|
PSPACE-complete Problems
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. The problems that are PSPACE-complete can be thought of as the hardest problems in PSPACE, the class of decision problems solvable in polynomial space, because a solution to any one such problem could easily be used to solve any other problem in PSPACE. Problems known to be PSPACE-complete include determining properties of regular expressions and context-sensitive grammars, determining the truth of quantified Boolean formulas, step-by-step changes between solutions of combinatorial optimization problems, and many puzzles and games. Theory A problem is defined to be PSPACE-complete if it can be solved using a polynomial amount of memory (it belongs to PSPACE) and every problem in PSPACE can be t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Word Problem (computability)
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natural number is prime. Another example is the problem, "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" A decision procedure for a decision problem is an algorithmic method that answers the yes-no question on all inputs, and a decision problem is called decidable if there is a decision procedure for it. For example, the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" is decidable since there is a decision procedure called long division that gives the steps for determining whether ''x'' evenly divides ''y'' and the correct answer, ''YES'' or ''NO'', accordingly. Some of the most important problems in mathematics are undecidable, e.g. the halting problem. The field of computational co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 arranged in an 8×8 grid. The players, referred to as White and Black in chess, "White" and "Black", each control sixteen Chess piece, pieces: one king (chess), king, one queen (chess), queen, two rook (chess), rooks, two bishop (chess), bishops, two knight (chess), knights, and eight pawn (chess), pawns, with each type of piece having a different pattern of movement. An enemy piece may be captured (removed from the board) by moving one's own piece onto the square it occupies. The object of the game is to "checkmate" (threaten with inescapable capture) the enemy king. There are also several ways a game can end in a draw (chess), draw. The recorded history of chess goes back to at least the emergence of chaturanga—also thought to be an ancesto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reversi
Reversi is a strategy board game for two players, played on an 8×8 uncheckered board. It was invented in 1883. ''Othello'', a variant with a fixed initial setup of the board, was patented in 1971. Basics Two players compete, using 64 identical game pieces ("disks") that are light on one side and dark on the other. Each player chooses one color to use throughout the game. Players take turns placing one disk on an empty square, with their assigned color facing up. After a play is made, any disks of the opponent's color that lie in a straight line bounded by the one just played and another one in the current player's color are turned over. When all playable empty squares are filled, the player with more disks showing in their own color wins the game. History Original version Englishmen Lewis Waterman and John W. Mollett both claim to have invented the game of reversi in 1883, each denouncing the other as a fraud. The game gained considerable popularity in England at the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hex (board Game)
Hex (also called Nash) is a two player abstract strategy board game in which players attempt to connect opposite sides of a rhombus-shaped board made of hexagonal cells. Hex was invented by mathematician and poet Piet Hein in 1942 and later rediscovered and popularized by John Nash. It is traditionally played on an 11×11 rhombus board, although 13×13 and 19×19 boards are also popular. The board is composed of hexagons called ''cells'' or ''hexes''. Each player is assigned a pair of opposite sides of the board, which they must try to connect by alternately placing a stone of their color onto any empty hex. Once placed, the stones are never moved or removed. A player wins when they successfully connect their sides together through a chain of adjacent stones. Draws are impossible in Hex due to the topology of the game board. Despite the simplicity of its rules, the game has deep strategy and sharp tactics. It also has profound mathematical underpinnings related to the Brouwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Game
In computational complexity theory, a generalized game is a game or puzzle that has been generalized so that it can be played on a board or grid of any size. For example, generalized chess is the game of chess played on an n\times n board, with 2n pieces on each side. Generalized Sudoku includes Sudokus constructed on an n\times n grid. Complexity theory studies the asymptotic difficulty of problems, so generalizations of games are needed, as games on a fixed size of board are finite problems. For many generalized games which last for a number of moves polynomial in the size of the board, the problem of determining if there is a win for the first player in a given position is PSPACE-complete. Generalized hex and reversi are PSPACE-complete. For many generalized games which may last for a number of moves exponential in the size of the board, the problem of determining if there is a win for the first player in a given position is EXPTIME-complete. Generalized chess, go (with Ja ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Game Theory
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Research in this field has primarily focused on two-player games in which a ''position'' evolves through alternating ''moves'', each governed by well-defined rules, with the aim of achieving a specific winning condition. Unlike game theory, economic game theory, combinatorial game theory generally avoids the study of games of chance or games involving imperfect information, preferring instead games in which the current state and the full set of available moves are always known to both players. However, as mathematical techniques develop, the scope of analyzable games expands, and the boundaries of the field continue to evolve. Authors typically define the term "game" at the outset of academic papers, with definitions tailored to the specific game under analysis rather than reflecting the field’s full scope. Combinatorics, Comb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation (graph Theory)
In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph. Oriented graphs A directed graph is called an oriented graph if none of its pairs of vertices is linked by two mutually symmetric edges. Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of and may be arrows of the graph). A tournament is an orientation of a complete graph. A polytree is an orientation of an undirected tree. Sumner's conjecture states that every tournament with vertices contains every polytree with vertices. The number of non-isomorphic oriented graphs with vertices (for ) is : 1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, … . Tournaments are in one-to-one correspondence with complete directed graphs (graphs in which there is a directed edge in one or both directions between every pair of distinct vertices). A complete directed graph can be con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nondeterministic Constraint Logic
In theoretical computer science, nondeterministic constraint logic is a combinatorial system in which an orientation is given to the edges of a weighted undirected graph, subject to certain constraints. One can change this orientation by steps in which a single edge is reversed, subject to the same constraints. This is a form of reversible logic in that each sequence of edge orientation changes can be undone. Reconfiguration problems for constraint logic, asking for a sequence of moves to connect certain states, connect all states, or reverse a specified edge have been proven to be PSPACE-complete. These hardness results form the basis for proofs that various games and puzzles are PSPACE-hard or PSPACE-complete. Constraint graphs In the simplest version of nondeterministic constraint logic, each edge of an undirected graph has weight either one or two. (The weights may also be represented graphically by drawing edges of weight one as red and edges of weight two as blue.) The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space
In computer science, a state space is a discrete space 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 and game theory. For instance, the toy problem 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 starting at 1 and are incremented over time has an infinite discrete state space. The angular position of an undamped pendulum is a continuous (and therefore infinite) state space. Definition 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 (mathematics), set of states * ''A'' is a set of arcs connecting the states * ''S'' is a nonempty subset of ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reconfiguration
In discrete mathematics and theoretical computer science, reconfiguration problems are computational problems involving reachability or Connectivity (graph theory), connectivity of state spaces. Types of problems Here, a state space is a discrete set of configurations of a system or solutions of a combinatorial problem, called states, together with a set of allowed moves linking one state to another. Reconfiguration problems may ask: *For a given class of problems, is the state space always connected? That is, can one transform every pair of states into each other with a sequence of moves? If not, what is the computational complexity of determining whether the state space for a particular problem is connected? *What is the diameter of the state space, the smallest number such that every two states can be transformed into each other with at most moves? *Given two states, what is the complexity of determining whether they can be transformed into each other, or of finding the shortes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Satisfiability Problem
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) asks whether there exists an Interpretation (logic), interpretation that Satisfiability, satisfies a given Boolean logic, Boolean Formula (mathematical logic), formula. In other words, it asks whether the formula's variables can be consistently replaced by the values TRUE or FALSE to make the formula evaluate to TRUE. If this is the case, the formula is called ''satisfiable'', else ''unsatisfiable''. For example, the formula "''a'' AND NOT ''b''" is satisfiable because one can find the values ''a'' = TRUE and ''b'' = FALSE, which make (''a'' AND NOT ''b'') = TRUE. In contrast, "''a'' AND NOT ''a''" is unsatisfiable. SAT is the first problem that was proven to be NP-complete—this is the Cook–Levin theorem. This means that all problems in the complexity class NP (complexity), NP, which includes a wide range of natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
