Limitation in collective learnings and its possible remedy

A legality condition is derived theoretically for any given mapping relations to be learned collectively by a two-layered perceptron. When the mapping is legal and the number of mapping relations to be learned is not greater than the number of input nodes, a simple one-step method can be used to reach the goal of collective learning. This legality condition can be generalized to be used in the design of a 'universal' multilayered learning machine.<>