random-number-generator probability
Updated Fri, 20 May 2022 20:18:16 GMT

On The Next Bit Test

I would like to know what $O(v(n))$ really means in detailed and simple words please. I found it everywhere in the literature I am reviewing but I cannot find what the intuition of it (especially if it means the Big O Notation, then what it has to do with the Probability here; also what is $v$, it is never defined except in another paper as a constant -but maybe not related directly to v in this formula-).

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Many Thanks.

References 1 , 2


But this notation is defined (informally) in the first paper.

The notation $O(\nu(n))$ is used for any function, $f(n)$, that vanishes faster than the inverse of any polynomial, that is for every polynomial, $\mathrm{poly} (n)$, and $n$ large enough, $f(n) \leq 1/\mathrm{poly}(n)$

Therefore, what it means is no probabilistic polynomial time (PPT) algorithm $A$ can guess the next bit at inverse polynomially decreasing error rate.

Given any PPT algorithm $A$ this error probability decays super-polynomially, for example at a rate $\exp(-\log^2 n)$ which goes to zero faster than any polynomial.

Claim: $\exp(-\log^2 n)$ is less than $n^{-c}$ for any constant $c$ for $n$ large enough.

Proof: Look at the reciprocal. $$\exp(\log^2 n) > n^c$$ if and only if $$\log^2 n > c \log n$$ which will clearly happen as soon as $$\frac{\log^2 n}{\log n}>c$$ i.e., as soon as $\log n>c$.

Note that $\exp(\log^{1+\epsilon} n)$ is superpolynomial for all $\epsilon>0.$

Edit: The superpolynomial convergence is essentially what is referred to as negligibility.

Comments (3)

  • +0 – Thanks and +1. So a successful predictor should predict with a probability greater than or equal $1-O((n))$, How is the latter evaluated numerically. That's from where should I derive the $O((n))$ numerically? A simple example of a simple algorithm will be very much appreciated. — Sep 01, 2020 at 09:57  
  • +0 – Is O(v(n)) the error rate? And how is it derived? Practically from the experiment/simulation? Thanks. — Sep 01, 2020 at 10:38  
  • +2 – It is implicitly negligibility. Adding this into answer will increase the awareness. — Sep 01, 2020 at 18:17