# Blum–Shub–Smale machine

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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 often referred to as Real RAM model. BSS machines are more powerful than Turing machines (which in a sense are restricted to storing rational numbers only).

## Definition

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

The instructions of M can be of the following types:

• Computation${\displaystyle (x_{0})}$: a substitution ${\displaystyle x_{0}:=g_{k}(x)}$ is performed, where ${\displaystyle g_{k}}$ is an arbitrary rational function (a quotient of two polynomial functions with arbitrary real coefficients); copy registers r and w may be changed, either by ${\displaystyle r:=0}$ or ${\displaystyle r:=r+1}$ and similarly for w. The next instruction is k+1.
• Branch${\displaystyle (x_{0},l)}$: if ${\displaystyle x_{0}\geq 0}$ then goto l else goto k+1.
• Copy(${\displaystyle x_{r},x_{w}}$): the content of the "read" register ${\displaystyle x_{r}}$ is copied into the "write" register ${\displaystyle x_{w}}$; the next instruction is k+1