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I've been trying to wrap my head around the proof provided in this answer. I understand that $P$ is a class where languages can be decided by a Turing Machine and that $P/poly$ is a bigger class that also includes languages that can be decided by circuits.

Take a language $L$ which is not in $\mathsf{E} = \bigcup_{c=1}^\infty \mathsf{TIME}(2^{cn})$. Now consider the language $L' = \{1^m : m \in L\}$. Then $L'$ is clearly in $\mathsf{P/poly}$, but it's not in $\mathsf{P}$: if it were decidable in time $O(m^k)$, then we could decide $L$ in time $O((2^n)^k)$, and so $L$ would be in $\mathsf{E}$. Our decision procedure works as follows: on input $m$ of length $n = \log m$, we run the algorithm for $L'$ on the input $1^m$. This runs in time $O(m^k) = O((2^n)^k)$.

It remains to ensure that $L'$ is decidable. To that end, all we need to do is to choose some $L \notin \mathsf{E}$ which is decidable, for that makes $L'$ trivially decidable: given an input, if it's not of the form $1^m$, reject; otherwise, answer according to whether $m \in L$.

The existence of a decidable language $L \notin \mathsf{E}$ is guaranteed by the time hierarchy theorem.

I have a couple of questions about the understanding of this proof.

Then I am confused as to how we first define language $L$ to not be in $E$ and then we show that actually is in $E$ since $L^{'}$ can be computed within the limits...

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  • $\begingroup$ $P$ and $E$ are not the same. By the time-hierarchy theorem $P \subseteq \text{TIME}(2^{n}) \subsetneq \text{TIME}(2^{2n}) \subseteq E$. $\endgroup$
    – Steven
    Nov 11, 2023 at 19:16
  • $\begingroup$ Oh thank you! But still, $L$ is firstly said to not be in $E$ but then we show that is runs in $O((2^{n})^{k})$ and hence is in $E$? $\endgroup$
    – Meki21
    Nov 11, 2023 at 19:20
  • $\begingroup$ Are you familiar with proofs by contradiction? The answer is choosing $L \not\in E$ and then showing that $L' \not\in P$ by contradiction. To do so it assumes that $L' \in P$ and concludes that this would imply $L \in E$. Since this is indeed a contradiction (as $L \not \in E$), it must be $L' \not\in P$. $\endgroup$
    – Steven
    Nov 11, 2023 at 19:30
  • $\begingroup$ Ohh I see it now - I didn’t recognise contradiction through this structure $\endgroup$
    – Meki21
    Nov 11, 2023 at 20:00
  • $\begingroup$ What exactly is your question? Please ask only one question per post. Please make sure to articulate your question explicitly. A question normally ends in '?'. $\endgroup$
    – D.W.
    Dec 8, 2023 at 20:56

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