A continuous approximation for the intersection of two hyper-spheres in the Boolean space

Braga and Aleksander (1994, 1995) presented geometrical procedures to describe the distribution of distances among vectors in the n-dimensional Boolean space when two arbitrary vectors ξ¹ and ξ² are maintained fixed and separated by a fixed Hamming distance h. Once this distribution is obtained, important features of the space which show how ξ¹ and ξ² are related to each other can be derived. Some of these features are: (1) overlap between the two classes defined by ξ¹ and ξ²; (2) intersection of two hyper-spheres with centres in ξ¹ and ξ² and (3) membership of the two corresponding classes. The model enables estimating how the two patterns relate to each other and also presents a more simple and accurate solution to the problem than that found in previous work (Kanerva, 1988). This paper will focus on describing a continuous model to estimate the intersection of two hyper-spheres. The expressions that will be presented fit very well with the discrete model and represent an alternative way for analysing properties of the Boolean space and to study artificial neural networks (ANNs) based on Boolean nodes