Constructive solvability and the P versus NP problem
arXiv:2406.16843
The paper defines a family of decision problems, proves that under a finiteness condition one of them belongs to NP, and shows that no sound, constructible formal theory can prove any of these problems is in P.
Abstract
The relation between the computational complexity class NP and other complexity classes is addressed in the context of provability and limitations on the possibility of finding sound axioms for formal theories. We construct a family D of decision problems and show that under a certain finiteness condition, D contains a problem which is in NP. Further, it is shown that if the term ``constructible theory'' is defined in a way satisfying a specific natural condition, then no constructible and sound theory verifies a solution algorithm for any of the problems in D. Arguably, this solves the P versus NP problem under a constructive interpretation. The relation to classical proofs of NP EXPTIME is discussed. These proofs tacitly use an assumption which may fail for problems in D.
10 pages, no figures. Development from previous version: Title change, correction of misprints and some minor changes of exposition