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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.
Formal definition
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 context-sensitive 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 co-PSPACE = PSPACE.
Relation among other classes
The following relations are known between PSPACE and the complexity classes NL, P, NP, PH, EXPTIME and EXPSPACE (note that ⊊ 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 PSPACE-Complete problems. See PSPACE-Complete for examples of problems that are suspected to be in PSPACE but not in NP.
Other characterizations
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 second-order 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 all-powerful prover trying to convince a randomized polynomial-time 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 PCTC, problems solvable by classical computers using closed timelike curves,[6] as well as to BQPCTC, problems solvable by quantum computers using closed timelike curves.[7]
PSPACE-completeness
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. A language B is PSPACE-complete if it is in PSPACE and it is PSPACE-hard, which means for all A ∈ PSPACE, A B, where A B means that there is a polynomial-time many-one reduction from A to B. PSPACE-complete problems are of great importance to studying PSPACE problems because they represent the most difficult problems in PSPACE. Finding a simple solution to a PSPACE-complete problem would mean we have a simple solution to all other problems in PSPACE because all PSPACE problems could be reduced to a PSPACE-complete problem.[8]
An example of a PSPACE-complete problem is the quantified Boolean formula problem (usually abbreviated to QBF or TQBF; the T stands for "true").[8]
References
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- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Section 8.2–8.3 (The Class PSPACE, PSPACE-completeness), pp. 281–294. - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Chapter 19: Polynomial space, pp. 455–490. - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Chapter 8: Space Complexity
External links
- Lecture slides on space complexity From University of Toronto
- Lecture slides on space complexity From Princeton University
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- ↑ Arora & Barak (2009) p.81
- ↑ Arora & Barak (2009) p.85
- ↑ Arora & Barak (2009) p.86
- ↑ Arora & Barak (2009) p.100
- ↑ QIP = PSPACE, Rahul Jain, Zhengfeng Ji, Sarvagya Upadhyay, John Watrous Template:ArXiv (July 2009)
- ↑ S. Aaronson, NP-complete problems and physical reality, SIGACT News, March 2005. arXiv:quant-ph/0502072.
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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