# Notes

## Section 4: Turing Machines

History [of Turing machines]

Turing machines were invented by Alan Turing in 1936 to serve as idealized models for the basic processes of mathematical calculation (see page 1128). As discussed on page 1110, Turing's main interest was in showing what his machines could in principle be made to do, not in finding out what simple examples of them actually did. Indeed, so far as I know, even though he had access to the necessary technology, Turing never explicitly simulated any Turing machine on a computer.

Since Turing's time, Turing machines have been extensively used as abstract models in theoretical computer science. But in almost no cases has the explicit behavior of simple Turing machines been considered. In the early 1960s, however, Marvin Minsky and others did work on finding the simplest Turing machines that could exhibit certain properties. Most of their effort was devoted to finding ingenious constructions for creating appropriate machines (see page 1119). But around 1961 they did systematically study all 4096 2-state 2-color machines, and simulated the behavior of some simple Turing machines on a computer. They found repetitive and nested behavior, but did not investigate enough examples to discover the more complex behavior shown in the main text.

As an offshoot of abstract studies of Turing machines, Tibor Radó in 1962 formulated what he called the Busy Beaver Problem: to find a Turing machine with a specified number of states that "keeps busy" for as many steps as possible before finally reaching a particular "halt state" (numbered 0 below). (A variant of the problem asks for the maximum number of black cells that are left when the machine halts.) By 1966 the results for 2, 3 and 4 states had been found: the maximum numbers of steps are 6, 21 and 107, respectively, with 4, 5 and 13 final black cells. Rules achieving these bounds are:

The result for 5 states is still unknown, but a machine taking 47,176,870 steps and leaving 4098 black cells was found by Heiner Marxen and Jürgen Buntrock in 1990. Its rule is:

The pictures below show (a) the first 500 steps of evolution, (b) the first million steps in compressed form and (c) the number of black cells obtained at each step. Perhaps not surprisingly for a system optimized to run as long as possible, the machine operates in a rather systematic and regular way. With 6 states, a machine is known that takes about 3.002 × 101730 steps to halt, and leaves about 1.29 ×10865 black cells. (See also page 1144.)

From Stephen Wolfram: A New Kind of Science [citation]