# Minsky machine

A Minsky machine is a finite-state automaton with access to a number of unbounded registers or counters. These registers can be thought of as monosymbolic stacks (i.e. stacks of all the same symbol.)

### General Minsky machine

In a general Minsky machine, two operations are available on each register:

• increment it; and
• decrement it unless it is zero, in which case follow an alternate state transition.

Minsky machines with two or more registers have been shown to be in the same computational class as Turing machines. Because of this, and because of their simplicity, they can be used to prove several esoteric programming languages to be Turing-complete as well.

The canonical reference is Minsky's book, Computation: Finite and Infinite Machines (Prentice-Hall International, 1967; ISBN 0131655639), in which he calls these machines program machines.

A proof of Turing-completeness is also given in the textbook Aho, Ullman, The Theory of Parsing, Translation, and Compiling, (1972; ISBN 0139145567).

### The section 14.2 Minsky machine

In section 14.2 of Minsky's book, he also describes a variation which is Turing-complete with only one register; this variation requires that the machine has operations to:

• multiply the register by a constant; and
• check to see if it is divisible by a constant, and if so, divide by that constant and effect an alternate state transition.

Multiplying and dividing a number ${\displaystyle 2^{x}3^{y}}$ by prime numbers ${\displaystyle 2}$ and ${\displaystyle 3}$ is computationally equivalent to incrementing and decrementing ${\displaystyle x}$ and ${\displaystyle y}$ respectively. Checking for divisibility is the same as checking if ${\displaystyle x}$ and ${\displaystyle y}$ are equal to zero, since ${\displaystyle p^{0}}$ is not divisible by ${\displaystyle p}$ (as long as ${\displaystyle p\neq 1}$). Therefore, a Minsky machine with functions for multiplication and conditional division can simulate any two-register Minsky machine, which can simulate any Turing machine and is thus Turing complete.

The section 14.2 Minsky machine has been used to demonstrate the Turing-completeness of FRACTRAN.