

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
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OFFSET

1,1


COMMENTS

In general a tag as defined by Emil Leon Post, is given by a 4tuple (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. 197215.
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



