Blum–Shub–Smale machine

In

BSS machines are more powerful than Turing machines, because the latter are by definition restricted to a finite set of symbols.^[2] 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 ${\displaystyle I}$ 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 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: 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); 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${\displaystyle (l)}$: if ${\displaystyle x_{0}\geq 0}$ then goto ${\displaystyle l}$; else goto k+1.
• Copy(${\displaystyle x_{r},x_{w}}$): the content of the "read" register ${\displaystyle x_{r}}$ is copied into the "written" register ${\displaystyle x_{w}}$; the next instruction is k+1

Theory

Blum, Shub and Smale defined the

See also

• Complexity and Real Computation
• General purpose analog computer
• Hypercomputation
• Real computer
• Quantum finite automaton

References