LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Remarks on Relativisations and DNA Encodings 135
Moreover, I-compliance is equivalent to enjoying the first property required
by solidity, which we called solidity/1. Therefore, if a language L is solid, or L
is solid relative to a language in which all strings in L are factors, then L is also
I-compliant.
Finally, it is obvious that if L is solid/1 then it is solid/1 relative to any
language.
The following simple lemma considers another fact about solidity/1 properties,
and will also be used later. It applies, for instance, to the case of M = L,
or M = L 2 ,orM = L ∗ .
Lemma 1. A language L is solid/1 iff L is solid/1 relative to a language M in
which all strings in L are factors.
Proof. Solidity/1 is obviously stronger than any relative solidity/1; on the other
hand, relative solidity/1 of M verifies that no word of L is proper substring of
any other.
Proposition 1. A language L is solid relative to L ∗ iff L is solid relative to L 2 .
Proof. It is immediate to verify that solidity relative to L ∗ implies solidity relative
to L 2 ,sinceL 2 is a subset of L ∗ . We prove the other direction by contradiction:
if L is not solid relative to L ∗ then either L falsifies condition 1, and
the same would be relatively to L 2 ,orawordinL 2 falsifies relative solidity, and
this itself would be the contradiction, or a word w = ypuqz ∈ L n+1 ,withpq non
null and n>1, would contradict solidity. This last case would falsify solidity
relative to L n in one of the ways that we will see, and so lead to contradiction
for L 2 .Ifw has a parsing in L n+1 according to which either yp has a prefix or qz
has a suffix in L, then we could build from w, by dropping such prefix or suffix,
a word which would falsify solidity in L n . If no such parsing exists, then pu or
uq falsify condition 1 of relative solidity, by spanning on at least one word of L
in the string w.
Proposition 2. A language L is I-free iff L is solid relative to L ∗ .
Proof. Proposition 1 allows us to reduce the statement to consider only equivalence
with solidity relative to L 2 . We know that I-freedom implies I-compliance,
which in turns implies solidity/1 and, by Lemma 1, solidity/1 relative to M = L 2 .
By absurd, if L is I-free but would not enjoy second property required by the
relative solidity, then we would have a string w ∈ L 2 such that w = ypuqz with
u non null and pu, uq ∈ L. This would contradict solidity/1 relative to L 2 ,in
case yz, orp or q, would be null, or I-freedom in other cases; for instance, if y
and q are non null, word pu ∈ L would falsify I-freedom.
In the other direction, the solidity relative to L 2 implies the I-freedom, otherwise
we would have a word st = yuz ∈ L 2 such that u ∈ L and both y and z
are non null; u would contradict property 1 of relative solidity, if it is a substring
of s or t, or property 2 otherwise.