# Implications between $\mathsf{L}=\mathsf{P}$ and $\mathsf{NL}=\mathsf{NP}$?

If we can prove that $\mathsf{L}=\mathsf{P}$, does it imply that $\mathsf{NL}=\mathsf{NP}$ ?

I thought it is the case, but I cannot prove it (also for the converse).

• Proving the converse would be pretty hard... – domotorp Nov 5 '13 at 14:38
• The converse boils to whether NL=P implies L=P. This is not necessarily true unless L=NL. – Mohammad Al-Turkistany Nov 5 '13 at 15:46
• I posted a related question about the relationships between P vs L, NP vs NL, BPP vs BPL, ⊕P vs ⊕L. If you're interested, please feel free to take a look. Thank you! cstheory.stackexchange.com/questions/31073/… – Michael Wehar Jan 17 '19 at 21:55