# Implications of $NL=P$

What would be some implications of $$NL$$$$=P$$? Would it be possible to get recommendations on good sources/papers I can read to learn more about this? Thank you

• No, Non deterministic logarithmic space. I believe implications of P=NP is widely available. – user86212 Mar 31 '18 at 23:49
• That's what I thought, but someone tried to 'correct' this into 'NP'. Let me clarify this. – Discrete lizard Apr 1 '18 at 6:53
• You might want to take a look at consequences of $NC=P$. This is a weaker statement, but more widely discussed (parallelism of P-complete problems). – Ariel Apr 1 '18 at 15:11

$$NL = P$$ means that every language in $$P$$ can be decided by a nondeterministic Turing machine using at most $$O(\log n)$$ space, where $$n$$ is the size of the input.
Savitch's theorem states for all $$f(n) \in \Omega(\log n)$$: $$NSPACE(f(n))\subseteq SPACE(f(n)^2)$$.
Therefore $$NL = P$$ implies that every language in $$P$$ can be decided by a deterministic Turing machine using at most $$O((\log n)^2)$$ space.