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I am a beginner in the computer science track. I have some problems with the following problems

Problem 1: Assume that $NP \ne coNP$. If $L_1, L_2 \in NP$, is $\bar{L_1} \cap L_2$ necessarily in $coNP$.

My attempt: I intend to choose $L_1, L_2$ such that $L_1= \emptyset $, $L_2$ is in $NP$ complete. Then $\bar{L_1} \cap L_2 = L_2$. $L_2$ is in $NP$ complete. Becaues $NP \ne coNP$ then the complement of $L_2$ (denoted by $\overline{L_2}$) is not in $NP$. Then $L_2$ is not in $coNP$.

Could you please help me with checking my work?

Problem 2: What is wrong with the following proof of $P \neq N P$ ?

Suppose that $P=NP$. Then, there exists an algorithm $A$ and a polynomial $p(n)$ such that SAT is decided by $A$ in time $O(p(n))$. Assume that $p(n)=O\left(n^{37}\right)$. By the Time Hierarchy Theorem, there exists a problem $P \in \operatorname{DTIME}\left(n^{38}\right)$ such that $P \notin \operatorname{DTIME}\left(n^{37} \log n^{37}\right)=$ DTIME $\left(n^{37} \log n\right)$. Since SAT is NP-complete, we can reduce $P$ to SAT and decide it in time $O\left(n^{37}\right)$. But we have just shown that $P$ requires time $\omega\left(n^{37} \log n\right)$. This leads to a contradiction, hence the assumption $\mathrm{P}=$ NP must be false.

My attempts: I think the proof is wrong because based on Time Hierarchy Theorem, we have $$\operatorname{DTIME}(o(n^{37})) \subsetneq \operatorname{DTIME}(n^{37} \log n^{37} )$$

So I think that the statement: "there exists a problem $P \in \operatorname{DTIME}\left(n^{38}\right)$ such that $P \notin \operatorname{DTIME}\left(n^{37} \log n^{37}\right)=$ DTIME $\left(n^{37} \log n\right)$ " is a wrong statement.

Could you kindly verify my assessment of this proof and offer any further insights you may have on the matter? I genuinely appreciate your time and assistance in helping me navigate these challenging problems.

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  • $\begingroup$ Please ask only one question per post. We generally discourage "please check whether my answer is correct", as the answer is unlikely to be useful to future visitors. $\endgroup$
    – D.W.
    Oct 28, 2023 at 4:09

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