### Abstract

The problem of sparse fuzzy rule bases is introduced. Because of the high computational complexity of the original compositional rule of inference (CRI) method, it is strongly suggested that the number of rules in a final fuzzy knowledge base is drastically reduced. Various methods of analogical reasoning available in the literature are reviewed. The mapping style interpretation of fuzzy rules leads to the idea of approximating the fuzzy mapping by using classical approximation techniques. Graduality, measurability, and distance in the fuzzy sense are introduced. These notions enable the introduction of the concept of similarity between two fuzzy terms, by their closeness derived from their distance. The fundamental equation of linear rule interpolation is given, its solution gives the final formulas used for interpolating pairs of rules by their α-cuts, using the resolution principle. The method is extended to multiple dimensional variable spaces, by the normalization of all dimensions. Finally, some further methods are shown that generalize the previous idea, where various approximation techniques are used for the α-cuts and so, various approximations of the fuzzy mapping R: X → Y.

Original language | English |
---|---|

Pages (from-to) | 197-225 |

Number of pages | 29 |

Journal | International Journal of Approximate Reasoning |

Volume | 9 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1993 |

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### Keywords

- Approximate reasoning
- approximation of fuzzy mapping
- fuzzy distance of fuzzy sets
- fuzzy rule base
- interpolation
- resolution principle
- sparse rules

### ASJC Scopus subject areas

- Artificial Intelligence
- Computer Science Applications
- Information Systems
- Information Systems and Management
- Statistics, Probability and Uncertainty
- Electrical and Electronic Engineering
- Statistics and Probability

### Cite this

**Approximate reasoning by linear rule interpolation and general approximation.** / Kóczy, L.; Hirota, Kaoru.

Research output: Contribution to journal › Article

*International Journal of Approximate Reasoning*, vol. 9, no. 3, pp. 197-225. https://doi.org/10.1016/0888-613X(93)90010-B

}

TY - JOUR

T1 - Approximate reasoning by linear rule interpolation and general approximation

AU - Kóczy, L.

AU - Hirota, Kaoru

PY - 1993

Y1 - 1993

N2 - The problem of sparse fuzzy rule bases is introduced. Because of the high computational complexity of the original compositional rule of inference (CRI) method, it is strongly suggested that the number of rules in a final fuzzy knowledge base is drastically reduced. Various methods of analogical reasoning available in the literature are reviewed. The mapping style interpretation of fuzzy rules leads to the idea of approximating the fuzzy mapping by using classical approximation techniques. Graduality, measurability, and distance in the fuzzy sense are introduced. These notions enable the introduction of the concept of similarity between two fuzzy terms, by their closeness derived from their distance. The fundamental equation of linear rule interpolation is given, its solution gives the final formulas used for interpolating pairs of rules by their α-cuts, using the resolution principle. The method is extended to multiple dimensional variable spaces, by the normalization of all dimensions. Finally, some further methods are shown that generalize the previous idea, where various approximation techniques are used for the α-cuts and so, various approximations of the fuzzy mapping R: X → Y.

AB - The problem of sparse fuzzy rule bases is introduced. Because of the high computational complexity of the original compositional rule of inference (CRI) method, it is strongly suggested that the number of rules in a final fuzzy knowledge base is drastically reduced. Various methods of analogical reasoning available in the literature are reviewed. The mapping style interpretation of fuzzy rules leads to the idea of approximating the fuzzy mapping by using classical approximation techniques. Graduality, measurability, and distance in the fuzzy sense are introduced. These notions enable the introduction of the concept of similarity between two fuzzy terms, by their closeness derived from their distance. The fundamental equation of linear rule interpolation is given, its solution gives the final formulas used for interpolating pairs of rules by their α-cuts, using the resolution principle. The method is extended to multiple dimensional variable spaces, by the normalization of all dimensions. Finally, some further methods are shown that generalize the previous idea, where various approximation techniques are used for the α-cuts and so, various approximations of the fuzzy mapping R: X → Y.

KW - Approximate reasoning

KW - approximation of fuzzy mapping

KW - fuzzy distance of fuzzy sets

KW - fuzzy rule base

KW - interpolation

KW - resolution principle

KW - sparse rules

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U2 - 10.1016/0888-613X(93)90010-B

DO - 10.1016/0888-613X(93)90010-B

M3 - Article

VL - 9

SP - 197

EP - 225

JO - International Journal of Approximate Reasoning

JF - International Journal of Approximate Reasoning

SN - 0888-613X

IS - 3

ER -