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**How to build a knowledge space by querying an expert.**
*(English)*
Zbl 0725.92029

The authors are concerned with the problem of building knowledge structures - KS - for particular domains (a KS is defined as the collection of all possible knowledge states, while the knowledge state of an individual is formalized as the subset of notions he has mastered; finally, a notion is identified with an equivalence class of questions/problems, testing just that notion). The algebraic foundation of this approach was investigated by J.-P. Doignon and J.-C. Falmagne [Int. J. Man-Mach. Stud. 23, 175-196 (1985; Zbl 0581.68066)]. If the KS is closed under union and intersection, then it can be equivalently specified by a quasi-order on the set of problems. When the - less realistic - assumption of closure under intersection is dropped, a representation of KSs by surmise systems - a variant of AND/OR graphs - is possible. The KSs which are closed under union are called knowledge spaces (KSP).

The point of the current paper is to derive an alternative representation for KSP. Here, quasi-orders are used as relations on the power set of the set of problems; this representation is at the basis of a procedure that translates the responses of an expert to a set of queries of a specific form into a corresponding KSP. Birkhoff’s theorem [G. Birkhoff, Duke Math. J. 3, 443-454 (1937; Zbl 0017.19403)], which plays a central role in the quasi-order representation, can be derived from a Galois connection between the collection of KSs and the collection of binary relations on the set of problems. The main result of the paper under review is an extension of Birkhoff’s theorem by establishing a more general Galois connection between KSs and relations on the power set of the set of problems.

The point of the current paper is to derive an alternative representation for KSP. Here, quasi-orders are used as relations on the power set of the set of problems; this representation is at the basis of a procedure that translates the responses of an expert to a set of queries of a specific form into a corresponding KSP. Birkhoff’s theorem [G. Birkhoff, Duke Math. J. 3, 443-454 (1937; Zbl 0017.19403)], which plays a central role in the quasi-order representation, can be derived from a Galois connection between the collection of KSs and the collection of binary relations on the set of problems. The main result of the paper under review is an extension of Birkhoff’s theorem by establishing a more general Galois connection between KSs and relations on the power set of the set of problems.

Reviewer: S.Luchian (Bucureşti)

### MSC:

91E10 | Cognitive psychology |

06A15 | Galois correspondences, closure operators (in relation to ordered sets) |

68T30 | Knowledge representation |

06A06 | Partial orders, general |

68T05 | Learning and adaptive systems in artificial intelligence |

91E40 | Memory and learning in psychology |

### Keywords:

knowledge structures; knowledge state; quasi-order on the set of problems; surmise systems; knowledge spaces; quasi-order representation; extension of Birkhoff’s theorem; Galois connection; power set of the set of problems
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\textit{M. Koppen}, J. Math. Psychol. 34, No. 3, 311--331 (1990; Zbl 0725.92029)

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### References:

[1] | Birkoff, G., Rings of sets, Duke mathematical journal, 3, 443-454, (1937) |

[2] | Birkhoff, G., Lattice theory, () · Zbl 0126.03801 |

[3] | Burrigana, L., Organization by rules in finite sequences, (1988), Submitted for publication |

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[5] | Doignon, J.-P.; Falmagne, J.-C., Spaces for the assessment f knowledge, International journal of man-machine studies, 23, 175-196, (1985) · Zbl 0581.68066 |

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[9] | Kambouri, M.; Koppen, M.; Villano, M.; Falmagne, J.-C., Knowledge assessment: tapping human expertise, (1989), In progress |

[10] | Koppen, M., Extracting human expertise for constructing knowledge spaces: an algorithm, (1989), Submitted for publication · Zbl 0768.92029 |

[11] | Monjardet, B., Tresses, fuseaux, préordres et topologies, Mathématiques et sciences humaines, 30, 11-22, (1970) |

[12] | Muller, C., A procedure to facilitate an Expert’s judgements on a set of rules, (), in press |

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