A Fuzzy Inference Methodology Based on the Fuzzification of Set Inclusion

Nowadays, people start to accept fuzzy rule–based systems as flexible and convenient tools to solve a myriad of ill–defined but otherwise (for humans) straightforward tasks such as controlling fluid levels in a reactor, automatical lens focussing in cameras and adjusting an aircraft’s navigation to the change of winds and so on. Contrary to the intuition often seen as the feeding ground of fuzzy rule–based systems—namely, that they realize an extension of the Modus Ponens (MP) rule of inference to an environment with more than two truth–values—most actual applications rely at the base level on common interpolation techniques or similarity assessments to simulate the process of “calculating with words” perceived at the user level. It is doubtful whether these somewhat opportunistic approaches will perform well when more challenging requirements (e.g. aspects of logical consistency; incorporation of varying facets of uncertainty) are imposed in order to implement a successful artificial reasoning unit. Therefore, in this paper, starting from the notion of a fuzzy restriction (i.e. the basic building block of our rule–based system) we list some elementary consistency requirements that a fuzzy inference system should satisfy. Subsequently we describe a reasoning methodology based on a measure of fulfilment of the antecedent clause of an if–then rule. Inclusion–based approximate reasoning, as we coined it in [7], outperforms the traditional scheme based on the Compositional Rule of Inference (CRI) in terms of both complexity and of logical soundness. In terms of semantics it also offers a better solution to the implementation of analogical reasoning than similarity measures are able to do.