LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

230 Nataˇsa Jonoska, Shiping Liao, and Nadrian C. Seeman
Since the shifting is performed one at a time, σi potentially has to be repeated
many times. This is obtained by setting the “end of input” of the form
(σi,...,σi,...,σi) which indicates that the next shift is the function σi. The
terminal states D and s3 make distinction between whether the shift σi has to
be repeated (state s3) or the shifting of the i’th coordinates is completed (state
D). When i = k the end tile at state D changes such that (σk,...,σk) is substituted
with TR which is the transducer that translates symbols from k-tuples
back into symbols from Σ.
Step 6 The transition from k-tuples of symbols to a single symbol is done with
the following simple transitions and only one state (note that after shifting, each
n-tuple contains at most one 1):
�
0 if y contains no 1’s
(q0, y) ↦→
1 if y contains a 1
The start tile for TR is [βl,TR,f,q0] indicating that the next computation has
to be performed with tiles for function f andtheendtileforTR is [q0,TR,α,βr]
fixing the input for f in the standard form. After the application of TR, the
input reads the unary representation of (g1(x),...,gk(x)).
Step 7 The composition is completed using tiles for f. These tiles exist by the
inductive hypothesis.
3.3 Recursion
In order to perform the recursion, we observe that the composition and translation
into n-tuples is sufficient. Denote with x the n-tuple (x1,...,xn). Then
we have h(x, 0) = f(x) for some primitive recursive function f. By induction we
assume that there is a set of tiles that performs the computation for the function
f. Forh(x,t+1)=g(h(x,t), x,t), also by the inductive hypothesis we assume
that there is a set of tiles that performs the computation of g(y, x,t).
Set up the input. We use the following input 01 t+1 ν¯x010 where ¯x denotes
the unary representation of the n-tuple x. Wedenotethiswith(t +1| x, 0).
The symbol ν is used to separate the computational input from the counter of
the recursion. Since this input is not standard, we show that we can get to this
input from a standard input for h which is given with an ordered (n + 1)-tuple
(x,y). First we add a 0 unary represented with string 010 at the end of the
input to obtain (x,y,0). This is obtained with the transducer T n+1
add(0) depicted
in Figure 7. Note that for k = 1 the translation function Tk (Figure 6) is in
fact the identity function, hence, there is a set of prototiles that represents the
function Id(x,y,0). Consider the composition Id(U n+1
n+2 ,U1 n+2,...,U n n+2,U n+2
n+2 ).
This composition transforms (x,y,0) into (y, x, 0). We obtain the desired entry
for the recursion by marking the end of the first coordinate input (used as a
counter for the recursion) with symbol ν. This is obtained by adding the following
transitions to Id(x, 0): (i0, 0) ↦→ (0,i0), (i0, 1) ↦→ (1,i1), (i1, 1) ↦→ (1,i1) and