Closure properties of the compositional rule of inference.

*(English)*Zbl 0704.03006Author’s abstract: “This paper deals with the compositional rule of inference: if x is P and x and y are R, then y is Q, where Q is defined in terms of P and R. P, Q and R are fuzzy sets defined with membership functions \(\chi_ P\), \(\chi_ Q\) and \(\chi_ R\) from a certain class of functions C. It is shown that when the parameters of these three functions are chosen equal, then either \(\chi_ P\), \(\chi_ Q\) and \(\chi_ R\) are the same, or \(\chi_ Q\) lies between \(\chi_ P\) and \(\chi_ R\). In this case it can be shown where approximately \(\chi_ Q\) lies between \(\chi_ P\) and \(\chi_ R\). Unfortunately its position depends on the original parameters of \(\chi_ P\) and \(\chi_ R\).”

Reviewer: S.Rudeanu

##### MSC:

03B52 | Fuzzy logic; logic of vagueness |

##### Keywords:

compositional rule of inference
PDF
BibTeX
XML
Cite

\textit{H. Hellendoorn}, Fuzzy Sets Syst. 35, No. 2, 163--183 (1990; Zbl 0704.03006)

Full Text:
DOI

##### References:

[1] | Bellman, R.E.; Zadeh, L.A., Local and fuzzy logics, (), 105-165 · Zbl 0382.03017 |

[2] | Bonissone, P.P.; Decker, K.S., Selecting uncertainty calculi and granularity: an experiment in trading-off precision and complexity, (), 217-247 |

[3] | Bonissone, P.P., Summarizing and propagating uncertain information with triangular norms, Internat. J. approximate reasoning, 1, 71-101, (1987) |

[4] | Dubois, D.; Prade, H., Fuzzy real algebra: some results, Fuzzy sets and systems, 2, 327-348, (1979) · Zbl 0412.03035 |

[5] | Dubois, D.; Prade, H., Possibility theory - an approach to computerized processing of uncertainty, (1988), Plenum Press New York |

[6] | Goguen, J.A., L-fuzzy sets, J. math. anal. appl., 18, 145-174, (1967) · Zbl 0145.24404 |

[7] | Zadeh, L.A., Fuzzy sets, Inform. and control, 8, 338-353, (1965) · Zbl 0139.24606 |

[8] | Zadeh, L.A., Fuzzy logic and approximate reasoning, Synthese, 30, 407-428, (1975) · Zbl 0319.02016 |

[9] | Zadeh, L.A., Calculus of fuzzy restrictions, (), 1-39 · Zbl 0327.02018 |

[10] | Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning I, Inform sci., 8, 199-249, (1975) · Zbl 0397.68071 |

[11] | Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning II, Inform sci., 8, 301-357, (1975) · Zbl 0404.68074 |

[12] | Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning III, Inform sci., 9, 43-80, (1975) · Zbl 0404.68075 |

[13] | Zadeh, L.A., A theory of approximate reasoning, (), 149-194 |

[14] | Zadeh, L.A., The role of fuzzy logic in the management of uncertainty in expert systems, Fuzzy sets and systems, 11, 199-227, (1983) · Zbl 0553.68049 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.