An equivalence between inductive learning and pseudo-Boolean logic simplification: a rule generation and reduction scheme

To provide confidence in a rule generation and reduction scheme, it is necessary to show an equivalence between at least two alternative approaches that provide the same reduced rule sets. For this purpose we present two rule reduction schemes in fuzzy knowledge based systems. These two schemes are based on: (1) a pseudo-Boolean analysis, and (2) the ID3 algorithm. Here, we show an equivalence between these two rule reduction schemes. As a consequence of this equivalence, we propose to substitute an online and automated expert system shell that uses the ID3 algorithm in place of the manual pseudo-Boolean simplification method for the purposes of rule reduction in fuzzy knowledge bases. For both approaches, we assign pseudo-Boolean variables to the intervals, identified as semantic intervals, on the base-axis of a membership representation. The pseudo-Boolean simplification of disjunctions of conjunctions formulae (DCF) is efficiently learnable from examples in the sense of Valiant (1984, 1985). Positive and negative examples are provided to illustrate that the equivalence holds for all nonnull nodes in decision trees under the condition of extending all the examples to possible attribute combinations. This approach supports an interactive learning environment for the design of fuzzy expert systems where a rule reduction algorithm constitutes an important component of the design process. Another important component is a rule generation algorithm. Two illustrative experiments are provided to demonstrate the rule generation and reduction