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^ Arora, Sanjeev Barak, Boaz (2009), Computational Complexity: A Modern Approach, Cambridge University Press, p. 92, ISBN 978-6-9.(1979), "Section 7.4: Polynomial Space Completeness", Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. In some cases, such as for chess, these extensions are artificial. Given a regular expression R board instead. Main article: List of PSPACE-complete problems Formal languages Ī 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. Other types of reductions, such as many-one reductions that always increase the length of the transformed input, have also been considered. It is not known whether these two types of reductions lead to different classes of PSPACE-complete problems. 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. 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. 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 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. 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.