Barriers in Complexity Theory

Several proof methods have been successfully employed over the years to prove several important results in Complexity Theory, but no one has been able to settle the question P =?NP. This article surveys three of the main proof methods used in Complexity Theory, and then shows that they cannot be used to prove P =?NP, thus exposing technical barriers to the advance of Complexity Theory. The first of those is diagonalization, which is shown to prove that the Halting Problem is not computable. Then, the concept of relativization is used to prove that diagonalization alone cannot prove P =?NP. Circuits are then introduced as a way of settling P =?NP. Then, the concept of a natural proof is introduced, and is proven to not be able to solve P =?NP. The third method, is algebrization, which is closely related to some results that were proven after the introduction of natural proofs, but that fail to relativize and naturalize. A result, IP = PSPACE, that does not relativize and is used together with non-naturalizing techniques to overcome the barriers of relativization and naturalization in proving lower bounds is exposed and then algebrization is introduced as a framework to understand why the methods used in IP = PSPACE can’t settle P =?NP.