State space

Vacuum World, a shortest path problem with a finite state space

In the theory of discrete dynamical systems, a state space is the set of all possible configurations of a system.^[1] For example, a system in queueing theory defining the number of customers in a line would have state space {0, 1, 2, 3, ...}. State spaces can be either infinite or finite. An example of a finite state space is that of the toy problem Vacuum World, in which there are a limited set of configurations that the vacuum and dirt can be in.

Definition

The states space is a directed graph where each possible state of a dynamical system is represented by a vertex, and there is a directed edge from a to b if and only if ƒ(a) = b where the function f defines the dynamical system.

State spaces are useful in computer science as a simple model of machines. Formally, a state space can be defined as a tuple [N, A, S, G] where:

• N is a set of states
• A is a set of arcs connecting the states
• S is a nonempty subset of N that contains start states
• G is a nonempty subset of N that contains the goal states.

Properties

A state space has some common properties:

• complexity, where branching factor is important
• structure of the space, see also graph theory:
  • directionality of arcs
  • tree
  • rooted graph

In many games the effective state space is small compared to all reachable states. For instance, in chess the effective state space is the set of positions that can be reached by game-legal moves. This is far smaller than the set of positions that can be achieved by placing combinations of the available chess pieces directly on the board.

State space search explores a state space.

See also

• State space representation for information about continuous state space in control engineering.
• State space (physics) for information about continuous state space in physics.
• Phase space for information about phase state (like continuous state space) in physics and mathematics.
• Probability space for information about state space in probability.
• Game complexity theory, which relies on the state space of game outcomes
• Cognitive Model#Dynamical systems for information about state space with a dynamical systems model of cognition.
• State space planning
• Probability space for information about state space in probability.
• Game complexity theory, which relies on the state space of game outcomes
• State (computer science)

References

• Equivalence Relations on Finite Dynamical Systems, Laubenbacher, R. Pareigis, B., ADVANCES IN APPLIED MATHEMATICS, 2001, VOL 26; PART 3, pages 237–251
• State-space search: algorithms, complexity, extensions, and applications, Weixiong Zhang, Springer, 1999, ISBN 978-0-387-98832-0