statespace

State space is the set of all possible states that a system can occupy. In dynamical systems and control theory, a state represents the minimal information needed to determine the system’s future behavior given its inputs. The state space is the collection of all admissible state vectors, and it can be finite or infinite, discrete or continuous, depending on the model and its variables.

In many models, the evolution of the system is described by a transition rule that maps one state to another, possibly subject to inputs or disturbances. For example, a discrete-time system might use x_{t+1} = f(x_t, u_t), while a continuous-time system uses differential equations x' = f(x, u). The transition model defines how actions or controls move the system through the state space.

In artificial intelligence and planning, the state space comprises all possible configurations of the world. Transitions correspond to actions, and a problem is usually specified by an initial state, a goal condition, a transition model, and possibly a cost or reward function. The state space can be represented as a graph where nodes are states and edges correspond to valid actions, enabling search algorithms such as breadth-first search, depth-first search, Dijkstra’s, or A*.

Common issues include state explosion in large or continuous domains, and partial observability, which leads to belief states rather than exact states. Real-world applications span robotics, scheduling, game playing, and control systems.