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| {{unsolved|computer science|Is '''P''' {{=}} '''PSPACE''' ?}}
| | Home improvement is not an area reserved for the experts; nearly anyone can tackle a home project. Improvements can give personality and character to your house. You can probably tackle most of the jobs yourself. If you adored this write-up and you would such as to obtain more information pertaining to [http://www.homeimprovementdaily.com homes ideas] kindly see our web site. For some help, follow the advice in this article.<br><br>Know what style you are looking for before beginning any type of home improvement. Identifying your decorating style will have a serious impact on the choices that you make. Without a plan, you can easily end up with mixed styles. Mistakes like this one can result in you having to backtrack and are also a waste of money.<br><br>You should avoid furniture with a busy upholstery. If you buy busy-looking pieces, you'll end up decorating your rooms around them. Buy your furniture in solid tones such as green, brown, blue, black or beige and use your accessories to benefit your furniture. Patterns are easier to take in small doses such as blankets and toss pillows.<br><br>Get a motion activated light for your front porch to help you keep your energy costs low, while keeping your yard safely lit. These types of lights can be manually turned on and the sensitivity can also be adjusted.<br><br>Remember to think about landscaping when thinking of home improvement. People look at the front of your home when they first see it, and this can leave a lasting impression. Ensure your grass is cut and clean, and think about planting shrubbery and some other flowers to make it look great.<br><br>Your backyard is an important resource. Backyards and outdoor areas can be converted into an extended part of your overall living space. This area would be great for entertaining, barbecuing or just relaxing.<br><br>Write down all the items that you need prior to shopping from a home improvement project. A list of everything you will need will keep you from taking multiple trips. It will also keep you better organized.<br><br>Add some personality like a nice bold print if you think your room doesn't have personality. You could add animal printed rugs, pillows or paintings.<br><br>Place peepholes on your doors. Then, you can know who is on the other side of the door before you open it. Peepholes are a great security feature and a cheap home improvement project. All you will need is a power drill and some extra time. A peephole will give you that extra sense of reassurance when someone knocks.<br><br>Any decent home improvement project will include laying brick and purchasing as many as necessary beforehand would make things flow much quicker. When you have to do additions to early brick work, you may not be able to find bricks that are new that match the original ones. Brick styles are constantly changing. To avoid problems with brick-matching in the future, create a store of extra original blocks, so you have replacments when needed.<br><br>If you'd like to brighten up a room in your house, look into replacing dingy older outlet covers with brand new ones. Your outlets will look discolored and old over time. So, changing them out for new, fresh covers, can have quite an visual impact on any room.<br><br>You should now be aware that home improvement is not nearly as daunting as it first appears. With a bit of good advice and know how, you are well on your way to cost savings, increased re-sale value and a much more enjoyable home. |
| {{Infobox Complexity Class
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| |class=PSPACE
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| |image=[[File:Complexity subsets pspace.svg|200px]]
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| |long-name=Polynomial space
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| |description=
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| |wheredefined=
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| |external-urls={{CZoo|PSPACE|P#pspace}}
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| <!--|dtime=<math>n^{\Omega(1)}</math>, <math>2^{n^{O(1)}}</math>-->
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| |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]] }}
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| 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 ==
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| 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>
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| :<math>\mathbf{PSPACE} = \bigcup_{k\in\mathbb{N}} \mathbf{SPACE}(n^k). </math>
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| '''PSPACE''' is a strict superset of the set of [[context-sensitive language]]s.
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| 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'''. | |
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| == Relation among other classes ==
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| [[Image:Complexity subsets pspace.svg|300px|thumb|right|A representation of the relation among complexity classes]]
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| 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 ⊈):
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| :<math>\mathbf{NL} \subseteq \mathbf{P} \subseteq \mathbf{NP} \subseteq \mathbf{PH} \subseteq \mathbf{PSPACE}</math>
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| :<math>\mathbf{PSPACE} \subseteq \mathbf{EXPTIME} \subseteq \mathbf{EXPSPACE}</math>
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| :<math>\mathbf{NL} \subsetneq \mathbf{PSPACE} \subsetneq \mathbf{EXPSPACE}</math>
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| :<math>\mathbf{P}\subsetneq \mathbf{EXPTIME}</math>
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| 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.
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| 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.
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| 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'''.
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| == Other characterizations ==
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| 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>
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| 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]]'''.
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| 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.
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| '''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>
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| '''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>
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| == PSPACE-completeness ==
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| {{main|PSPACE-complete}}
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| 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>
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| 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/>
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| == References ==
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| {{reflist}}
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| * {{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 }}
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| * {{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. 281–294.
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| * {{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. 455–490.
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| * {{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
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| * {{CZoo|PSPACE|P#pspace}}
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| ==External links==
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| * [http://www.cs.utoronto.ca/~fpitt/CSC363/20079/lectures/LN11.txt Lecture slides on space complexity From University of Toronto]
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| * [http://www.cs.princeton.edu/~wayne/kleinberg-tardos/09PSPACE.pdf Lecture slides on space complexity From Princeton University]
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| {{ComplexityClasses}}
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| {{DEFAULTSORT:Pspace}}
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| [[Category:Complexity classes]]
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Home improvement is not an area reserved for the experts; nearly anyone can tackle a home project. Improvements can give personality and character to your house. You can probably tackle most of the jobs yourself. If you adored this write-up and you would such as to obtain more information pertaining to homes ideas kindly see our web site. For some help, follow the advice in this article.
Know what style you are looking for before beginning any type of home improvement. Identifying your decorating style will have a serious impact on the choices that you make. Without a plan, you can easily end up with mixed styles. Mistakes like this one can result in you having to backtrack and are also a waste of money.
You should avoid furniture with a busy upholstery. If you buy busy-looking pieces, you'll end up decorating your rooms around them. Buy your furniture in solid tones such as green, brown, blue, black or beige and use your accessories to benefit your furniture. Patterns are easier to take in small doses such as blankets and toss pillows.
Get a motion activated light for your front porch to help you keep your energy costs low, while keeping your yard safely lit. These types of lights can be manually turned on and the sensitivity can also be adjusted.
Remember to think about landscaping when thinking of home improvement. People look at the front of your home when they first see it, and this can leave a lasting impression. Ensure your grass is cut and clean, and think about planting shrubbery and some other flowers to make it look great.
Your backyard is an important resource. Backyards and outdoor areas can be converted into an extended part of your overall living space. This area would be great for entertaining, barbecuing or just relaxing.
Write down all the items that you need prior to shopping from a home improvement project. A list of everything you will need will keep you from taking multiple trips. It will also keep you better organized.
Add some personality like a nice bold print if you think your room doesn't have personality. You could add animal printed rugs, pillows or paintings.
Place peepholes on your doors. Then, you can know who is on the other side of the door before you open it. Peepholes are a great security feature and a cheap home improvement project. All you will need is a power drill and some extra time. A peephole will give you that extra sense of reassurance when someone knocks.
Any decent home improvement project will include laying brick and purchasing as many as necessary beforehand would make things flow much quicker. When you have to do additions to early brick work, you may not be able to find bricks that are new that match the original ones. Brick styles are constantly changing. To avoid problems with brick-matching in the future, create a store of extra original blocks, so you have replacments when needed.
If you'd like to brighten up a room in your house, look into replacing dingy older outlet covers with brand new ones. Your outlets will look discolored and old over time. So, changing them out for new, fresh covers, can have quite an visual impact on any room.
You should now be aware that home improvement is not nearly as daunting as it first appears. With a bit of good advice and know how, you are well on your way to cost savings, increased re-sale value and a much more enjoyable home.