en>Yobot: Categories more at one line and/or other fixes using AWB (9466)

2013-09-08T11:02:46Z

Categories more at one line and/or other fixes using AWB (9466)

← Older revision		Revision as of 13:02, 8 September 2013
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	~~Let me initial begin by introducing myself~~. ~~My name~~ is [~~http:~~//~~Depts~~.~~washington~~.~~edu~~/~~handbook~~/~~syndromesBoth~~/~~ch9_ulcers.html Boyd Butts] although it~~ is ~~not~~ the ~~name~~ on ~~my beginning certificate~~. ~~South Dakota is exactly~~ ~~std testing~~ at ~~home where me std testing [http:~~//~~www~~.~~youporntime~~.~~com~~/~~user~~/~~KBiaggini home std test kit] at home over~~ the ~~counter std test and my spouse reside~~. ~~The factor she adores most~~ is ~~[http:~~//~~Gaymenshealth~~.~~org~~/~~std~~-~~information~~/~~herpes~~/ ~~body building] and now she is attempting~~ to ~~make cash with it~~. ~~Hiring is her day job now~~ and ~~she will not change it anytime soon.~~<br><br>~~my web-site~~ ..~~. std~~ ~~[http:~~//~~www~~.~~gaysphere~~.~~net~~/~~blog~~/~~262619 over the counter std test] testing at home; [http:~~//~~exchange~~.~~craftemergency~~.~~org/thankyous/can~~-~~men~~-be-~~sexually-harassed-ladies This Web page~~],		{{Unreferenced\|date=August 2009}}
			In [[computational complexity theory]], '''(SAT, ε-UNSAT)''' is a language that is used in the proof of the [[PCP theorem]], which relates the language [[NP (complexity)\|NP]] to [[probabilistically checkable proof]] systems.

			For a given [[conjunctive normal form\|3-CNF]] formula, Φ, and a constant, ε < 1, Φ is in (SAT, ε-UNSAT) if it is satisfiable and not in (SAT, ε-UNSAT) if the maximum number of satisfiable clauses ([[MAX-3SAT]]) is less than or equal to (1-ε) times the number of [[Clause (logic)\|clauses]] in Φ. If neither of these conditions are true, the membership of Φ in (SAT, ε-UNSAT) is undefined.

			== Complexity ==

			It can be shown that (SAT, ε-UNSAT) characterizes PCP(O(log n), O(1)).

			<math>L \in \mbox{PCP}(O(\log n),O(1))</math>, then <math>L \le ( \mbox{SAT}, \epsilon-\mbox{UNSAT})</math>. (See [[PCP theorem]] for more information)<br>
			Let each bit in the proof ''y'' be <math> y_1, y_2, \ldots, y_m</math>.

			First, it is necessary to encode when the verifier accepts in 3CNF clauses <math>\phi=\bigwedge_r \phi_r</math>.
			Next, for each random string ''r'', construct a sub-formula <math>\phi_r</math>.
			For a fixed ''r'', its possible to determine all the variables queried,
			Enumerate each random string ''r'', and add a clause <math>\phi_r = f_r (y_{i_1}, y_{i_2}, \ldots , y_{i_q}) \ </math>, where <math>f_r</math> is true if and only if the PCP system accepts on reading the given random bits ''r''. There are at most <math>2^q</math> SAT clauses. After these clauses are converted into 3CNF clauses, there are at most <math>q 2 ^ q</math> clauses.

			If <math>x \in L</math>, then there is a proof ''y'' such that is accepted for every random string ''r''. Therefore <math>\phi(y_1,y_2,\ldots,y_m)</math> is satisfiable.<br>
			If <math>x \notin L</math>, then for every assignment to <math>y_1, y_2, \ldots, y_m</math> the corresponding proof causes the verifier to reject for half of the random strings ''r''. For each ''r'' that is rejected one of the clauses in <math>f_r</math> fails. Therefore at least <math>\epsilon = \frac{1}{2 q 2^q}</math> fraction of the clauses fail.

			Therefore <math>L \le (\mbox{SAT}, \epsilon-\mbox{UNSAT})</math>.

			For <math>(SAT, \epsilon-UNSAT) \in PCP(O(\log n), O(1))</math>, let the proof that the PCP system reads be a satisfying assignment for the input [[Conjunctive normal form\|3-CNF]], Φ. The system chooses <math>O(1/\epsilon)</math> clauses of the proof to check if they are truly satisfied. Note that only <math>\log n</math> random bits are needed to choose one of <math>n</math> clauses, and thus only <math>O(\log n/\epsilon) = O(\log n)</math> total random bits are needed. (Remember that ε is a constant.) For each clause to be checked, only 3 bits need to be read, and thus only <math>O(3/\epsilon) = O(1)</math> (a constant number) of bits from the proof need to be read. The system rejects if any of the clauses are not satisfied. If Φ is satisfiable, then there exists a proof (a truly satisfying assignment) that the system will always accept. If Φ is not in (SAT, ε-UNSAT), this means that an ε fraction of the clauses is not satisfiable. The probability that this system will accept in this case is <math>(1-\epsilon)^{1/\epsilon} \leq 1/e < 1/2</math>. Therefore, <math>(SAT, \epsilon-UNSAT) \in PCP(O(\log n), O(1))</math>.

			(SAT, ε-UNSAT) is an [[NP-hard]] language. As part of the proof of the [[PCP theorem]], (SAT, ε-UNSAT) has also been shown to be in PCP(O(log n), O(1)).

			{{DEFAULTSORT:Sat, Unsat}}
			[[Category:NP-complete problems]]

42.3.204.104: small typo (switching places of x, t)

2012-01-25T18:02:15Z

small typo (switching places of x, t)

New page

Let me initial begin by introducing myself. My name is [http://Depts.washington.edu/handbook/syndromesBoth/ch9_ulcers.html Boyd Butts] although it is not the name on my beginning certificate. South Dakota is exactly std testing at home where me std testing [http://www.youporntime.com/user/KBiaggini home std test kit] at home over the counter std test and my spouse reside. The factor she adores most is [http://Gaymenshealth.org/std-information/herpes/ body building] and now she is attempting to make cash with it. Hiring is her day job now and she will not change it anytime soon.<br><br>my web-site ... std [http://www.gaysphere.net/blog/262619 over the counter std test] testing at home; [http://exchange.craftemergency.org/thankyous/can-men-be-sexually-harassed-ladies This Web page],

Predicate abstraction - Revision history

en>Yobot: Categories more at one line and/or other fixes using AWB (9466)

42.3.204.104: small typo (switching places of x, t)