LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Transducers with Programmable Input by DNA Self-assembly 227
are depicted. Note that the only input accepted by these functions is the correct
representation of x or x. The zero transducer has three states, q0 is initial
and q2 is terminal. The transducer for the function s(x) contains a transition
that adds a symbol 0 to the input if the last 0 is changed into 1. This transition
utilizes the “end of input” symbol α. Thestateq0is the initial and q3
and ! are terminal states for this transducer. The increase of space needed as
a result of computation is provided with the end tile (denoted !) for q3. This
tile is of the form ! =!α =[q3,βb,α,βr] which corresponds to the transition
(q3,λ) ↦→ (α, !). (In DNA implementation, this corresponds to a tile whose bot-
tom side has no sticky ends.) The i-th projection function U n i
accepts as inputs
strings 0 ···01 x1+1 0 s1 1 x2+1 0 ···01 xn+1 0 ···0, i.e, unary representations of x and
it has 2n + 1 states. Transitions starting from every state change 1’s into 0’s
except at the state qi where the i-th entry of 1’s is copied.
0 0
0 0
0 0
10
s0 s1 s2 s3 1 0
1 0
1 0
0 0
0 0
initial terminal
z(x)
initial
1 1
1 1
S(x)
0 1
00
1 0
s0 s1 s1’ s2 initial
1 0 0 0 1 0
α 0
0 0 1 1
n
U (x , x , ..., x n )
i 1 2
λ 0
terminal terminal
1 1
!
s i s i ’ s n
0 0 1 0
1 0
1 0
0 0
sn’ 0 0
terminal
Fig. 5. The three initial functions of the primitive recursive functions
3.2 Composition
Let h be the composition h(x) =f(g1(x),...,gk(x)) where x =(x1,...,xn).
Since the input of h is x and there are k functions that need to be simultaneously
computed on that input, we use a new alphabet ¯ Σ = {(a1,...,ak) | ai ∈ Σ} as an
aid. By the inductive hypothesis, there are sets of prototiles P (gi) thatperform
computations for gi with input x for each i =1,...,k. Also there is a set of
prototiles P (f) that simulates the computation for f. The composition function
h is obtained with the following steps. Each step is explained below.
1. Check for correct input.
2. Translate the input symbol a into k-tuples (a,a,...,a)fora ∈ Σ.