7,21]-=-), although lattices have been proposed as mathematical structures for representing ontology, FCA provides a scientific and algorithmic basis for it, as well as an understanding that lattice structure... |

102 | A Deductive System for Nonmonotonic Reasoning
- Eiter, Leone, et al.
- 1997
(Show Context)
Citation Context ...atter work has also led to the proposal of a non-monotonic reasoning paradigm on (possibly infinite) formal contexts [9], which is in the spirit of the recent evolving answer set programming paradigm =-=[6, 20]. -=-Also worth noting are potential connections between our work and that of Lamarche [11] and Plotkin [14]. The work reported here can be viewed as part of a unique research program [8, 25–27,19] explo... |

101 |
Topology via logic
- Vickers
- 1989
(Show Context)
Citation Context ...are allowed. It is well-known in logic that an infinite conjunction p in the antecedent of an entailment p → q destroys compactness. For readers interested in discussions along this line, we refer t=-=o [23]-=- for general and intuitive examples, and to [10] for related hardcore theory. We now summarize relevant results from [26]. Recall that a complete lattice is called algebraic if each of its element is ... |

96 |
Handbook of categorical algebra 1. Basic category theory. Encyclopedia of Mathematics and its applications
- Borceux
- 1994
(Show Context)
Citation Context ...nifested in its ability in guiding research to the discovery of categorically natural, but otherwise non-obvious missing entities in a conceptual picture. Our category-theoretical terminology follows =-=[5]. A ca-=-tegory C consists of (i) a class |C| of objects of the category, (ii) for all A, B ∈ |C|, a set C(A, B) of morphisms from A to B, (iii) for all A, B, C ∈ |C|, a composition operation ◦ : C(B, C)... |

94 |
Knowledge Representation: Logical
- Sowa
- 2000
(Show Context)
Citation Context ...lied as a formal method for automated web-menu design, where the top layers of the concept lattice naturally provide a menu hierarchy for the navigation of a website. In ontological engineering (e.g. =-=[7,21]-=-), although lattices have been proposed as mathematical structures for representing ontology, FCA provides a scientific and algorithmic basis for it, as well as an understanding that lattice structure... |

86 |
autonomous categories and linear logic
- Barr
- 1991
(Show Context)
Citation Context ...otion of approximable concepts at the same time. A rich collection of constructions including product and function space is immediately possible. In contrast, work in related areas such as Chu spaces =-=[2, 15] o-=-r information flow [3] have traditionally focused on formal contexts alone; our work achieves an integration of concept lattices in the same framework. ⋆ The first author was supported by a Postdoc-... |

54 |
Using Information Systems to Solve Recursive Domain Equations Effectively", LNCS 173
- Larsen
- 1984
(Show Context)
Citation Context ... is eqivalent to the category of Scott information systems with trivial consistency predicate and approximable mappings as morphisms. The corresponding definitions are as follows, and can be found in =-=[12, 17,18,22,26,28]. An in-=-formation system (with trivial consistency predicate) A is a pair (A, ⊢A), where A is the token set and ⊢A Fin(A) ×Fin(A) is the entailment relation, and furthermore the conditions (is1) a ∈ X ... |

37 |
Logic of Domains
- Zhang
- 1991
(Show Context)
Citation Context ... is eqivalent to the category of Scott information systems with trivial consistency predicate and approximable mappings as morphisms. The corresponding definitions are as follows, and can be found in =-=[12, 17,18,22,26,28]. An in-=-formation system (with trivial consistency predicate) A is a pair (A, ⊢A), where A is the token set and ⊢A Fin(A) ×Fin(A) is the entailment relation, and furthermore the conditions (is1) a ∈ X ... |

35 |
The Category of Constraint Systems is Cartesian-Closed
- Saraswat
- 1992
(Show Context)
Citation Context ... is eqivalent to the category of Scott information systems with trivial consistency predicate and approximable mappings as morphisms. The corresponding definitions are as follows, and can be found in =-=[12, 17,18,22,26,28]. An in-=-formation system (with trivial consistency predicate) A is a pair (A, ⊢A), where A is the token set and ⊢A Fin(A) ×Fin(A) is the entailment relation, and furthermore the conditions (is1) a ∈ X ... |

33 |
Categories for the working mathematician, Springer Graduate Texts
- MacLane
- 1971
(Show Context)
Citation Context ... main contribution of this paper is the introduction of an appropriate notion of morphisms on formal contexts and the proof that the following three categories are equivalent in the categorical sense =-=[13]-=-: 1. the category of formal contexts and context morphisms Cxt introduced here for the first time, 2. the category of complete algebraic lattices and Scott continuous functions, and 3. the category of... |

17 | Clausal logic and logic programming in algebraic domains
- Rounds, Zhang
(Show Context)
Citation Context ... generality with the corresponding structure being the complete algebraic lattices. Work along a similar line for the finite case was spelled out in [8] in a different guise using the logic RZ due to =-=[16]-=-. The latter work has also led to the proposal of a non-monotonic reasoning paradigm on (possibly infinite) formal contexts [9], which is in the spirit of the recent evolving answer set programming pa... |

12 | Approximable concepts, Chu spaces and information systems. Theory and Application of Categories
- Zhang, Shen
- 2006
(Show Context)
Citation Context ... items of knowledge or information should either be finitely representable or approximable by such finitely representable items. This idea motivated a new notion by Zhang and Shen in a separate paper =-=[26]-=-, called approximable concept. Approximable concept lattices derived from this notion are exactly the complete algebraic ones; and every (classical) formal concept is approximable. Furthermore, in cas... |

8 | Default Reasoning over Domains and Concept Hierarchies
- Hitzler
(Show Context)
Citation Context ...e was spelled out in [8] in a different guise using the logic RZ due to [16]. The latter work has also led to the proposal of a non-monotonic reasoning paradigm on (possibly infinite) formal contexts =-=[9]-=-, which is in the spirit of the recent evolving answer set programming paradigm [6, 20]. Also worth noting are potential connections between our work and that of Lamarche [11] and Plotkin [14]. The wo... |

8 | Chu Spaces from the Representational Viewpoint. Volume of Annals of Pure and Applied Logic honoring R
- Pratt
- 1999
(Show Context)
Citation Context ...otion of approximable concepts at the same time. A rich collection of constructions including product and function space is immediately possible. In contrast, work in related areas such as Chu spaces =-=[2, 15] o-=-r information flow [3] have traditionally focused on formal contexts alone; our work achieves an integration of concept lattices in the same framework. ⋆ The first author was supported by a Postdoc-... |

6 |
Chu spaces, formal concepts, and domains
- Zhang
(Show Context)
Citation Context ...inite cases. In this paper we establish a systematic connection between formal concept analysis and domain theory as a categorical equivalence, enriching the link between the two areas as outlined in =-=[25]-=-. Domain theory is a subject of extensive study in theoretical computer science and programming languages. Its basic idea of partial information and successive approximation suggests that for infinite... |

5 | M.: Formal concept analysis and resolution in algebraic domains
- Hitzler, Wendt
(Show Context)
Citation Context ...systems and formal contexts was established in full generality with the corresponding structure being the complete algebraic lattices. Work along a similar line for the finite case was spelled out in =-=[8]-=- in a different guise using the logic RZ due to [16]. The latter work has also led to the proposal of a non-monotonic reasoning paradigm on (possibly infinite) formal contexts [9], which is in the spi... |

2 |
From Chu spaces to cpos. Theory and Formal Methods of Computing 94
- Lamarche
- 1994
(Show Context)
Citation Context ...infinite) formal contexts [9], which is in the spirit of the recent evolving answer set programming paradigm [6, 20]. Also worth noting are potential connections between our work and that of Lamarche =-=[11] a-=-nd Plotkin [14]. The work reported here can be viewed as part of a unique research program [8, 25–27,19] exploiting the synergies among some recurring themes in several independently developing and ... |

2 |
Concept lattices, clustering, and visualization using L ATEX
- Shen, Tian, et al.
- 2004
(Show Context)
Citation Context ...is cartesian closed, and as a result a rich collection of constructions including product and function space is immediately made possible. Our paper can be viewed as part of a unique research program =-=[8,25,26, 27,19]-=- exploiting the synergies among the following recurring themes in several independently developing and yet somewhat related areas: – functional dependency X → Y in databases, – association rules X ⇒ Y... |

1 |
Notes on the Chu construction and recursion, manuscript
- Plotkin
- 1993
(Show Context)
Citation Context ...contexts [9], which is in the spirit of the recent evolving answer set programming paradigm [6, 20]. Also worth noting are potential connections between our work and that of Lamarche [11] and Plotkin =-=[14]. -=-The work reported here can be viewed as part of a unique research program [8, 25–27,19] exploiting the synergies among some recurring themes in several independently developing and yet somewhat rela... |

1 |
A note on strongly finite sequent structures
- Spreen
- 2004
(Show Context)
Citation Context |