LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

280 Manfred Kudlek
Definition 5. Consider
{[x]}⊗{[y]} = �
{[aξaη] | aξ ∈ [x],aη∈ [y]}.
a∈V
Note that {[x]}⊗∅= ∅⊗{[x]} = ∅⊗∅= ∅ and
{[x]}⊗{[λ]} = {[λ]}⊗{[x]} = ∅ .
Also this operation is commutative but not associative.
Lemma 3. The operation ⊗ is a commutative but not an associative.
Proof. Commutatitivity is obvious from the definition. But
({[ac]}⊗{[ab]}) ⊗{[bc]} = {[acab]}⊗{[bc]} = {[bacabc], [cabacb]},
{[ac]}⊗({[ab]}⊗{[bc]}) ={[ac]}⊗{[babc]} = {[acabcb], [cacbab]},
showing that ⊗ is not associative. ✷
Thus, we define another operation ◦, using the shuffle operation , as follows.
Definition 6.
{[x]}◦{[y]} = {[τ] | τ ∈{ξ} {η},ξ ∈ [x],η ∈ [y]} = {[τ] | τ ∈ [x] [y]}.
This operation may be called the shuffle of cyclic words.
Lemma 4. The operation ◦ is a commutative and associative.
Proof. Commutativity is obvious since is a commutative operation.
[σ] ∈ ({[x]}◦{[y]}) ◦{[z]} ⇔∃τ ∈ [x] [y] [σ] ∈{[τ]}◦{[z]}
⇔∃τ ∈ [x] [y] σ ∈ [τ] [z]
⇔ σ ∈ ([x] [y]) [z] =[x] ([y] [z])
⇔∃ρ ∈ [y] [z] σ ∈ [x] [ρ]
⇔∃ρ ∈ [y] [z] [σ] ∈{[x]}◦{[ρ]}
⇔ [σ] ∈{[x]}◦({[y]}◦{[z]}).
Therefore, ◦ is associative. ✷
Consequently, the structure MC =(2 CV , ◦, {[λ]}) is a monoid, and the structure
SC =(2 CV , ∪, ◦, ∅, {[λ]}) isanω-complete semiring.
Thus, systems of equations may be defined [10]. Since ◦ is commutative, the
classes of rational, linear and algebraic sets coincide. The class of such sets will
be denoted by RAT(◦).
Proposition 1. Any M ∈ RAT(◦) can be generated by a right-linear ◦grammar
G = (∆, Σ, S, P ) where ∆ is a finite set of variables, Σ ⊂ CV a
finite set of constants, S ∈ ∆, andP a finite set of productions of the forms
X→Y ◦{[x]}, X→{[x]} with [x] ∈ Σ, and[x] =[λ] only for S→{[λ]} and S
does not appear on any other right hand side of a production.