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In [[computational complexity theory]], a [[complexity class]] ''B'' is said to be '''low''' for a complexity class ''A'' if ''A''<sup>''B''</sup> = ''A''; that is, ''A'' with an [[oracle machine|oracle]] for ''B'' is equal to ''A''. Such a statement implies that an [[abstract machine]] which solves problems in ''A'' achieves no additional power if it is given the ability to solve problems in ''B'' at unit cost. In particular, this means that if ''B'' is low for ''A'' then ''B'' is contained in ''A''. Informally, lowness means that problems in ''B'' are not only solvable by machines which can solve problems in ''A'', but are "easy to solve." An ''A'' machine can simulate many oracle queries to ''B'' without exceeding its resource bounds.
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Results and relationships that establish one class as low for another are often called '''lowness''' results.
 
== Results ==
 
Several natural complexity classes are known to be low for themselves. For instance, [[P (complexity)|P]] is low for itself (that is, P<sup>P</sup> = P) because polynomial-time algorithms are closed under composition: a polynomial-time algorithm can make polynomially many queries to other polynomial-time algorithms, while retaining a polynomial running time. [[PSPACE]] is also low for itself, and this can be established by exactly the same argument. Similarly, [[L (complexity)|L]] is low for itself because it can simulate log space oracle queries in log space, reusing the same space for each query.
 
[[Bounded-error probabilistic polynomial|BPP]] is also low for itself and the same arguments almost work for BPP, but one has to account for errors, making it slightly harder to show that BPP is low for itself. Similarly, the argument for BPP almost goes through for [[BQP]], but we have to additionally show that quantum queries can be performed in coherent superposition.<ref name=BernVazi>Bernstein and Vazirani, Quantum complexity theory, [[SIAM Journal on Computing]], 26(5):1411-1473, 1997. [http://www.cs.berkeley.edu/~vazirani/bv.ps]</ref>
Every class which is low for itself is closed under [[complement (complexity)|complement]], provided that it is powerful enough to negate the boolean result. This implies that '''NP''' isn't low for itself unless '''NP''' = co-'''NP''', which is considered unlikely because it implies that the [[polynomial hierarchy]] collapses to the first level, whereas it is widely believed that the hierarchy is infinite. The converse to this statement is not true. If a class is closed under complement, it does not mean that the class is low for itself. An example of such a class is [[EXP]], which is closed under complement, but is not low for itself.
 
Some of the more complex and famous results regarding lowness of classes include:
* Both [[Parity P]] (<math>{\oplus}\hbox{P}</math>) and [[Bounded-error probabilistic polynomial|BPP]] are low for themselves. These were important in showing [[Toda's theorem]].<ref>[http://www.cs.rutgers.edu/~allender/538/murata3.pdf]</ref>
* [[BQP]] is low for [[PP (complexity)|PP]] <ref>L. Fortnow and J. D. Rogers. Complexity limitations on quantum computation. In ''Proceedings of IEEE Complexity '98'', p.202-209. 1998. {{arxiv|cs.CC/9811023}}. </ref> In other words, a program based around taking the majority decision of an unbounded number of iterations of a poly-time [[randomized algorithm]] can easily solve all the problems that a [[quantum computer]] can solve efficiently.
* The [[graph isomorphism problem]] is low for [[Parity P]] (<math>{\oplus}\hbox{P}</math>).<ref>V. Arvind and P. Kurur. Graph isomorphism is in SPP. {{ECCC|2002|02|037}}. 2002.</ref> This means that if we can determine whether an '''NP''' machine has an even or odd number of accepting paths, we can easily solve graph isomorphism. In fact, it was later shown that graph isomorphism is low for '''[[ZPP (complexity)|ZPP]]<sup>NP</sup>'''.<ref>Vikraman Arvind and Johannes Köbler. Graph Isomorphism Is Low for ZPP(NP) and Other Lowness Results. ''Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science'', ISBN 3-540-67141-2, p.431-442. 2000.</ref>
* [[Amplified PP]] is low for [[PP (complexity)|PP]].<ref>L. Li. On the Counting Functions. PhD thesis, University of Chicago. 1993.</ref>
 
== Applications ==
 
Lowness is particularly valuable in relativization arguments, where it can be used to establish that the power of a class does not change in the "relativized universe" where a particular oracle machine is available for free. This allows us to reason about it in the same manner we normally would. For example, in the relativized universe of '''BQP''', '''PP''' is still closed under union and intersection. It's also useful when seeking to expand the power of a machine with oracles, because lowness results determine when the machine's power remains the same.
 
==See also==
* [[Low (computability)]]
 
== References ==
 
<references />
 
[[Category:Computational complexity theory]]

Revision as of 16:59, 26 February 2014

Singapore has elevated a tax on foreign property consumers as a part of new short-term measures to chill its residential housing market which has seen continued sturdy demand regardless of previous efforts to curb costs.

Town-state of Singapore is the place to dwell! Also it's a country the place funding in its the property market yields good return. With a top class transit system, world class health care and first-fee worldwide schools, the city state has become an excellent place for expats to buy a property and lift kids. c) Developers of future EC sale sites from the Authorities Land Gross sales programme will only be allowed to launch models on the market 15 months from the date of award of the sites or after the physical completion of basis works, whichever is earlier. b) Reducing the Mortgage-to-Value (LTV) restrict to eighty% for all housing loans supplied by monetary institutions regulated by the Financial Authority of Singapore (MAS) New Condo Waterfront @ Faber COULD

Many brokers in Singapore need to be able to constantly be in contact with potential clients and new condos in singapore - Read the Full Report, properties which can be up available in the market. Being able to access this database from wherever is essential of their business improvement process. Most property owners as nicely today put up their requirements on websites that host databases like iProperty Singapore, to extend their possibilities of promoting a property or renting them out. The property tax is calculated by multiplying the Annual Worth (AV) of the property with the prevailing property tax price When the sale just starts, I'm not in a hurry to buy. It is completely different from a personal sale open for privileged clients for someday only. Personal Condominium @ EC Costs! From Only $8xxPSF! Belysa EC @ Pasir Ris

Decrease the Loan-to-Value (LTV) restrict for housing loans granted by financial establishments regulated by MAS to those patrons from the present eighty% to 70%. The measures will take quick effect on 30 August 2010. The prolonged SSD won't have an effect on HDB lessees as the required Minimum Occupation Period for HDB flats is at the very least three years. Introducing a Seller's Stamp Duty (SSD) on all residential properties and residential lands which can be bought on or after 20 February 2010 and offered inside 1 yr from the date of purchase ; and The SSD is not going to be relevant to HDB flats as they are already subject to a minimum occupation period of not less than one yr. Loans granted by HDB for HDB flats (together with DBSS flats) will still have an LTV cap of 90%. 7xxpsf view this challenge

SingaporePropertyCycle.com.sg is a Singapore Property web site devoted to offering traders with the graphs and charts to trace the Singapore Property Market Cycle and make better actual property investment selections. SINGAPORE – The Singapore authorities immediately introduced new measures to maintain a steady and sustainable property market, which can take effect by tomorrow, January 14. If there's any new condo launch or property in singapore you search curiosity in, let us know or you'll be able to drop in a mail or a call to us and we will attend to you at our best level Singapore Property Measures - 14 Sep 2009 SG$ 199 Singapore - Home - Sale Singapore Real Property Alternate Property Index, SPI, Singapore Actual Property Alternate and SRX are trade names of SSPL. Revised Rules

If the tenant fails to pay hire as stated within the contract (sometimes 14 days after it is due), the landlord can get hold of a writ of misery from the Court. The Court would then authorize the sheriff to seize and promote goods contained in the property and recuperate unpaid lease and legal costs from the tenant. Landlords also have the precise to repossess the property. District 20, Freehold Strata Cluster home District 10, Freehold house District 10, Freehold Apartment Housing Improvement Board (HDB) Housing Loans To develop your property portfolio in Australia with the leading home lender in Australia for the past ten years Upcoming co ndo In 2013 Octorber Paris Ris EC @ Paris Ris searching for indication of curiosity. Waterwoods Punggol EC @ Punggol FREEHOLD INDUSTRIAL DEVELOPMENT

Zest Carlton – Make it your home in Australia Zest Carlton is a condominium development at Melbourne. Strategically positioned in the bustling heard of Carlton and approximately 3km away from Central Enterprise District (CBD). It takes a few minutes to Sant Ritz near Potong Pasir MRT is now open for VVIP reserving! Sant Ritz has launched on 05 April 2013. Sant Ritz is an upcoming challenge located simply 3 minutes stroll away from Potong Pasir MRT. Sant Ritz european inspired architecture is designed The Meyerise is probably the most prestigious freehold Condominium in Meyer Street District 15! Click on here to register your curiosity for The Meyerise now! At The Meyerise, every factor is uniquely made to compliment the opposite, from the excellent structure to