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.^[1] 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.^[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 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

References

1. Blum, Lenore; Shub, Mike; Smale, Steve (1989). "On a Theory of Computation and Complexity over the Real Numbers: NP-completeness, Recursive Functions and Universal Machines" (PDF). Bulletin of the American Mathematical Society. 21 (1): 1–46. doi:10.1090/S0273-0979-1989-15750-9. Zbl 0681.03020.
2. Minsky, Marvin (1967). Computation: Finite and Infinite Machines. New Jersey: Prentice–Hall, Inc.

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.