LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Formal Properties of Gene Assembly 211
holds: v2 = wu2, andthenalsou1 = v1w. Sinceγu = γv, the substring w must
be a legal string (if only one occurrence of a pointer p is in w, thenκ would
overlap with p either in u or v, but not in both). Now
u = v1wκ ′ u2(κ − 1)κ and v = v1 κ ′ wu2(κ − 1)κ.
By the form of the substrings wκ ′ in u and κ ′ w in v in the above, w must be
an image ϱκ(α) =w for some α ∈ Θ � κ−2. By Lemma 1, dom(w) =[p, q] forsome
pointers p ≤ q, and by Lemma 2(ii), either w = w ′ q or w = qw ′ for a substring
w ′ . Moreover, as shown in the proof of Lemma 2, in the former case w(q +1)
is a substring of both u and v and in the latter case, (q +1)w is a substring of
both u and v. It then follows that q = κ − 1, and so ϱκ(Mκ−1) occurs twice in
u and v - this is impossible since u and v are realistic. ✷
Acknowledgements. Research supported partly by the MolCoNet project,
IST-2001-32008. T. Harju gratefully acknowledges the support of the Academy
of Finland under project 39802. G. Rozenberg gratefully acknowledges partial
support by NSF grant 0121422.
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