Septuagint: Difference between revisions

Revision as of 22:49, 3 February 2014

In computational complexity theory, the complexity class #P (pronounced "number P" or, sometimes "sharp P" or "hashtag P") is the set of the counting problems associated with the decision problems in the set NP. More formally, #P is the class of function problems of the form "compute ƒ(x)", where ƒ is the number of accepting paths of a nondeterministic Turing machine running in polynomial time. Unlike most well-known complexity classes, it is not a class of decision problems but a class of function problems.

An NP problem is often of the form "Are there any solutions that satisfy certain constraints?" For example:

Are there any subsets of a list of integers that add up to zero? (subset sum problem)
Are there any Hamiltonian cycles in a given graph with cost less than 100? (traveling salesman problem)
Are there any variable assignments that satisfy a given CNF formula? (Boolean satisfiability problem)

The corresponding #P problems ask "how many" rather than "are there any". For example:

How many subsets of a list of integers add up to zero?
How many Hamiltonian cycles in a given graph have cost less than 100?
How many variable assignments satisfy a given CNF formula?

Clearly, a #P problem must be at least as hard as the corresponding NP problem. If it's easy to count answers, then it must be easy to tell whether there are any answers – just count them and see whether the count is greater than zero.

One consequence of Toda's theorem is that a polynomial-time machine with a #P oracle (P^#P) can solve all problems in PH, the entire polynomial hierarchy. In fact, the polynomial-time machine only needs to make one #P query to solve any problem in PH. This is an indication of the extreme difficulty of solving #P-complete problems exactly.

Surprisingly, some #P problems that are believed to be difficult correspond to easy P problems. For more information on this, see #P-complete.

The closest decision problem class to #P is PP, which asks whether a majority (more than half) of the computation paths accept. This finds the most significant bit in the #P problem answer. The decision problem class ⊕P instead asks for the least significant bit of the #P answer.

The complexity class #P was first defined by Leslie Valiant in a 1979 article on the computation of the permanent, in which he proved that permanent is #P-complete.^[1]

Larry Stockmeyer has proved that for every #P problem P there exists a randomized algorithm using oracle for SAT, which given an instance a of P and ε > 0 returns with high probability a number x such that $(1 - ϵ) P (a) \leq x \leq (1 + ϵ) P (a)$ . The runtime of the algorithm is polynomial in a and 1/ε. The algorithm is based on leftover hash lemma.

References

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External links

Template:CZoo

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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

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

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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+{{Correct title|title=#P|reason=hash}}
+In [[computational complexity theory]], the complexity class '''#P''' (pronounced "number P" or, sometimes "sharp P" or "hashtag P") is the set of the counting problems associated with the [[decision problem]]s in the set '''[[NP (complexity)|NP]]'''. More formally, '''#P''' is the class of function problems of the form "compute ''&fnof;''(''x'')", where ''&fnof;'' is the number of accepting paths of a [[nondeterministic Turing machine]] running in polynomial time.  Unlike most well-known complexity classes, it is not a class of [[decision problem]]s but a class of [[function problem]]s.
+An '''NP''' problem is often of the form "Are there any solutions that satisfy certain constraints?" For example:
+* Are there any subsets of a list of integers that add up to zero? ([[subset sum problem]])
+* Are there any [[Hamiltonian cycle]]s in a given [[graph theory|graph]] with cost less than 100? ([[traveling salesman problem]])
+* Are there any variable assignments that satisfy a given [[conjunctive normal form|CNF]] formula? ([[Boolean satisfiability problem]])
+The corresponding '''#P''' problems ask "how many" rather than "are there any".  For example:
+* How many subsets of a list of integers add up to zero?
+* How many Hamiltonian cycles in a given graph have cost less than 100?
+* How many variable assignments satisfy a given CNF formula?
+Clearly, a '''#P''' problem must be at least as hard as the corresponding '''NP''' problem.  If it's easy to count answers, then it must be easy to tell whether there are any answers – just count them and see whether the count is greater than zero.
+One consequence of [[Toda's theorem]] is that a polynomial-time machine with a '''#P''' [[oracle machine|oracle]] ('''P'''<sup>'''#P'''</sup>) can solve all problems in '''[[PH (complexity)|PH]]''', the entire [[polynomial hierarchy]]. In fact, the polynomial-time machine only needs to make one '''#P''' query to solve any problem in '''PH'''. This is an indication of the extreme difficulty of solving '''#P'''-complete problems exactly.
+Surprisingly, some '''#P''' problems that are believed to be difficult correspond to easy '''[[P (complexity)|P]]''' problems.  For more information on this, see [[sharp-P-complete|#P-complete]].
+The closest decision problem class to '''#P''' is '''[[PP (complexity)|PP]]''', which asks whether a majority (more than half) of the computation paths accept. This finds the most significant bit in the '''#P''' problem answer. The decision problem class '''[[Parity P|&oplus;P]]''' instead asks for the least significant bit of the '''#P''' answer.
+The complexity class '''#P''' was first defined by [[Leslie Valiant]] in a 1979 article on the computation of the [[permanent]], in which he proved that [[permanent is sharp-P-complete|permanent is #P-complete]].<ref>{{cite journal
+  | author = Leslie G. Valiant
+  | title = The Complexity of Computing the Permanent
+  | journal = Theoretical Computer Science
+  | volume = 8
+  | pages = 189–201
+  | publisher = [[Elsevier]]
+  | location =
+  | date = 1979
+  | doi = 10.1016/0304-3975(79)90044-6
+  | issue = 2}}</ref>
+[[Larry Stockmeyer]] has proved that for every #P problem ''P'' there exists a randomized algorithm using oracle for SAT, which given an instance ''a'' of ''P'' and ''ε'' > 0 returns with high probability a number ''x'' such that <math>(1-\epsilon) P(a) \leq x \leq (1+\epsilon) P(a)</math>. The runtime of the algorithm is polynomial in ''a'' and 1/''ε''. The algorithm is based on [[leftover hash lemma]].
+== References ==
+<references/>
+== External links ==
+* {{CZoo|Class #P|Symbols#sharpp}}
+{{ComplexityClasses}}
+[[Category:Complexity classes]]

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