P vs NP: Arguments against P=NP


by Jaime Sevilla Molina Jun 14 2016 updated Jul 8 2016

Why we believe P and NP are different


Natural proofs

If $~$P \neq NP,$~$ then there are results showing that lower bounds in complexity are inherently harder to prove in a [natural_proof technical yet natural sense]. In other words, if $~$P \neq NP$~$ then proving $~$P \neq NP$~$ is hard.

The opposite proposition for $~$P=NP$~$ is also expected to be true. That is, it would make sense that if $~$P=NP,$~$ then it should be easier to prove, since we could build far more efficient [ theorem provers].

Empirical argument

We have been trying to get a fast algorithm for [ $~$NP$~$-complete] problems since the 70s, without success. And take into account that "we" does not only comprise a minor group of theoretical computer scientists, but a whole industry trying to get faster algorithms for commercial purposes.

One could also argue more weakly that if $~$P=NP$~$, then evolution could have made use of this advantage to sped up its search process, or create more efficient minds to solve $~$NP$~$-complete problems at thoughtspeed.

The arithmetical hierarchy argument

Some authors have drawn an analogy between the [ polynomial hierarchy] and the arithmetical hierarchy.

There are results showing that the [ arithmetical hierarchy is strict], and if the analogy holds then we would have that the polynomial hierarchy is strict as well, [collapseofthepolynomialhierarchy which automatically implies $~$P \neq NP$~$].