Dynamic programming state ($s$)

A dynamic programming state is an element of a state space that summarizes all information needed for a remaining mathematical optimization problem. Histories can share a value function entry only when their allowed continuations and future rewards agree, or have the same conditional laws in a stochastic control problem. Time or the remaining item index may be part of the summary. If past decisions still affect feasibility or rewards, enlarge the dynamic programming state; forgetting those effects invalidates the Bellman equation.