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When I studied computer science 10 years ago, it was still an open question whether $\mathsf{L}$ and $\mathsf{NP}$ are truly different classes. Is that still the case or has the inequality been proven in the meantime?

(I know this is a somewhat stupid question, but there's nothing about it in the complexity zoo, and googling L and NP is surprisingly ineffective ;-))

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    $\begingroup$ books.google.co.il/… You can check out the paragraph about separating $L, NP$ here (section 3 in the linked page). $\endgroup$ – Ariel May 6 '16 at 7:38
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    $\begingroup$ Possible, but unlikely. $\endgroup$ – Yuval Filmus May 6 '16 at 8:21
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$L \subseteq NL \subseteq P \subseteq NP \subseteq PSPACE \subseteq EXP$

From space hierarchy and time hierarchy theorems we can prove that

$L \subsetneq PSPACE$

$NL \subsetneq NPSPACE$

Note that we know $PSPACE = NPSPACE$.

$P \subsetneq EXP$

$NP \subsetneq NEXP$

Note that we don't know if $EXP \subsetneq NEXP$.

Whether the other inclusions are strict, are still open problems. Even though there are no proofs, many, but not all, believe most of the inclusions to be strict.

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    $\begingroup$ Note that ​ NPSPACE = PSPACE . ​ ​ ​ ​ $\endgroup$ – user12859 May 6 '16 at 8:39
  • $\begingroup$ True but I was using only space and time hierarchy theorems. And I did not mention NPSPACE in the inclusion list. $\endgroup$ – Shreesh May 6 '16 at 9:39
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    $\begingroup$ Anyway I have added a note that NPSPACE=PSPACE. $\endgroup$ – Shreesh May 6 '16 at 9:42

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