Collatz Sequence Computed by a Tag System

initial

3

time

0

The Collatz sequence starts with a positive integer

n

. If

n

is even, then divide it by 2. If

n

is odd, then multiply it by 3 and add 1. The sequence is the result of iterating this process.

We can modify this sequence by deleting all the

3n+1

numbers in the sequence by noticing that if

n

is odd, then

3n+1

is even and the next number in the modified sequence should be

(3n+1)/2

.

Liesbeth De Mol described a set of rules for a tag system that computes this slightly modified view of the Collatz sequence. An

m

-tag system is a machine that starts with a finite string of symbols, removes the first

m

of them, and then appends some other string to the end, according to the value of the first symbol of the string at each click of the clock.

At each time step, the machine here removes two symbols from the beginning and then, depending on the first symbol, appends a string according to the rules:

abc

,

ba

,

caaa

.

The colors pink, red, and yellow stand for

a

,

b

, and

c

, respectively.

Each number

k

of the sequence (including the initial condition) is encoded as a string of

k

successive "

a

" symbols (or pink cells).