In computation theory, the Blum–Shub–Smale machine, or BSS machine, is a model of computation introduced by Lenore Blum, Michael Shub and Stephen Smale, intended to describe computations over the real numbers. Essentially, a BSS machine is a Random Access Machine with registers that can store arbitrary real numbers and that can compute rational functions over reals in a single time step. It is closely related to the Real RAM model. BSS machines are more powerful than Turing machines, because the latter are by definition restricted to a finite set of symbols. A Turing machine can represent a countable set (such as the rational numbers) by strings of symbols, but this does not extend to the uncountable real numbers.

Definition

A BSS machine M is given by a list I of N+1 instructions (to be described below), indexed. A configuration of M is a tuple (k,r,w,x), where k is the index of the instruction to be executed next, r and w are registers holding non-negative integers, and is a list of real numbers, with all but finitely many being zero. The list x is thought of as holding the contents of all registers of M. The computation begins with configuration (0,0,0,x) and ends whenever k=N; the final content of x is said to be the output of the machine. The instructions of M can be of the following types:

Theory

Blum, Shub and Smale defined the complexity classes P (polynomial time) and NP (nondeterministic polynomial time) in the BSS model. Here NP is defined by adding an existentially-quantified input to a problem. They give a problem which is NP-complete for the class NP so defined: existence of roots of quartic polynomials. This is an analogue of the Cook-Levin Theorem for real numbers.