A theory of NP-completeness and ill-conditioning for approximate real computations

We develop a complexity theory for approximate real computations. We first produce a theory for exact computations but with condition numbers. The input size depends on a condition number, which is not assumed known by the machine. The theory admits deterministic and nondeterministic polynomial time recognizable problems. We prove that P is not NP in this theory if and only if P is not NP in the BSS theory over the reals. Then we develop a theory with weak and strong approximate computations. This theory is intended to model actual numerical computations that are usually performed in floating point arithmetic. It admits classes P and NP and also an NP-complete problem. We relate the P vs NP question in this new theory to the classical P vs NP problem. Date: March 12, 2018. 2010 Mathematics Subject Classification. 68Q15; 68Q05, 68Q17 . This work was partially supported by the Smale Institute. 1 ar X iv :1 80 3. 03 60 0v 1 [ cs .C C ] 9 M ar 2 01 8 2 A THEORY OF NP-COMPLETENESS...