A336287 Period of orbit of Post's tag system ({0,1},{(0,1),(1,00110)},3,100^n).

2, 0, 28, 0, 20, 0, 28, 0, 28, 144, 28, 0, 28, 0, 10556, 426, 10556, 0, 10556, 426, 10556, 104, 28, 0, 10556, 426, 10556, 104, 28, 0, 10556, 104, 10556, 426, 28, 426, 28, 0, 28, 426, 28, 1896, 10556, 1896, 10556, 0, 28, 426, 28, 426, 10556, 0, 28, 1896, 10556

Offset: 1

Author

A.H.M. Smeets, Jul 16 2020

Keywords

• nonn

Comments

In general a tag as defined by Emil Leon Post, is given by a 4-tuple (Sigma,AF,n,w0), where Sigma is some (nonempty) set of symbols called the alphabet, AF is the associated function (sometimes also called set of production rules) AF: Sigma -> Sigma*, n is the deletion number and w0 the initial string.

From the starting sequence we obtain a new string in each step by adjoining the string associated to the prefix symbol of the string, where after the prefix n symbols are removed from the string.

The decision problem is: will the tag end up in an empty string, a(n) = 0 or not, a(n) <> 0?

a(n) is an even number. Proof: for each cycle the number of associations (productions) 0 -> 1 must equal the number of associations (productions) 1 -> 00110 applied within a cycle.

Links

• Sean A. Irvine, Java program (github)
• Emil L. Post, Formal reductions of the general combinatorial decision problem., American Journal of Mathematics, Vol. 65, No. 2 (Apr., 1943), pp. 197-215.
• Eric Weisstein's World of Mathematics, Tag System

Crossrefs

Cf. A291793, A292091, A336327.

Programs

Python

def step(w):
    i = 0
    while w[0] != alfabet[i]:
        i = i+1
    w = w+suffix[i]
    return w[n:len(w)]
alfabet, suffix, n, ws, w0, m = "01", ["1","00110"], 3, "100", "", 0
while m >= 0:
    w0, m = w0+ws, m+1
    w, ww, i, a = w0, w0, 0, 0
    while w != "" and a == 0:
        w, i = step(w), i+1
        if i%100000 == 0:
            ww = w
        else:
            if w == ww or w == "":
                if w != "":
                    a = i%100000
                print(m,a)