Sub-symbolically managing pieces of symbolical functions for sorting

We present a hybrid system for managing both symbolic and sub-symbolic knowledge in a uniform way. Our aim is to solve problems where some gap in formal theories occurs which stops one from getting a fully symbolical solution. The idea is to use neural modules to functionally connect pieces of symbolic knowledge, such as mathematical formulas and deductive rules. The whole system is trained through a backpropagation learning algorithm where all (symbolic or sub-symbolic) free parameters are updated piping back the error through each component of the system. The structure of this system is very general, possibly varying over time and managing fuzzy variables and decision trees. We use as a test-bed the problem of sorting a file, where suitable suggestions on next sorting moves are supplied by the network also on the basis of the hints provided by some conventional sorters. A comprehensive discussion of system performance is provided in order to understand behaviors and capabilities of the proposed hybrid system