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I want to give examples to explain want I want to know first. Let $G \colon s \mapsto G(s)$ be a PRG.

  1. Let $F_{1} \colon s \mapsto G(s) \Vert b$, where $b = \bigoplus_{k = 1}^{|G(s)|} G(s)[k]$. Obviously, $F_{1}$ is not a PRG, since the last bit can be predicted. But if we remove the last bit, it is a pseudorandom string.

  2. Let $F_{2} \colon s \mapsto G(s) \Vert G(s)$, $F_{2}$ cannot be PRG, too. but if only choose $|G(s)|$ bits properly, it can be pseudorandom.

So, if $F \colon \{0,1\}^l \to \{0,1\}^n$ is a polytime deterministic generator. Perhaps, $F$ is not a PRG, but $F$ may be closed to a PRG. Namely, there exists a compression algorithm $A \colon \{0,1\}^{n} \to \{0,1\}^m$, where $n \geq m > l$, such that $A \circ F$ is a PRG. Is it well-defined? Can we use something like $n - \min_{A} m$ to measure how is $F$ closed to PRGs? Is there a standard way to measure the maximum random bits of all the outputs of $F$?

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You might be looking for computational min-entropy.

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