Consider states, inputs and outputs of a system, interrelated in the following way: the present state depends on the previous state and the previous input, and the present output depends on the present state and the present input. The functional equation approach of dynamic programming is applied to the problem of determining procedures which enable one, starting from an initial situation in which only the set \(S\) of all possible states is given, to reach a desired state.
During the treatment a concept of ambiguity is defined, viz. the limit for \(k\to\infty\) of \(\alpha_k(S) = \min_I \max_O \alpha_{k-1} (S_{IO})\) where \(I\) is an input, \(O\) an output, and \(S_{IO}\) is the current set of possible states, which depends on \(S\) and on all inputs and outputs up to now.