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According to Wikipedia and other sources, the question whether $L=P$ is an open problem, and of course everyone is familiar with the problem of whether $P=NP$. However, I found absolutely no information online regarding a possible equality between $L$ and $NP$.

Such an equality doesn't directly violate the space-hierarchy theorem or the time-hierarchy theorem, and so I don't have any idea how to disprove it.

Dean Gurvitz
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  • Have you checked the complexity zoo? – Raphael Jan 19 '19 at 10:26
  • A quick skim of the sections for $L$ and $NP there doesn't answer my question, and in general I think it is elementary enough to deserve its own cs.stackexhange. I did find some somewhat relevant and answered questions in cs.theory, but they were a lot more complicated and with a broader scope. – Dean Gurvitz Jan 19 '19 at 10:29
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    It's an open question: https://rjlipton.wordpress.com/2011/11/11/taking-passes-at-np-versus-l/ – Pål GD Jan 19 '19 at 10:30
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    See also 47522 (If P=NP, then is L=NL?) – Pål GD Jan 19 '19 at 10:32
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    The only thing we know is $\text{L} \subsetneq \text{PSPACE}$. – Pål GD Jan 19 '19 at 10:33
  • @PålGD That's stronger than David's answer -- maybe you want to add something there? Should we close this here as duplicate then? – Raphael Jan 19 '19 at 10:36
  • @DeanGurvitz I find it hard to navigate the zoo myself, just thought to point you there. Afaik it reflects the current state of research so if a relation isn't on there, it's most likely unknown. – Raphael Jan 19 '19 at 10:38
  • Thank you for your help, I do think that an explicit answer to this question would be helpful, and unfortunately isn't a part of David's answer on the linked cs.stackexchange post. – Dean Gurvitz Jan 19 '19 at 10:38
  • Okay, I made my links into a short answer. I agree that the question per se is not a duplicate, but the answer exists on stackexchange. But then again, for which question is that not the case? – Pål GD Jan 19 '19 at 10:42
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    @PålGD Not only "for which question is that not the case?", but also the scope/level of the other questions linked is much broader/higher, and I sincerely think that this post has a right to exist merely because it is a relatively basic and focused question whose answer is hard to find otherwise. – Dean Gurvitz Jan 19 '19 at 10:51

1 Answers1

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The question whether $L = NP$ is an open problem [1], so yes, it is possible. However, it is considered unlikely, or in other words, most believe that $L \subsetneq P \subsetneq NP \subsetneq PSPACE$, but we only know that $L \subsetneq PSPACE$ [2, 3].

References:

Pål GD
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