On Ladner’s Result for a Class of Real Machines with Restricted Use of Constants

Abstract

We study the question whether there are analogues of Ladner’s result in the computational model of Blum, Shub and Smale. It is known that in the complex and the additive BSS model a pure analogue holds, i.e. there are non-complete problems in NP ∖ P assuming NP ≠ P. In the (full) real number model only a non-uniform version is known. We define a new variant which seems relatively close to the full real number model. In this variant inputs can be treated as in the full model whereas real machine constants can be used in a restricted way only. Our main result shows that in this restricted model Ladner’s result holds. Our techniques analyze a class P/const that has been known previously to be crucial for this kind of results. By topological arguments relying on the polyhedral structure of certain sets of machine constants we show that this class coincides with the new restricted version of ${\rm P}_{\mathbb R},$ thus implying Ladner’s result.