computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...

, a state space is a

discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...

representing the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of

artificial intelligence Artificial intelligence (AI) is the capability of computer, computational systems to perform tasks typically associated with human intelligence, such as learning, reasoning, problem-solving, perception, and decision-making. It is a field of re ...

and

game theory Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...

. For instance, the

toy problem In scientific disciplines, a toy problem or a puzzlelike problem is a problem that is not of immediate scientific interest, yet is used as an expository device to illustrate a trait that may be shared by other, more complicated, instances of the ...

Vacuum World has a discrete finite state space in which there are a limited set of configurations that the vacuum and dirt can be in. A "counter" system, where states are the

natural numbers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

starting at 1 and are incremented over time has an infinite discrete state space. The angular position of an undamped

pendulum A pendulum is a device made of a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate i ...

is a continuous (and therefore infinite) state space.

Definition

State spaces are useful in

as a simple model of machines. Formally, a state space can be defined as a

tuple In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is o ...

'N'', ''A'', ''S'', ''G''where: * ''N'' is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of states * ''A'' is a set of arcs connecting the states * ''S'' is a nonempty

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

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 In computing, tree data structures, and game theory, the branching factor is the number of children at each node, the outdegree. If this value is not uniform, an ''average branching factor'' can be calculated. For example, in chess, if a "node ...

is important * structure of the space, see also

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

: ** directionality of arcs ** tree ** rooted graph For example, the Vacuum World has a branching factor of 4, as the vacuum cleaner can end up in 1 of 4 adjacent squares after moving (assuming it cannot stay in the same square nor move diagonally). The arcs of Vacuum World are bidirectional, since any square can be reached from any adjacent square, and the state space is not a tree since it is possible to enter a loop by moving between any 4 adjacent squares. State spaces can be either infinite or finite, and discrete or continuous.

Size

The size of the state space for a given system is the number of possible configurations of the space.

Finite

If the size of the state space is finite, calculating the size of the state space is a

combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

problem. For example, in the

eight queens puzzle The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; thus, a solution requires that no two queens share the same row, column, or diagonal. There are 92 solutions. ...

, the state space can be calculated by counting all possible ways to place 8 pieces on an 8x8 chessboard. This is the same as choosing 8 positions without replacement from a set of 64, or :

\binom = 4,426,165,368

This is significantly greater than the number of legal configurations of the queens, 92. In many games the effective state space is small compared to all reachable/legal states. This property is also observed in

chess Chess is a board game for two players. It is an abstract strategy game that involves Perfect information, no hidden information and no elements of game of chance, chance. It is played on a square chessboard, board consisting of 64 squares arran ...

, where 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.

Infinite

All continuous state spaces can be described by a corresponding

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

and are therefore infinite. Discrete state spaces can also have (

countably In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...

) infinite size, such as the state space of the time-dependent "counter" system, similar to the system in

queueing theory Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because th ...

defining the number of customers in a line, which would have state space .

Exploration

Exploring a state space is the process of enumerating possible states in search of a goal state. The state space of Pacman, for example, contains a goal state whenever all food pellets have been eaten, and is explored by moving Pacman around the board.

Search states

A search state is a compressed representation of a world state in a state space, and is used for exploration. Search states are used because a state space often encodes more information than is necessary to explore the space. Compressing each world state to only information needed for exploration improves efficiency by reducing the number of states in the search. For example, a state in the Pacman space includes information about the direction Pacman is facing (up, down, left, or right). Since it does not cost anything to change directions in Pacman, search states for Pacman would not include this information and reduce the size of the search space by a factor of 4, one for each direction Pacman could be facing.

Methods

Standard search algorithms are effective in exploring discrete state spaces. The following algorithms exhibit both completeness and optimality in searching a state space: *

Breadth-first search Breadth-first search (BFS) is an algorithm for searching a tree data structure for a node that satisfies a given property. It starts at the tree root and explores all nodes at the present depth prior to moving on to the nodes at the next dept ...

* A* search * Uniform cost search These methods do not extend naturally to exploring continuous state spaces. Exploring a continuous state space in search of a given goal state is equivalent to optimizing an arbitrary

which is not always possible; see

mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...

References

{{DEFAULTSORT:State Space Models of computation Dynamical systems Reconfiguration