LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Transducers with Programmable Input by DNA Self-assembly 229
states have superscripts i which are avoided to simplify the reading, we denote
y =(a1,...,ak) andy ′ =(a ′ 1 ,...,a′ k ))
�
(y,s0) for ai =0
(s0, y) ↦→
(y,s1) for ai =1
when 1 is found in the i-th coordinate, change to s1
�
(y,s1) for ai =1
(s1, y) ↦→
(y ′ ,s2) for ai =0,a ′ i = γ
when all 1’s are scanned, change the first 0 into γ
(s2, y) ↦→ (y,s2)
Note that the starting tile for each Mi (i �= k) is[βl,Mi,Mi+1,s i 0]. For i = k,
Mi+1 is substituted with σ2 which is the function that start shifting with the
second coordinate. Similarly, for i = k the end of input (α,...,α)intheendtile
is substituted with (σ2,σ2,...,σ2).
Step 5 Now the shifting of the i-th input to the right of the i − 1 input is
achieved with the function σi. This function corresponds to a transducer with
the following transitions (all states have superscripts i which are avoided to
simplify the reading, hence we denote y =(a1,...,ak) andy ′ =(a ′ 1 ,...,a′ k )):
⎧
⎨ (y,s0) for ai =0,ai−1 �= γ
(s0, y) ↦→ (y,D) for ai =0,ai−1 = γ
⎩
(y ′ ,s1) for ai =1,aj = a ′ j ,a′ i =0(j �= i)
start the shift with s1 unless there is a γ in the i − 1 coordinate
before there is 1 in the i-th coordinate, in that case
go
�
to the final state D
(y,s1) for ai =1
(s1, y) ↦→
(y,s2) for ai = γ,a ′ i =1,aj = a ′ j (j �= i)
copy the 1s, change the end γ into 1 and go to s2
(s2, y) ↦→ (y ′ ,s3) forai =0,α,a ′ i = γ
record the end symbol γ and change to s3, note that ai �
cannot be 1
(y,s3) for ai �= σi, y �= λ
(s3, y) ↦→
(y ′ , !y ′) for y = λ, a′ j = σi (j =1,...,k)
copy the input while there are input symbols, if no input
is to be found indicate the end of input and go to the next shifting,
expand space if necessary
if i