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In complexity theory, a decision problem is PSPACE-complete if it is in the complexity class PSPACE, and every problem in PSPACE can be reduced to it in polynomial time (see complete (complexity)). The problems that are PSPACE-complete can be thought of as the hardest problems in PSPACE, because a solution to any one such problem could easily be used to solve any other problem in PSPACE. These problems are widely suspected to be outside of the more famous complexity classes P and NP, but that is not known. It is known that they lie outside of the small class NC.



The first known PSPACE-complete problem was the word problem for deterministic context-sensitive grammars. In the word problem for context-sensitive grammars, one is given a set of grammatical transformations which can increase, but cannot decrease, the length of a sentence, and wishes to determine if a given sentence could be produced by these transformations. The technical condition of "determinism" (implying roughly that each transformation makes it obvious that it was used) ensures that this process can be solved in polynomial space, and S.-Y. Kuroda's 1964 work "Classes of languages and linear-bounded automata"[1] showed that any (possibly non-deterministic) program computable in linear space could be converted into the parsing of a context-sensitive grammar, in a way which preserves determinism.

In 1970, Savitch's theorem showed that PSPACE is closed under nondeterminism, implying that even non-deterministic context-sensitive grammars are in PSPACE.

But the archetypal PSPACE-complete problem is generally taken to be the quantified Boolean formula problem (usually abbreviated to QBF or TQBF; the T stands for "true"), a generalization of the first known NP-complete problem, the Boolean satisfiability problem (SAT). The satisfiability problem 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:

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