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PSPACE-complete
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The hardest problems in PSPACE are the PSPACE-Complete problems.
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This is supposed to be hard, at least PSPACE-complete.
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Bylander (1992) shows simple STRIPS planning to be PSPACE-complete.
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Not surprisingly, many puzzles turn out to be NP-complete, and many games turn out to be PSPACE-complete.
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The PSPACE-complete problems are widely suspected to be outside the more famous complexity classes P and NP, but that is not known.
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It is widely believed that PSPACE-complete problems are strictly harder than any problem in NP, although this has not yet been proved.
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If both quantifiers are allowed, the problem is called the quantified Boolean formula problem (QBF), which can be shown to be PSPACE-complete.
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By contrast, generalized games that can last for a number of moves that is polynomial in the size of the board are often PSPACE-complete.
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Membership of a string in a language defined by an arbitrary context-sensitive grammar, or by an arbitrary deterministic context-sensitive grammar, is a PSPACE-complete problem.
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Note that the definition of PSPACE-completeness is based on asymptotic complexity: the time it takes to solve a problem of size n, in the limit as n grows without bound.
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Correspondingly, the PSPACE-complete problem there resembles games: 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?
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Examples of games that are PSPACE-complete (when generalized so that they can be played on an n × n board) are the games Hex and Reversi and the solitaire games Rush Hour, Mahjong, Atomix, and Sokoban.
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That means a finite game like checkers (which is played on an 8 × 8 board) could never be PSPACE-complete, since they can be solved in constant time and space using a very large lookup table (checkers is already solved in this way).
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See Game complexity for more games whose completeness for PSPACE or other complexity classes has been determined.
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PSPACE-complete - Wikipedia, the free encyclopedia
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Solvable with polynomial space. PSPACE-complete The hardest problems in PSPACE.
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Enough to say that NP-complete problems are contained within PSPACE-complete problems.
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Joseph C. Culberson has also proved that Sokoban, a block pushing game, is PSPACE-complete.
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With some more work it can be shown to be PSPACE-complete.[2]
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The PSPACE-complete problems are widely suspected to be outside the more famous complexity classes P and NP, but that is not known.
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