LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

1
01
s 0
Transducers with Programmable Input by DNA Self-assembly 223
0 00 0 10 1 β 0 α
1
01
s 1
(a) (b)
s 0
1
β
s 1
0 γ
00 0
01
10 1
11 0
λ 1
s0 s1 s2 Fig. 2. An encoder that produces the modified frequency modulation code.
and up we have colors encoding input and output symbols. Then a computation
with T is obtained by a process of assembling the tiles such that the abutted
edges are colored with the same color. Only translation, and no rotations of the
tiles are allowed. We describe this process in a better detail below.
2.1 Finite State Machines with Tile Assembly
Tiles: A Wang tile is a unit square with colored edges. A finite set of distinct
unit squares with colored edges are called Wang prototiles. We assume that from
each prototile there are arbitrarily large number of copies that we call tiles. A
tile τ with left edge colored l, bottom edge colored b, top edge colored t and right
edge colored r is denoted with τ =[l, b, t, r]. No rotation of the tiles is allowed.
Two tiles τ =[l, b, t, r] andτ ′ =[l ′ ,b ′ ,t ′ ,r ′ ] can be placed next to each other, τ
to the left of τ ′ iff r = l ′ ,andτ ′ on top of τ iff t = b ′ .
– Computational tiles. For a transducer T with a transition of form (q, a) ↦→
(a ′ ,q ′ ) we associate a prototile [q, a, a ′ ,q ′ ] as presented in Figure 3. If there
are m transitions in the transducer, we associate m such prototiles. These
tiles will be called computational tiles.
q
a’ tile: [ q,a,a’,q’ ]
a
q’
transition: (q,a )
δ
( a’,q’)
00 1
Fig. 3. A computational tile for a transducer.
– Input and output tiles. Additional colors called border are added to the set
of colors. These colors will not be part of the assembly, but will represent
the boundary of the computational assembly. Hence the left border is distinct
from the right border. We denote these with βl,βb,βr,βt for left, bottom,
right and top border. We assume that each input word is from Σ ∗ α
where α is a new symbol “end of input” that does not belong to Σ. For
(c)
01
10 0
11 1