# Probability amplication in OWF without hardcore bits

We know that OWF $f$ such that none of its bits are hardcore exists. We also know that given an algorithm that solves a problem with non-negligible probability, we can repeat it many times and take the majority of the answers. Now, to predict $x$ given $f(x)$, we deﬁne predictors $P_i$ for $i$ ∈ $[1..n]$, where $P_i$ predicts the $i$th bit of $x$ with non-negligible probability. We would like to guess all n bits by running $P_1.....P_n$ several times each and taking the majority for each bit.

I want to know does this contradicts the definition of $f$?

The problem with the approach you suggest appears if you write down formally what you suggest: you want to repeat many times each predictor, and evaluate a majority over their predictions. However, the randomness must be taken over a random choice of input. Your predictors give approximately good answers, when averaging over all inputs to $f$. This means that there is no way to implement the strategy you suggest for a fixed input $x$.
Essentially, you can imagine a OWF where each $i$th bit predictor would give a good answer for a non-negligible fraction of the inputs (say, some subset $S_i$ of the domain of $f$), but that does not mean that inputs from $S_i$ are inputs for which the other predictors will give good results.
Here is a quick counter example: suppose that we have a length-preserving OWF $g$, with domain $\{0,1\}^n$, whose outputs are uniformly distributed. You can construct such a function easily using a PRG, which in turn can be based on any OWF. Now, let $f$ be defined as follow: consider a natural ordering of the bitstrings in $\{0,1\}^n$, and divide the domain in $n$ blocks $(B_1, \ldots, B_n)$, each containing $2^n/n$ strings. On input $x$, we define $f(x)$ to be the value obtained by replacing the $i$th bit of $g(x)$ by $0$ if the $i$th bit of $x$ is $0$, where $i$ is the index of the block $B_i$ that $x$ belongs to.
I claim that $f$ is one-way if $g$ is. Here is why: on a challenge $y$, given access to an algorithm $A$ that inverts $g$ with non-negligible probability, call $A$ on all values $(y_0,\ldots,y_n)$, where $y_i$ is obtained by replacing the $i$th bit of $y$ by $1$ (and $y_0$ is just $y$). When receiving $A$'th answers $(x_0,\ldots,x_n)$, evaluate $f$ on $(x_0,\ldots,x_n)$; if one of the evaluations (say, on $x_i$) returns $y$, stop and return $x_i$; otherwise, return $\bot$. As one of the $y_i$ is exactly the correct output of $g$ on a preimage of $y$ by $f$, it is easy to see that the probability of successfully inverting is exactly the success probability of $A$.
I claim that there are predictors $P_i$ predicting the $i$th bit of $x$ with non-negligible probability. $P_i$ simply works as follows: on input $y$, return the $i$th bit of $y$. Why does it work? Because a random input belongs to $B_i$ and has his $i$th bit equal to $0$ with probability $1/2n$, in which case the $i$th bit of the output is $0$ and the predictor always succeeds. Otherwise, the $i$th bit is uniformly distributed and the predictor succeeds with probability $1/2$, hence it has non-negligible success probability overall.