PSPACE-complete

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.

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 transformed in polynomial time into an equivalent instance of the given problem.

The PSPACE-complete problems are widely suspected to be outside the more famous complexity classes P (polynomial time) and NP (non-deterministic polynomial time), but that is not known. It is known that they lie outside of the class NC, a class of problems with highly efficient parallel algorithms, because problems in NC can be solved in an amount of space polynomial in the logarithm of the input size, and the class of problems solvable in such a small amount of space is strictly contained in PSPACE by the space hierarchy theorem.

The transformations that are usually considered in defining PSPACE-completeness are polynomial-time many-one reductions, transformations that take a single instance of a problem of one type into an equivalent single instance of a problem of a different type. However, it is also possible to define completeness using Turing reductions, in which one problem can be solved in a polynomial number of calls to a subroutine for the other problem. It is not known whether these two types of reductions lead to different classes of PSPACE-complete problems. Other types of reductions, such as many-one reductions that always increase the length of the transformed input, have also been considered.

A version of the Berman–Hartmanis conjecture for PSPACE-complete sets states that all such sets look alike, in the sense that they can all be transformed into each other by polynomial-time bijections.

Given a regular expression ${\displaystyle R}$, determining whether it generates every string over its alphabet is PSPACE-complete.

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 Kuroda (1964) showed that every (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.