OFFSET
1,1
COMMENTS
Post's tag system maps a word w over {0,1} to w', where if w begins with 0, w' is obtained by appending 00 to w and deleting the first three letters, or if w begins with 1, w' is obtained by appending 1101 to w and deleting the first three letters.
The empty word is included in the count.
Here, following Asveld, a(n)=0 if the orbit ends at the empty word. On the other hand, Shallit defines a(n) to be 1 if that happens, which gives a different sequence, A284121.
From A.H.M. Smeets, Jul 16 2020: (Start)
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) 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.
Here, the period lengths a(n) refer to the tags ({0,1},{(0,00),(1,1101)},3,100^n).
a(n) is an even number. Proof: for each cycle the number of associations (productions) 0 -> 00 must equal the number of associations (productions) 1 -> 1101 applied within a cycle. (End)
LINKS
Lars Blomberg, Table of n, a(n) for n = 1..6075 (corrected for n=165 by A.H.M. Smeets)
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.
Lars Blomberg, Histogram over non-zero terms
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
EXAMPLE
For n = 2 the orbit of (100)^2 = 100100 consists of a preperiod of length 15, followed by a periodic portion of length 6.
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", ["00", "1101"], 3, "100", "", 0
while m < 83:
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%1000 == 0:
ww = w
else:
if w == ww or w == "":
if w != "":
a = i%1000
print(m, a) # A.H.M. Smeets, Jul 16 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(50)-a(83) from Lars Blomberg, Sep 08 2017
STATUS
approved