Decomposition and approximation of underdetermined data

For any underdetermined source, we consider its decomposition as product of sources generating symbols 0,1, and the indefinite symbol $*$. Also, we learn best approximate (in a prescribed sense) decomposition if correct decomposition is impossible. We prove that the best approximate decomposition always exists (for the decomposable source, it coincides with its decomposition), and it may be constructed by a polynomial algorithm. For some problems relating to simplifications and equivalent transformations of decompositions, polynomial algorithms are offered. In closing, we state that any underdetermined source has a decomposition in some more general form.