Blum–Shub–Smale machine

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 $0,1,\dots ,N$. 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 $x=(x_0,x_1,\dots )$ 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:

• Computation: a substitution $x_0:=g_k(x)$ is performed, where $g_k$ is an arbitrary rational function (a quotient of two polynomial functions with arbitrary real coefficients); registers $r$ and $w$ may be changed, either by $r:=0$ or $r:=r+1$ and similarly for $w$. The next instruction is $k+1$.
• Branch($l$): if $x_0\ge 0$ then goto $l$; else goto $k+1$.
• Copy($x_r,x_w$): the content of the "read" register $x_r$ is copied into the "written" register $x_w$; the next instruction is $k+1$.

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.

See also

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

Further reading

• Blum, Lenore; Cucker, Felipe; Shub, Mike; Smale, Steve (1998). Complexity and Real Computation. Springer New York. doi:10.1007/978-1-4612-0701-6. ISBN 978-0-387-98281-6. S2CID 12510680. Retrieved 23 March 2022.
• Bürgisser, Peter (2000). Completeness and reduction in algebraic complexity theory. Algorithms and Computation in Mathematics. Vol. 7. Berlin: Springer-Verlag. ISBN 3-540-66752-0. Zbl 0948.68082.
• Grädel, E. (2007). "Finite Model Theory and Descriptive Complexity". Finite Model Theory and Its Applications (PDF). Springer-Verlag. pp. 125–230. Zbl 1133.03001.