Recurrent neural networks to approximate the semantics of acceptable logic programs

In [9] we have shown how to construct a 3-layer recurrent neural network (RNN) that computes the iteration of the meaning function T-p of a given propositional logic program, what corresponds to the computation of the semantics of the program. In this paper we define a notion of approximation for interpretations and prove that there exists a feed forward neural network (FNN) that approximates the calculation of T-p, for a,given (first order) acceptable logic program with an injective level mapping arbitrarily veil. By extending the FNN by recurrent connections we get a RNN whose iteration approximates the fixed point of T-p. The proof is found by taking advantage of the fact that for acceptable logic programs, T-p is a contraction mapping on the complete metric space of the interpretations for the program. Mapping this metric space to the metric space IR the real valued function f(p) corresponding to T-p turns out to he continuous and a contraction and for this reason can be approximated by an indicated class of FNN.