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Undid revision 592337649 by 82.34.44.23 (talk) "Compass" hs other meanings, e.g a geometrical instrument, or a synonym of "scope".
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{{unsolved|computer science|Is '''P''' {{=}} '''PSPACE''' ?}}
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{{Infobox Complexity Class
|class=PSPACE
|image=[[File:Complexity subsets pspace.svg|200px]]
|long-name=Polynomial space
|description=
|wheredefined=
|external-urls={{CZoo|PSPACE|P#pspace}}
<!--|dtime=<math>n^{\Omega(1)}</math>, <math>2^{n^{O(1)}}</math>-->
|complete-class='''[[PSPACE-complete]]''' |complement-class=''self'' |proper-supersets=[[EXPSPACE]]<ref name="spacehierarchy">[[Space hierarchy theorem]]</ref> |improper-supersets=[[AlmostPSPACE]],<ref>Definition of Almost-PSPACE. PSPACE ⊆ PSPACE^A for every A.</ref> [[EXPTIME]], '''[[RG (complexity)|RG]]''', [[QPSPACE]]<ref name="zoodiagram">Greg Kuperberg, Complexity Zoology: Active Inclusion Diagram, 2006, http://www.math.ucdavis.edu/~greg/zoology/diagram.xml</ref> |equals='''AP''',<ref>[http://portal.acm.org/citation.cfm?id=322243&coll=Portal&dl=ACM&CFID=60283170&CFTOKEN=44928981 Chandra, A.K. and Kozen, D.C. and Stockmeyer, L.J., 'Alternation'], Journal of the ACM, Volume 28, Issue 1, pp. 114-133, 1981.</ref> '''[[BPPSPACE]]''',<ref>Complexity Zoo, [http://qwiki.stanford.edu/wiki/Complexity_Zoo:A#almostpspace]. Retrieved Mars 25, 2009.</ref> '''[[BQPSPACE]]''',<ref>J. Watrous, [https://cs.uwaterloo.ca/~watrous/Papers/SpaceBoundedQuantumComplexity.pdf Space-bounded quantum complexity], ''J. Comput. Sys. Sci.'' '''59''':2 (1999), pp. 281–326.</ref> '''[[IP (complexity)|IP]]''',<ref>Adi Shamir. [http://portal.acm.org/citation.cfm?doid=146585.146609 IP = PSPACE]. ''Journal of the ACM'', volume 39, issue 4, p.869–877. October 1992.</ref> '''NPSPACE''',<ref>[[Savitch's theorem]]</ref> '''[[PPSPACE]]''',<ref name="papadimitriou85games">{{cite journal|authorlink=Christos Papadimitriou |first=Christos|last=Papadimitriou|title=Games against Nature | journal="Journal of Computer and System Sciences"|volume=31|year=1985}}</ref> [[SAPTIME]]<ref name="papadimitriou85games"/> |related=[[PTIME]] |notequals=P-close, P/log |improper-subsets='''[[CH (complexity)|CH]]''',<ref name="wagner85complexity">{{cite journal|first=K. W.| last=Wagner|title=The complexity of combinatorial problems with succinct representation|journal=Informatica|volume=23|year=1986|pages=325–356}}</ref> P^'''[[PP (complexity)|PP]]''',<ref name="toda89onthe">[[Toda's theorem]]: {{cite journal|first=S.|last=Toda|title=On the computational power of PP and ⊕P|journal=FOCS 1989|year=1989|pages=514–519}}</ref> P^'''[[Sharp-P|#P]]''',<ref name="toda89onthe"/> '''[[QSZK]]''', RG[1] |proper-subsets='''[[NL (complexity)|NL]]'''<ref name="spacehierarchy"/> |properties=Syntactic |low-with=''self'' |low-for=''self'' |closed-reductions=[[Polynomial-time Turing reduction|Poly-time]] |closed-operations= |canonical-problems=[[QSAT]] |models=[[Alternating Turing machine]], [[Turing machine]] }}


In [[computational complexity theory]], '''PSPACE''' is the set of all [[decision problem]]s that can be solved by a [[Turing machine]] using a [[polynomial]] amount of space.
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== Formal definition ==
[http://www.xn--80aeerfhjjg1bv8C.xn--p1ai/%D0%A1%D0%BE%D0%B4%D0%B5%D1%80%D0%B6%D0%B0%D0%BD%D0%B8%D0%B5/master-art-calorie-banana-these-five-tips xn--80aeerfhjjg1bv8c.xn--p1ai]In fact, complete elimination of animal products while supplementing with calcium and Vitamin B-12 is quite healthy. If you order the actual Mc - Donald's smoothie (12 ounces) inside strawberry banana flavor, the calorie content articles are pretty reasonable at 210 calories.<br><br>You could make nut butter and banana sandwich, or a banana and peanut butter smoothie with almond milk, protein powder and cinnamon. Prepare some healthy snacks and take them with you each of the time, in your vehicle, whatever.<br><br>Based with a 2000 calorie diet it gives you you with 10% of the daily saturated fat needs and 5% of your respective carbohydrate requirements. Order the Venti Signature Hot Chocolate with 2% milk so you're [http://support.apdailydeals.com/entries/52328224-The-Human-Brain-On-Food-Things-To-Eat-For-Optimal-Brain-Health looking] at 390 calories, 11g of fat, and 62g of carbohydrates. If you do less exercise on any particular day, eat less. My recommendation: Warm up one'[https://mcpvp.zendesk.com/entries/51817170-3-Ways-To-Get-Rid-Of-Facial-Fatness how much calories a banana has] body before stretching, or perform mild stretches until warmed up.
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<ref name=AB81>Arora & Barak (2009) p.81</ref>
 
:<math>\mathbf{PSPACE} = \bigcup_{k\in\mathbb{N}} \mathbf{SPACE}(n^k). </math>
 
'''PSPACE''' is a strict superset of the set of [[context-sensitive language]]s.
 
It turns out that allowing the Turing machine to be [[Nondeterministic algorithm|nondeterministic]] does not add any extra power.  Because of [[Savitch's theorem]],<ref name=AB85>Arora & Barak (2009) p.85</ref> '''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).<ref name=AB86>Arora & Barak (2009) p.86</ref>  Also, the [[Complement (complexity)|complements]] of all problems in '''PSPACE''' are also in '''PSPACE''', meaning that '''co-PSPACE''' = '''PSPACE'''.
 
== Relation among other classes ==
[[Image:Complexity subsets pspace.svg|300px|thumb|right|A representation of the relation among complexity classes]]
The following relations are known between '''PSPACE''' and the complexity classes '''[[NL (complexity)|NL]]''', '''[[P (complexity)|P]]''', '''[[NP (complexity)|NP]]''', '''[[PH (complexity)|PH]]''', '''[[EXPTIME]]''' and '''[[EXPSPACE]]''' (note that ⊊ is not the same as ⊈):
 
:<math>\mathbf{NL} \subseteq \mathbf{P} \subseteq \mathbf{NP} \subseteq \mathbf{PH} \subseteq \mathbf{PSPACE}</math>
:<math>\mathbf{PSPACE} \subseteq \mathbf{EXPTIME} \subseteq \mathbf{EXPSPACE}</math>
:<math>\mathbf{NL} \subsetneq \mathbf{PSPACE} \subsetneq \mathbf{EXPSPACE}</math>
:<math>\mathbf{P}\subsetneq \mathbf{EXPTIME}</math>
 
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'''.<ref name=AB100>Arora & Barak (2009) p.100</ref>
 
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 (complexity)|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 (complexity)|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 (complexity)|QIP]]'''.<ref>'''QIP''' = '''PSPACE''', Rahul Jain, Zhengfeng Ji, Sarvagya Upadhyay, [[John Watrous (computer scientist)|John Watrous]] {{arXiv|0907.4737}} (July 2009)</ref>
 
'''PSPACE''' is also equal to '''P<sub>CTC</sub>''', problems solvable by classical computers using [[closed timelike curve]]s,<ref>S. Aaronson, [http://arxiv.org/abs/quant-ph/0502072 NP-complete problems and physical reality], ''SIGACT News'', March 2005. [http://arxiv.org/abs/quant-ph/0502072 arXiv:quant-ph/0502072].</ref> as well as to '''BQP<sub>CTC</sub>''', problems solvable by [[quantum computer]]s using closed timelike curves.<ref>{{cite journal | doi=10.1098/rspa.2008.0350|bibcode = 2009RSPSA.465..631A | title=Closed timelike curves make quantum and classical computing equivalent | year=2009 | last1=Watrous | first1=John | last2=Aaronson | first2=Scott | journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | volume=465 | issue=2102 | pages=631 |arxiv = 0808.2669 }}</ref>
 
== PSPACE-completeness ==
{{main|PSPACE-complete}}
A language ''B'' is ''[[PSPACE-complete]]'' if it is in '''PSPACE''' and it is PSPACE-hard, which means for all A ∈ '''PSPACE''', A <math>\leq_p</math> B, where A <math>\leq_p</math> 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.<ref name=AB83>Arora & Barak (2009) p.83</ref>
 
An example of a '''PSPACE'''-complete problem is the [[quantified Boolean formula problem]] (usually abbreviated to '''QBF''' or '''TQBF'''; the '''T''' stands for "true").<ref name=AB83/>
 
== References ==
{{reflist}}
* {{cite book | zbl=1193.68112 | last1=Arora | first1=Sanjeev | authorlink1=Sanjeev Arora | last2=Barak | first2=Boaz | title=Computational complexity. A modern approach | publisher=[[Cambridge University Press]] | year=2009 | isbn=978-0-521-42426-4 }}
* {{cite book|authorlink = Michael Sipser | first=Michael | last=Sipser | year = 1997 | title = Introduction to the Theory of Computation | publisher = PWS Publishing | isbn = 0-534-94728-X}} Section 8.2–8.3 (The Class PSPACE, PSPACE-completeness), pp.&nbsp;281–294.
* {{cite book|authorlink = Christos Papadimitriou | first=Christos | last=Papadimitriou | year = 1993 | title = Computational Complexity | publisher = Addison Wesley | edition = 1st | isbn = 0-201-53082-1}} Chapter 19: Polynomial space, pp.&nbsp;455–490.
* {{cite book| first = Michael | last=Sipser | authorlink = Michael Sipser | year = 2006 | title = Introduction to the Theory of Computation | publisher = Thomson Course Technology| edition = 2nd edition | isbn = 0-534-95097-3}} Chapter 8: Space Complexity
 
* {{CZoo|PSPACE|P#pspace}}
 
==External links==
* [http://www.cs.utoronto.ca/~fpitt/CSC363/20079/lectures/LN11.txt Lecture slides on space complexity From University of Toronto]
* [http://www.cs.princeton.edu/~wayne/kleinberg-tardos/09PSPACE.pdf Lecture slides on space complexity From Princeton University]
 
{{ComplexityClasses}}
 
{{DEFAULTSORT:Pspace}}
[[Category:Complexity classes]]

Latest revision as of 11:04, 10 January 2015

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That 150 calories saved per day would take you a little over 23 days to reduce a pound. Bananas are an excellent source of dietary fiber, in order that they aid the digestive system. This minimizes you from overeating and therefore to enhance your metabolism caused from the burning effect from the foods which you consume. Weight gainers please be sure that you simply will put on weight if you consume three or four bananas every single day with milk. When we're young, were told by our parents, families, and schools about how important it is always to eat properly and maintain a structured diet. You should stay also in foods that includes a high level of fats, especially foods with 'saturated' fats.

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