# Tag Info

### Proof that circuit upper bounds for $E$ imply $P \neq NP$

I don't think the paper in the other answer contains an answer to your question. Indeed I am not really sure a proof has been published, because the result follows from other well known results. The ...
• 26.7k
Accepted

### How many negations do we need to compute monotone functions?

Markov proved that any function of $n$ inputs can be computed with only $\lceil \log (n+1)\rceil$ negations. An efficient constructive version was described by Fisher. See also an exposition of the ...
• 2,809
Accepted

### Is BPP vs. P a real problem after we know BPP lies in P/poly?

Not sure how much of an answer this is, I'm just indulging in some rumination. Question 1 could be equally asked about P $\neq$ NP and with a similar answer -- the techniques/ideas used to prove the ...
• 7,140
Accepted

### smallest circuit size using XOR gates

This is NP-hard. See: Joan Boyar, Philip Matthews, René Peralta. Logic Minimization Techniques with Applications to Cryptology. http://link.springer.com/article/10.1007/s00145-012-9124-7 The ...
• 26.7k
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• 8,546