LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

On Languages of Cyclic Words 281
Proposition 2. (Iteration Lemma) For any M ∈ RAT(◦) there exists a number
N ∈ IN such that for all [x] ∈ M with ||[x]|| >N there exist [u], [v], [w] ∈CV
such that the following holds:
[x] ∈{[u]}◦{[v]}◦{[w]}, ||[u] ◦ [v]|| ≤ N, ||[v]|| > 0,
∀k ≥ 0 {[u]}◦{[v]} k ◦{[w]} ⊆M.
Definition 7. For any word language L ⊆ V ∗ define the set of all equivalence
classes with respect to ∼ by κ(L) ={[x] | x ∈ L}. κ is chosen for the Greek word
κ´υκλoς meaning circle.
For any set M ∈CV let
γ(M) = �
[x] ⊆ V ∗ .
[x]∈M
γ is chosen for γραµµ´η meaning line in Greek.
Trivially, κγ(M) =M. But in general, γκ(L) �= L.
γκ(L) represents the closure of L under cyclic permutation.
Example 1. Let V = {a, b, c, d}. Then:
{[a]}◦{[b]} = {[ab]},
{[a]}◦{[a]}◦{[b]} = {[aa]}◦{[b]} = {[aab]},
{[a]}◦{[b]}◦{[c]} = {[ab]}◦{[c]} = {[a]}◦{[bc]} = {[abc], [acb]},
{[a]}◦{[b]}◦{[c]}◦{[d]} = {[a]}◦{[bc]}◦{[d]} = {[ab]}◦{[cd]}
= {[abcd], [abdc], [adbc], [acbd], [acdb], [adcb]},
{[abc]}◦{[d]} = {[abcd], [adbc], [abdc]},
{[ab]}◦{[ab]} = {[aabb], [abab]}.
Definition 8. (Primitive Cyclic Words) A cyclic word x is called primitive with
respect to ◦ iff it does not fulfill {[x]} ⊆{[y]} k for some y ∈ V ∗ and some k>1,
where the power is meant with respect to ◦.
Note that if x is primitive with respect to catenation ·, thenallξ ∈ [x] are
primitive with respect to ·, too.
Definition 9. Let A ⊆CV . Then the ⊕-algebraic closure L⊕(A) of the set A,
with ⊕∈{⊙, ⊗, ◦}, is defined by
A0 = A,
Aj+1 = Aj ∪ �
L⊕(A) =
[x],[y]∈Aj
∞�
Aj.
j=0
{[x]}⊕{[y]},
Furthermore, if X is any class of sets (languages), let ACL⊕(X) denotethe
class of all ⊕-algebraic closures of sets A ∈ X.