> > > > > > > > > > > > Because you do not check the lines in order. > > > > > > > > > > > > It is always your basic assumption that you > > > > > > > > > > > > first check the first line and after that > > > > > > > > > > > > the next line. That is wrong. > > > > > > > > > > > > That is necessary because you cannot find the n-th > > > > > > > > > > > line unless you know the line number n - 1 or some > > > > > > > > > > > equivalent mark. > > ... > > > > > > Why does it imply checking the previous lines? Why do you not > > > > > > answer > > > > > > that question? > > > > > > You cannot check line n without knowing line n-1. > > > > > But why does that imply *checking* line n-1? Why do you not answer that > > > > question? > > > > It implies knowing line n-1. It implies counting till that number. > > > And still no answer to my question, and what you write here is wrong. In > > the bijection of the rationals > 0 and the naturals I have so often shown, > > to know the rational maped to 66 you need only know the rationals mapped > > to lines 1, 2, 4, 8, 16, 32 and 33. So I need not to know the rational > > mapped to 65. > > > But my question was: "why is knowing a line the same as checking a line"? > > Because you originally asserted: "when checking line n of Cantor's list > > you need to check all previous lines". What is that? Nonsense or not? > > Whether or not it is nonsense, it is irrelevant, since the Cantor > argument applies to all lines simultaneoulsy with a single rule. > > -- > Virgil- Hide quoted text - > > - Show quoted text -
If it is known that if any NP-complete language is sparse (contains no more than a polynomial number of strings of length n), then P = NP. [BH08] improved this result, showing that if any language in NP has an NP-hard set of subexponential density, then coNP is contained in NP/ poly and thus, by [Yap82], PH collapses to the third level. Conceptually, a decision problem is a problem that takes as input some string, and outputs "yes" or "no". If there is an algorithm (say a Turing machine, or a computer program with unbounded memory) which is able to produce the correct answer for any input string of length Failed to parse (<math_output_error>): n
in at most c \cdot n^k steps, where k and Failed to parse (<math_output_error>): c are constants independent of the input string, then we say that the problem can be solved in polynomial time and we place it in the class P. Formally, P is defined as the set of all languages which can be decided by a deterministic polynomial-time Turing machine. That is,
P = {L:L = L(M) for some deterministic polynomial-time Turing machine M}
where L(M) = \{ w\in\Sigma^{*}: M \text{ accepts } w \}
and a deterministic polynomial-time Turing machine is a deterministic Turing machine M which satisfies the following two conditions:
1. M halts on all input w; and 2. there exists k \in N such that T_{M}(n)\in\; O(nk),
where T_{M}(n) = \max\{ t_{M}(w) : w\in\Sigma^{*}, \left|w \right| = n \} and tM(w) = number of steps M takes to halt on input w.
NP can be defined similarly using nondeterministic Turing machines (the traditional way). However, a modern approach to define NP is to use the concept of certificate and verifier. Formally, NP is defined as the set of languages over a finite alphabet that have a verifier that runs in polynomial time, where the notion of "verifier" is defined as follows.
Let Failed to parse (<math_output_error>): L
be a language over a finite alphabet, ?.
L\in\mathbf{NP} if, and only if, there exists a binary relation R \subset\Sigma^{*}\times\Sigma^{*} and a positive integer k such that the following two conditions are satisfied:
1. For all x\in\Sigma^{*}, x\in L \Leftrightarrow\exists y\in\Sigma^ {*} such that (x,y)\in R\; and \left|y\right|\in\;O(\left|x\right|^ {k}); and 2. the language L_{R} = \{ x\# y:(x,y)\in R\} over \Sigma\cup\{\#\} is decidable by a Turing machine in polynomial time.
A Turing machine that decides LR is called a verifier for L and a y such that (x,y)\in R is called a certificate of membership of x in L.
In general, a verifier does not have to be polynomial-time. However, for L to be in NP, there must be a verifier that runs in polynomial time. --MartinMichaelMusatov 07:18, 24 February 2009 (UTC) NPC: NP-Complete
The class of decision problems such that (1) they're in NP and (2) every problem in NP is reducible to them (under some notion of reduction). In other words, the hardest problems in NP.
Two notions of reduction from problem A to problem B are usually considered:
1. Karp or many-one reductions. Here a polynomial-time algorithm is given as input an instance of problem A, and must produce as output an instance of problem B. 2. Turing reductions, in this context also called Cook reductions. Here the algorithm for problem B can make arbitrarily many calls to an oracle for problem A.
Some examples of NP-complete problems are discussed under the entry for NP.
The classic reference on NPC is [GJ79].
Unless P = NP, NPC does not contain any sparse problems: that is, problems such that the number of 'yes' instances of size n is upper- bounded by a polynomial in n [Mah82].
A famous conjecture [BH77] asserts that all NP-complete problems are polynomial-time isomorphic -- i.e. between any two problems, there is a one-to-one and onto Karp reduction. If that's true, the NP-complete problems could be interpreted as mere "relabelings" of one another.
NP-complete problems are p-superterse unless P = NP [BKS95]. This means that, given k Boolean formulas F1,...,Fk, if you can rule out even one of the 2k possibilities in polynomial time (e.g., "if F1,...,Fk-1 are all unsatisfiable then Fk is satisfiable"), then P = NP.
------------------------------------------------------------------------------------------------------------ On Jun 30, 2:54 pm, Virgil <virg...@nowhere.com> wrote: > In article <KM22v0...@cwi.nl>, "Dik T. Winter" <Dik.Win...@cwi.nl> > wrote: > > > > > > > In article <8b2855cd-18d0-44b1-be59-8dd3d8dab...@v2g2000vbb.googlegroups.com> > > WM <mueck...@rz.fh-augsburg.de> writes: > > > On 22 Jun., 15:36, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > ... Results 1 - 10 for cantor p=np. (0.25 seconds)
Cantor's Non-Diagonal Proof « Gödel's Lost Letter and P=NP?No one? referred to Leopold Kronecker who thought Cantor a ?corrupter of youth?, and to Henri Poincaré who thought Cantor's set theory a malady that would ... rjlipton.wordpress.com/2009/04/18/cantors-non-diagonal-proof/
Re: Cantor's argument is erroneous Jun 14, 2009 ... Gravitational Cosmic Lensing os very handy for Musatov's proof P=NP<<» ... May 27 by Martin Musatov - 9 messages - 5 authors. Cantor's ... sci.tech-archive.net/Archive/sci.math/2009-06/msg01643.html
Musatov "null zero rrror!" sets dimensional time block, to ...P=NP For example, one of his results is the famous proof, in 1969, . <...> http://groups.google.com/group/comp.theory?lnk=rgr. Cantor's argument is ...
Cantor's, Godel's, Tarski's, and Turing's Apr 16, 2009 ... to revise current interpretations of Cantor's, Godel's, .... 4.4 P = NP under a constructive interpretation of Peano. Arithmetic ...
Cantor's argument is erroneous - sci.math | Google Groups Jun 13, 2009 ... Message from discussion Cantor's argument is erroneous ... So it seems likely enough that you, the guy flogging the P=NP ...
If not empty, NP?P is topologically large Zim and proves that, in a combination of Cantor and superset topologies, the set NP?P, if not empty, is of the second (Baire) category, while NP-complete ...
P, NP, Co-NP and weak systems of arithmetic - Elsevier(b) For some 1 e N, PA f U( ,, k 1 Q (m, k) p PA NP = CoNP. ...... function (x)Z defined in result 3.5, not the inverse functions of the Cantor pairing). ...
An algebraic version of Cantor-Bendixson analysisdescribe the Cantor- Bendixson pair (i.e. the derivative and nucleus) ... point p of S let p = p- n p where p = A{X c CS : p E X} , pO = {U 0S : p E U}. ...
Cook's Class Contains Pi « Gödel's Lost Letter and P=NPClearly, this can be computed in polynomial time to {p} bits of precision in space at most .... Cantor's Non-Diagonal Proof « Gödel's Lost Letter and P=NP ...
The Cantor-Lebesgue and Denjoy-Luzin properties for double systems ...Cantor--Lebesgue and Denjoy--Luzin properties. 299. We denote by .... lim rain (mp, np) = poo and. < 1. (3.2) flo,, d(x)-_ -7 (p = 1,2 .... ). ...
