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A336287 Period of orbit of Post's tag system ({0,1},{(0,1),(1,00110)},3,100^n). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

Table of n, a(n) for n=1..55.

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

PROG

(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)

CROSSREFS

Cf. A291793, A292091, A336327.

Sequence in context: A157305 A306416 A327601 * A337074 A156459 A007218

Adjacent sequences:  A336284 A336285 A336286 * A336288 A336289 A336290

KEYWORD

nonn

AUTHOR

A.H.M. Smeets, Jul 16 2020

STATUS

approved

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Last modified October 23 04:06 EDT 2021. Contains 348211 sequences. (Running on oeis4.)