## Rough Concept Analysis – theory development in the Mizar system (2004)

Venue: | Proc. of MKM 2004, Lecture Notes in Computer Science 3119 |

Citations: | 3 - 0 self |

### BibTeX

@INPROCEEDINGS{Grabowski04roughconcept,

author = {Adam Grabowski and Christoph Schwarzweller},

title = {Rough Concept Analysis – theory development in the Mizar system},

booktitle = {Proc. of MKM 2004, Lecture Notes in Computer Science 3119},

year = {2004},

pages = {130--144}

}

### OpenURL

### Abstract

Abstract. Theories play an important role in building mathematical knowledge repositories. Organizing knowledge in theories is an obvious approach to cope with the growing number of definitions, theorems, and proofs. However, they are also a matter of subject on their own: developing a new piece of mathematics often relies on extending or combining already developed theories in this way reusing definitions as well as theorems. We believe that this aspect of theory development is crucial for mathematical knowledge management. In this paper we investigate the facilities of the Mizar system concerning extending and combing theories based on structure and attribute definitions. As an example we consider the formation of rough concept analysis out of formal concept analysis and rough sets. 1

### Citations

894 | R.: Formal Concept Analysis: Mathematical Foundations - Ganter, Wille - 1997 |

779 |
Isabelle/HOL — A Proof Assistant for Higher-Order Logic
- Nipkow, Paulson, et al.
- 2002
(Show Context)
Citation Context ...ances (the latter correspond to an aggregate in Mizar). Theory declarations allow theories to be encapsulated and instantiated copies of the implicitly imported theory are generated. In Isabelle/Isar =-=[NPW02]-=- there are mechanisms of merging both theories and locales. The latter modules correspond to a context rather which is a narrower notion than we dealt in this paper. In Theorema [Buc01] theories can b... |

443 |
Restructuring lattice theory: An approach based on hierarchies of concepts
- Wille
- 1982
(Show Context)
Citation Context ...1,Ken96]. Rough concept analysis is a synthesis of rough sets and formal concept analysis. Rough sets [Paw82] approximate objects with respect to an indiscernibility relation. Formal concept analysis =-=[Wil82]-=- is an attributebased approach for representation and analysis of data. The combination of both thus establishes a theory for representing and analyzing uncertain data. Both rough sets and formal conc... |

107 |
Equivalence relations and classes of abstraction
- Raczkowski, Sadowski
- 1990
(Show Context)
Citation Context ...ence and with_tolerance which state that the InternalRel of the underlying RelStr is an equivalence resp. a tolerance relation (where a tolerance relation is a total reflexive symmetric relation, see =-=[RS90]-=-). With such defined notions, the basic definitions are as follows: definition mode Approximation_Space is with_equivalence non empty RelStr; mode Tolerance_Space is with_tolerance non empty RelStr; e... |

22 |
On the structure of Mizar types
- Bancerek
- 2003
(Show Context)
Citation Context ...ry close to the language used by mathematicians), we will illustrate them by living examples from Rough Concept Analysis and skipping most technical details. For details of the Mizar type system, see =-=[Ban03]-=-. Below we enumerate ten features that proved their usability in merging the theories of RST and FCA. Though they are Mizar specific we claim that any mathematical knowledge should support the general... |

20 | F.: C-CoRN, the constructive coq repository at Nijmegen
- Cruz-Filipe, Geuvers, et al.
- 2004
(Show Context)
Citation Context ... usage. The larger the database is, the easier linkages and connections can be studied. Even if MML is known as the largest library of formalized mathematics, it has a number of disadvantages. C-CoRN =-=[Wie04]-=-, the Constructive Coq Repository at Nijmegen, uses experience gathered when building MML. The number of people involved in this project is rather small (about 15 in total) as their library is central... |

14 | Verification of the Miller-Rabin probabilistic primality test
- Hurd
- 2003
(Show Context)
Citation Context ...in the theory of topological groups, and so on. Thus almost every theory relies on a number of (more) elementary theories. This phenomenon also occurs in more computer science related areas, see e.g. =-=[Hur03]-=- where the verification of a ⋆ The work of the second author was partially supported by grant BW 5100-5-0147-4.s2 A. Grabowski, C. Schwarzweller probabilistic primality test algorithm required proving... |

12 | Rough concept analysis: A synthesis of rough sets and formal concept analysis - Kent - 1996 |

9 | Mathematical Knowledge Management in Mizar
- Rudnicki, Trybulec
- 2001
(Show Context)
Citation Context ...zzy and rough sets, can be more successful if both are coded in similar manner or if they can be lifted to a common formal platform. In this paper we address only issues of what Trybulec and Rudnicki =-=[RT01]-=- classify as abstract mathematics. In classical mathematics theories are hardly merged, but rather inherited. It is one of Trybulec’s recent suggestions to cut MML into classical (not using the notion... |

8 | Basic properties of rough sets and rough membership function
- Grabowski
- 2004
(Show Context)
Citation Context ... independently developed theories. The plan of the paper is as follows. In the next section we briefly introduce rough sets and formal concept analysis and review their Mizar formalization as done in =-=[Gra03]-=- and [Sch98]. Section 3 describes the combination of these two theories into rough concept analysis. Mizar language features to do so are presented. In the last two sections we discuss the Mizar appro... |

8 | Concept approximations based on rough sets and similarity measures - Saquer, Deogun - 2001 |

7 | The Tecton Concept Library - Musser, Schupp, et al. - 1999 |

6 |
Approximations and rough sets based on tolerances
- Järvinen
(Show Context)
Citation Context ...emed to be less substantial than the other two. The generalization work went rather smoothly. As we discovered soon, similar investigations, but without any machine-motivations, were done by Järvinen =-=[Jär01]-=-. What is rather interesting, one of the referees among the RST community complained, that the presentation in [Gra03] is lacking its focus between approximation and tolerance spaces. Because every ap... |

6 | On the Integrity of a Repository of Formal Mathematics - Rudnicki, Trybulec - 2003 |

5 | Mathematical Knowledge Management in Theorema
- Buchberger
- 2001
(Show Context)
Citation Context ... Isabelle/Isar [NPW02] there are mechanisms of merging both theories and locales. The latter modules correspond to a context rather which is a narrower notion than we dealt in this paper. In Theorema =-=[Buc01]-=- theories can be constructed by using a special language construct. This allows to explicitly state which definitions, lemmas, and theorems are part of the theory. Thus every user can construct his ow... |

4 | Introduction to concept lattices
- Schwarzweller
- 1998
(Show Context)
Citation Context ...ly developed theories. The plan of the paper is as follows. In the next section we briefly introduce rough sets and formal concept analysis and review their Mizar formalization as done in [Gra03] and =-=[Sch98]-=-. Section 3 describes the combination of these two theories into rough concept analysis. Mizar language features to do so are presented. In the last two sections we discuss the Mizar approach for comb... |

1 | The Tecton Concept Description Language (Revised Version - Musser, Shao - 2002 |