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Fuzzy inferences and conditional possibility distributions. (English) Zbl 0633.68100
Let \(X_ i\) \((i=1,2)\) be two variables taking values in finite universes of discourse \(U_ i\) and associated with possibility distributions \(f_ i: U_ i\to [0,1]\). After an observation of \(X_ 1\), we can deduce a characterization of \(X_ 2\) by means of an inference process. We define \(q(x_ 2| x_ 1)\) for \(x_ i\in U_ i\) as some fuzzy implication and we introduce the possibility distribution \(\bar p(x_ 1,x_ 2)=f_ 1(x_ 1)\sqcap q(x_ 2| x_ 1)\), for a t-norm \(\sqcap\). We exhibit some reasons to say that \(q(\cdot | \cdot)\) and \(f(\cdot,\cdot)\) may be respectively regarded as a conditional and a joint possibility distribution, corresponding to marginal properties with regard to the use of a t-conorm \(\sqcup\). A property of non-interactivity of the variables, with respect to \(\sqcap\), is bound with the definition of \(q(\cdot | \cdot)\) and \(p(\cdot,\cdot)\). In this way, we suggest justifications of the choice of particular implications in inference processes.

MSC:
68T99 Artificial intelligence
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
03B52 Fuzzy logic; logic of vagueness
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