State-space search

State-space search is a process used in the field of computer science, including artificial intelligence (AI), in which successive configurations or states of an instance are considered, with the intention of finding a goal state with the desired property.

Problems are often modelled as a state space, a set of states that a problem can be in. The set of states forms a graph where two states are connected if there is an operation that can be performed to transform the first state into the second.

State-space search often differs from traditional computer science search methods because the state space is implicit: the typical state-space graph is much too large to generate and store in memory. Instead, nodes are generated as they are explored, and typically discarded thereafter. A solution to a combinatorial search instance may consist of the goal state itself, or of a path from some initial state to the goal state.

In state-space search, a state space is formally represented as a tuple ${\displaystyle S:\langle S,A,\operatorname {Action} (s),\operatorname {Result} (s,a),\operatorname {Cost} (s,a)\rangle }${\displaystyle S:\langle S,A,\operatorname {Action} (s),\operatorname {Result} (s,a),\operatorname {Cost} (s,a)\rangle }, in which:

• ${\displaystyle S}${\displaystyle S} is the set of all possible states;
• ${\displaystyle A}${\displaystyle A} is the set of possible actions, not related to a particular state but regarding all the state space;
• ${\displaystyle \operatorname {Action} (s)}${\displaystyle \operatorname {Action} (s)} is the function that establishes which action is possible to perform in a certain state;
• ${\displaystyle \operatorname {Result} (s,a)}${\displaystyle \operatorname {Result} (s,a)} is the function that returns the state reached performing action ${\displaystyle a}${\displaystyle a} in state ${\displaystyle s}${\displaystyle s};
• ${\displaystyle \operatorname {Cost} (s,a)}${\displaystyle \operatorname {Cost} (s,a)} is the cost of performing an action ${\displaystyle a}${\displaystyle a} in state ${\displaystyle s}${\displaystyle s}. In many state spaces, ${\displaystyle a}${\displaystyle a} is a constant, but this is not always true.

According to Poole and Mackworth, the following are uninformed state-space search methods, meaning that they do not have any prior information about the goal's location.^[1]

• Traditional depth-first search
• Breadth-first search
• Iterative deepening
• Lowest-cost-first search / Uniform-cost search (UCS)

These methods take the goal's location in the form of a heuristic function.^[2] Poole and Mackworth cite the following examples as informed search algorithms:

• Informed/Heuristic depth-first search
• Greedy best-first search
• A* search

• State space
• State-space planning
• Branch and bound – Method for making state-space search more efficient by pruning subsets of it

• Stuart J. Russell and Peter Norvig (1995). Artificial Intelligence: A Modern Approach. Prentice Hall.

• Search algorithms
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