LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

224 Nataˇsa Jonoska, Shiping Liao, and Nadrian C. Seeman
a ∈ Σ an input prototile τa = [c, βb,a,c] is added. There are |Σ| input
prototiles, each representing an input symbol. The left and right sides are
colored with the same color, connect color c. For the “end of input” symbol
α a prototile τα =[c, βb,α,βr] is constructed. The output tiles are essentially
the same as the input tiles, except they have the “top” border:
τ ′ a =[c′ ,a,βt,c ′ ]wherec ′ is another connect color. The output row of tiles
starts with τleft =[βl,β ′ t ,βt,c ′ ] and ends with τright =[c ′ β ′ t ,βt,βr]. For
DNA implementation, βt may be represented with a set of different motiffs
that will fascilitate the “read out” of the result. With these sets of input
and output tiles, every computation with T is obtained as a tiled rectangle
surrounded by boundary colors.
– Start tiles and accepting (end) tiles. Start of the input for the transducer T is
indicated with the start prototile ST =[βl,βb,T,c]whereT is a special color
specifying the transducer T. The input tiles can then be placed next to such
a starting tile. For the starting state s0 a starting prototile τ0 =[βl,T,η,s0]
is constructed. Then τ0 can be placed on top of ST. Thecolorη can be
equal to T if we want to iterate the same transducer or it can indicate
another transducer that should be applied after T. If the computation is to
be ended, η is equal to β ′ t, indicating the start of “top boundary”. For each
terminal state f ∈ F we associate a terminal prototile τf =[f, α, α, βr] if
another computation is required, otherwise, τf =[f,α,β ′ t ,βr] whichstops
the computation.
The set of tiles for executing a computation for transducer T2 that performs
division by 3 (see Figure 1 (b)) is depicted in Figure 4 (a).
Computation: The computation is performed by first assembling the input
starting with tile S and a sequence of input tiles ending with τα. The computation
of the transducer starts by assembling the computation tiles according to the
input state (to the left) and the input symbol (at the bottom). The computation
ends by assembling the end tile τf which can lie next to both the last input tile
and the last computational tile iff it ends with a terminal state. The output result
will be read from the sequence of the output colors assembled with the second
row of tiles and application of the output tiles. In this way one computation
with T is obtained with a tiled 3 × n rectangle (n >2) such that the sides of
the rectangle are colored with boundary colors. Denote all such 3 × n rectangles
with D(T). By construction, the four boundary colors are different and since no
rotation of Wang tiles is allowed, each boundary color can appear at only one
side of such a rectangle. For a rectangle ρ ∈ D(T) wedenotewρthe sequence
of colors opposite of boundary βb and w ′ ρ the sequence of colors opposite of
boundary βt. Then we have the following
Proposition 21 For a transducer T with input alphabet Σ, output alphabet Σ ′ ,
and any ρ ∈ D(T) the following hold:
(i) wρ ∈ L(T) and w ′ ρ ∈ Σ ′ .
(ii) T(wρ) =w ′ ρ