A337537 Period of orbit of Post's tag system ({0,1},{(0,0101100),(1,11000111100000)},10,(1+0^9)^n).

7, 7, 7, 7, 7, 308, 7, 308, 308, 112, 308, 308, 140, 308, 140, 3429251, 140, 308, 140, 802613, 3429251, 140, 140, 3429251, 802613, 3429251, 3429251, 3429251, 3429251, 3429251, 140, 140, 802613, 3429251, 802613, 802613, 140, 802613, 140, 802613, 802613, 3429251

Offset: 1

Author

A.H.M. Smeets, Aug 31 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?

This tag system was proposed by Liesbeth De Mol (p. 329).

a(n) == 0 (mod 7). Proof: for each cycle four times the number of associations (productions) 0 -> 0101100 must equal three times the number of associations (productions) 1 -> 11000111100000 applied within a cycle.

Links

• Liesbeth De Mol, Tracing unsolvability. A historical, mathematical and philosophical analysis with a special focus on tag systems, Ph.D. Thesis, Universiteit Gent (2007). See page 329.
• 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. A284119, A291793, A292091, A336287, A336327.