How do you calculate state-space complexity?

How do you calculate state-space complexity?

The state-space complexity of a game is the number of legal game positions reachable from the initial position of the game. When this is too hard to calculate, an upper bound can often be computed by also counting (some) illegal positions, meaning positions that can never arise in the course of a game.

What is the state-space of chess?

Chess has an estimated state-space complexity of 1046 , the estimated game tree complexity of 10123 is based on an average branching factor of 35 and an average game length of 80 ply .

How many total states are present in state-space of tic tac toe?

State spaces
57. State spaces are good representations for board games such as Tic-Tac-Toe. The state of a game can be described by the contents of the board and the player whose turn is next. The board can be represented as an array of 9 cells, each of which may contain an X or O or be empty.

Is Tic Tac Toe a NP problem?

For a small game like Tic-Tac-Toe the computer could play all possible future moves and still respond quickly as perceived by a human. For Nim it is possible to create an polynomial time algorithm. Qubic is NP-complete thus there is no polynomial time algorithm to calculate the next perfect move.

What is the complexity of decision trees in tic tac toe?

So complexity of decision trees in 3X3 Tic Tac Toe is 5 which is the number of digit of the leaf nodes (26,830).

Is chess a Pspace?

Not surprisingly, many puzzles turn out to be NP-complete, and many games turn out to be PSPACE-complete. Some other generalized games, such as chess, checkers (draughts), and Go are EXPTIME-complete because a game between two perfect players can be very long, so they are unlikely to be in PSPACE.

Is tic-tac-toe a NP problem?

What is the complexity of decision trees in tic-tac-toe?

What do we mean when we claim that a problem is PSPACE-complete?

decision problem
In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input length (polynomial space) and if every other problem that can be solved in polynomial space can be transformed to it in polynomial time.

Is sat in PSPACE?

Since SAT is in PSPACE and SAT is NP-complete, every problem in NP is in PSPACE. That is, Theorem. PSPACE is defined in terms of deterministic algorithms.

What is the complexity of maximum algorithm?

Return max and min. Time Complexity is O(n) and Space Complexity is O(1). For each pair, there are a total of three comparisons, first among the elements of the pair and the other two with min and max.

What is the complexity of the state space?

The state-space complexity of a game is the number of legal game positions reachable from the initial position of the game.

How is the complexity of a game calculated?

The state-space complexity of a game is the number of legal game positions reachable from the initial position of the game. When this is too hard to calculate, an upper bound can often be computed by also counting (some) illegal positions, meaning positions that can never arise in the course of a game.

Is the size of the state space a combinatorial problem?

The size of the state space for a given system is the number of possible configurations of the space. If the size of the state space is finite, calculating the size of the state space is a combinatorial problem.

Why is the game tree larger than the state space?

The game tree is typically vastly larger than the state space because the same positions can occur in many games by making moves in a different order (for example, in a tic-tac-toe game with two X and one O on the board, this position could have been reached in two different ways depending on where the first X was placed).