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The first known NP-complete problem was satisfiability (SAT). This is the problem of whether there are assignments of truth values to variables that make a boolean expression true. For example, one instance of SAT would be the question of whether the following is true:
The most basic PSPACE-complete problem is identical, except every other quantifier is a universal quantifier:
Notice that the NP-complete problem resembles a typical puzzle: is there some way to plug in values that solves the problem? The PSPACE-complete problem resembles a game: is there some move I can make, such that for all moves my opponent might make, there will then be some move I can make to win? The question alternates existential and universal quantifiers. Not surprisingly, many puzzles turn out to be NP-complete, and many games turn out to be PSPACE-complete.
The game of checkers (draughts) is PSPACE-complete when generalized so that it can be played on an n × n board. So are the generalized versions of the games cex and Reversi and the solitaire games Rush Hour, mahjongMahjong solitaire also known as Shanghai solitaire electronic or computerized mahjong MahJong solitaire solitaire Mahjong and, erroneously, as mahjong is a solitaire matching game that uses a set of Mahjong tiles rather than cards. The 144 tiles are arran, AtomixAtomix is the most popular magazine among video game consumers in Latin America. Published monthly, it was created by a group of video game fans in 1996. The original site was named Limit-X and it was the biggest Spanish-language online community covering and SokobanSokoban ( Japanese for "warehouse keeper") is a transport puzzle in which the player pushes boxes around a maze, viewed from above, and tries to put them in designated locations. Only one box may be pushed at a time, not two, and boxes cannot be pulled.. Some other generalized games, such as chessFor other meanings, see Chess (disambiguation). Chess (from the Persian word Shah is a board game for two players played on a square board divided into eight rows (or ranks and eight columns (or files creating 64 individual squares which alternate in colo and goGo is a strategic, two-player board game originating in ancient China between 2000 BC and 200 BC. Go is a popular game in East Asia. The development of Internet play has served to increase notably its popularity throughout the rest of the world, in recent are EXPTIME-complete because a game between two perfect players can be very long, so they are unlikely to be in PSPACE.
Note that the definition of PSPACE-complete is based on asymptotic complexity: the time it takes to solve a problem of size n, in the limit as n grows without bound. That means a game like checkers (which is played on an 8 × 8 board) could never be PSPACE-complete. That is why all the games were modified by playing them on an n × n board instead.
Another PSPACE-complete problem is the problem of deciding whether a given string is a member of the language defined by a given context-sensitive grammarA context-sensitive grammar is a formal grammar G N Σ, P S such that all rules in P are of the form : &alpha A beta; → αγβ with A in N (i. A is single nonterminal) and α and β in N U Σ) (i. alpha; and β st.