Introduction

Artificial Neural Networks classifiers have many advantages such as: noise tolerance, possibility of parallelization, better training with a small quantity of data . . . . Coupling neural networks with an explanation component will increase its usage for those applications. The explanation capacity of neural networks is solved by extracting knowledge incorporated in the trained network [Andrews et al., 1995]. We consider a single neuron (or perceptron) with Heaviside map as activation function (f (x) = 0 if x < 0 else 1). For a given perceptron with the connection weights vector W and the threshold θ, this means finding the different states where the neuron is active (wich could be reduced to the Knapsack problem.

With the existing algorithms, two forms of rules are reported in the literature : '

If (condition) then conclusion' form noted 'if then' rules, 'If (m of a set of conditions) then conclusion' form noted 'M of N '

The intermediate structures that we introduce are called MaxSubset list and generator list. The MaxSubset is a minimal structure used to represent the if-then rules while the generator list is some selected MaxSubsets from which we can derive all MaxSubsets and all MofN rules. We introduce heuristics to prune and reduce the candidate search space. These heuristics consist of sorting the incoming links according to the descending order, and then pruning the search space using the subset cardinality bounded by some determined values.

The MaxSubsets and generators Rules extraction approach

The general form of a MofN rule is 'if

m 1 of N 1 ∧ m 2 of N 2 ∧...∧m p of N p then conclusion' or ' i (m i of N i )

then conclusion'; for each subset N i of the inputs set, if m i elements are verified, the conclusion is true.

The common limit of previous approaches is the exclusive form of the extracted rules. Thus we introduce a novel approach called MaxSubset from which it is possible to generate both forms of rules. The MaxSubset approach follows operations (3), ( 4) and (5) of figure 1; while the existing known algorithms follow the path (1) for the if-then rules and (2) for the MofN rules. The processes (3), ( 4) and ( 5

Conclusion

This approach consists in extracting a minimal list of elements called MaxSubset list, and then generating rules in one of the standard forms : if-then or MofN. To our knowledge, it is the first approach which is able to propose to the user a generic representation of rules from which it is possible to derive both forms of rules.