On the Computing Power of $+$, $-$, and $\times$

TL;DR

This paper explores how the computational power of a model based on real functions depends on the set of functions used, showing that with basic operations it is either equivalent to polynomial time or contained within it.

Contribution

It introduces a generalized Blum-Shub-Smale model replacing field operations with semialgebraic functions and characterizes the resulting computational classes.

Findings

The class is always contained within the class for basic arithmetic operations.

If the function set includes addition and subtraction and can approximate small numbers, the class is either P or the basic operations class.

A dichotomy is established for classes generated by certain semialgebraic functions.

Abstract

Modify the Blum-Shub-Smale model of computation replacing the permitted computational primitives (the real field operations) with any finite set $B$ of real functions semialgebraic over the rationals. Consider the class of boolean decision problems that can be solved in polynomial time in the new model by machines with no machine constants. How does this class depend on $B$? We prove that it is always contained in the class obtained for $B = \{+, -, \times\}$. Moreover, if $B$ is a set of continuous semialgebraic functions containing $+$ and $-$, and such that arbitrarily small numbers can be computed using $B$, then we have the following dichotomy: either our class is $\mathsf P$ or it coincides with the class obtained for $B = \{+, -, \times\}$.