PSPACE
Unsolved problem in computer science: (more unsolved problems in computer science)

In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space.
Contents
Formal definition[edit]
If we denote by SPACE(t(n)), the set of all problems that can be solved by Turing machines using O(t(n)) space for some function t of the input size n, then we can define PSPACE formally as^{[1]}
PSPACE is a strict superset of the set of contextsensitive languages.
It turns out that allowing the Turing machine to be nondeterministic does not add any extra power. Because of Savitch's theorem,^{[2]} NPSPACE is equivalent to PSPACE, essentially because a deterministic Turing machine can simulate a nondeterministic Turing machine without needing much more space (even though it may use much more time).^{[3]} Also, the complements of all problems in PSPACE are also in PSPACE, meaning that coPSPACE = PSPACE.
Relation among other classes[edit]
The following relations are known between PSPACE and the complexity classes NL, P, NP, PH, EXPTIME and EXPSPACE (note that ⊊, meaning strict containment, is not the same as ⊈):
It is known that in the first and second line, at least one of the set containments must be strict, but it is not known which. It is widely suspected that all are strict.
The containments in the third line are both known to be strict. The first follows from direct diagonalization (the space hierarchy theorem, NL ⊊ NPSPACE) and the fact that PSPACE = NPSPACE via Savitch's theorem. The second follows simply from the space hierarchy theorem.
The hardest problems in PSPACE are the PSPACEComplete problems. See PSPACEComplete for examples of problems that are suspected to be in PSPACE but not in NP.
Closure properties[edit]
The class PSPACE is closed under operations union, complementation, and Kleene star.
Other characterizations[edit]
An alternative characterization of PSPACE is the set of problems decidable by an alternating Turing machine in polynomial time, sometimes called APTIME or just AP.^{[4]}
A logical characterization of PSPACE from descriptive complexity theory is that it is the set of problems expressible in secondorder logic with the addition of a transitive closure operator. A full transitive closure is not needed; a commutative transitive closure and even weaker forms suffice. It is the addition of this operator that (possibly) distinguishes PSPACE from PH.
A major result of complexity theory is that PSPACE can be characterized as all the languages recognizable by a particular interactive proof system, the one defining the class IP. In this system, there is an allpowerful prover trying to convince a randomized polynomialtime verifier that a string is in the language. It should be able to convince the verifier with high probability if the string is in the language, but should not be able to convince it except with low probability if the string is not in the language.
PSPACE can be characterized as the quantum complexity class QIP.^{[5]}
PSPACE is also equal to P_{CTC}, problems solvable by classical computers using closed timelike curves,^{[6]} as well as to BQP_{CTC}, problems solvable by quantum computers using closed timelike curves.^{[7]}
PSPACEcompleteness[edit]
A language B is PSPACEcomplete if it is in PSPACE and it is PSPACEhard, which means for all A ∈ PSPACE, , where means that there is a polynomialtime manyone reduction from A to B. PSPACEcomplete problems are of great importance to studying PSPACE problems because they represent the most difficult problems in PSPACE. Finding a simple solution to a PSPACEcomplete problem would mean we have a simple solution to all other problems in PSPACE because all PSPACE problems could be reduced to a PSPACEcomplete problem.^{[8]}
An example of a PSPACEcomplete problem is the quantified Boolean formula problem (usually abbreviated to QBF or TQBF; the T stands for "true").^{[8]}
References[edit]
 ^ Arora & Barak (2009) p.81
 ^ Arora & Barak (2009) p.85
 ^ Arora & Barak (2009) p.86
 ^ Arora & Barak (2009) p.100
 ^ Rahul Jain; Zhengfeng Ji; Sarvagya Upadhyay; John Watrous (July 2009). "QIP = PSPACE". arXiv:0907.4737.
 ^ S. Aaronson (March 2005). "NPcomplete problems and physical reality". SIGACT News. arXiv:quantph/0502072. Bibcode:2005quant.ph..2072A..
 ^ Watrous, John; Aaronson, Scott (2009). "Closed timelike curves make quantum and classical computing equivalent". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 465 (2102): 631. arXiv:0808.2669. Bibcode:2009RSPSA.465..631A. doi:10.1098/rspa.2008.0350.
 ^ ^{a} ^{b} Arora & Barak (2009) p.83
 Arora, Sanjeev; Barak, Boaz (2009). Computational complexity. A modern approach. Cambridge University Press. ISBN 9780521424264. Zbl 1193.68112.
 Sipser, Michael (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 053494728X. Section 8.2–8.3 (The Class PSPACE, PSPACEcompleteness), pp. 281–294.
 Papadimitriou, Christos (1993). Computational Complexity (1st ed.). Addison Wesley. ISBN 0201530821. Chapter 19: Polynomial space, pp. 455–490.
 Sipser, Michael (2006). Introduction to the Theory of Computation (2nd ed.). Thomson Course Technology. ISBN 0534950973. Chapter 8: Space Complexity
 Complexity Zoo: PSPACE