LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

116 Erzsébet Csuhaj-Varjú and Arto Salomaa
We have indicated by boldface the words not obtained by a complementarity
transition.
The beginning of the road of the first component is
110110110110011 ...
Denote by rj,j ≥ 1, the jth bit in the road.
Consider the increasing sequence of numbers, consisting of the positive powers
of 2 and 5:
2, 4, 5, 8, 16, 25, 32, 64, 125, 128, 256, 512, 625, 1024,...
For j ≥ 1, denote by mj the jth number in this sequence.
The next two lemmas are rather immediate consequences of the definitions
of the sequences rj and mj.
Lemma 1 The road of the first component begins with 11 and consists of occurrences
of 11 separated by an occurrence of 0 or an occurrence of 00.
The distribution of the occurrences of 0 and 00 depends on the sequence mj:
Lemma 2 For each j ≥ 1, rj = rj+1 =0exactly in case mj =2mj−1 =4mj−2.
The next lemma is our most important technical tool.
Lemma 3 The bits rj in the road of the first component do not constitute an
ultimately periodic sequence.
Proof of Lemma 3. Assume the contrary. By Lemma 2 we infer the existence of
two integers a (initial mess) andp (period) such that, for any i ≥ 0, the number
of powers of 2 between 5 a+ip and 5 a+(i+1)p is constant, say q. Let2 b be the
smallest power of 2 greater than 5 a .Thus,
and, for all i ≥ 1,
Denoting α =
5 a < 2 b < 2 b+1