LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

90 Rodica Ceterchi, Carlos Martín-Vide, and K.G. Subramanian
Similarly, using induction, from
(aba|b, abab|) ⊢r aba,
it follows that aba + ⊆ L(S1), and from
(aba|b, ab|a n ) ⊢r aba n+1 , for any n ≥ 1,
(aba|b, ab|ab) ⊢r aba 2 b,
(aba 2 |b, ab|ab) ⊢r aba n+1 b, for any n ≥ 2,
it follows that aba + b ⊆ L(S1).
For the other inclusion, L(S1) ⊆ L1, weusethecharacterizationofL(S1) as
the smallest language which contains the axioms, and is respected by the splicing
rules. It is a straightforward exercise to prove that L1 contains the axiom of S1,
and that it is respected by the splicing rule of S1.
To show that L1 /∈ SSH(2, 4), note that each word in L1 hasatmosttwo
occurrences of b. Suppose L1 were in SSH(2, 4). Then it would have been generated
by (and thus closed to) rules of one of the forms: (1,a;1,b), (1,b;1,a),
(1,a;1,a), (1,b;1,b). But we have:
(ab|ab, a|bab) ⊢ (1,a;1,b) abbab /∈ L1,
(aba|b, |abab) ⊢ (1,b;1,a) abaabab /∈ L1,
(ab|ab, |abab) ⊢ (1,a;1,a) ababab /∈ L1,
(aba|b, a|bab) ⊢ (1,b;1,b) ababab /∈ L1,
where we have marked the places where the cutting occurs, and all the words
obtained are not in L1 because they contain three occurrences of b.
The language L2 = b + ∪ abb + ∪ b + ab ∪ ab + ab belongs to SSH(2, 4), but not
to SSH(1, 3).
For the first assertion, L2 is generated by the (2, 4)-semi-simple H system
S2 =({a, b}, {abab}, {(1,a;1,b)}). The proof is along the same lines as the proof
of L1 ∈ SSH(1, 3).
For the second assertion, note that each word in L2 hasatmosttwooccurrences
of a. IfL2 were in SSH(1, 3) it would be closed to rules of one of the forms
(a, 1; b, 1), (b, 1; a, 1), (a, 1; a, 1), (b, 1; b, 1). But, even starting from the axiom,
we can choose the cutting places in such a way as to obtain words with three
occurrences of a, and thus not in L2:
(aba|b, ab|ab) ⊢ (a,1;b,1) abaab /∈ L2,
(abab|,a|bab) ⊢ (b,1;a,1) abab 2 ab /∈ L2,
(aba|b, a|bab) ⊢ (a,1;a,1) ababab /∈ L2,
(abab|,ab|ab) ⊢ (b,1;b,1) ababab /∈ L2.
The languages in the proof of Theorem 2 also provide examples of semi-simple
splicing languages which are not simple.
✷