LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

228 Nataˇsa Jonoska, Shiping Liao, and Nadrian C. Seeman
3. For i =1,...,k apply ¯ P (gi). The start prototiles are adjusted as follows: If
the current transducer Ti is to be followed by transducer Tj, then the start
tile for Ti is [βl, Ti, Tj,s0] wheres0 is the initial state for Ti.
4. For all i =1,...,k− 1inthei-th coordinate of the input mark the end of
the last 1.
5. For all i =2,...,k shift all 1’s in the i-th coordinate to the right of all 1’s in
the i − 1 coordinate. This ensures a proper input for the tiles that compute
f. Afterthis,thek-tuples contain at most one 1.
6. Translate back k-tuples into 1 if they contain a symbol 1, otherwise into 0.
7. Apply the tiles that perform the computation for function f.
Step 1 and 2 The input tiles are the same as described in Section 2. The translation
of the input from a symbol to a k-tuple is obtained with the transducer Tk
depicted in Figure 6. Note that this transducer is also checking the correctness
of the input.
0 (0,...0)
1 (1,...1)
0 (0,...0)
1 (1,...1)
s0 s1 s1’ s2 initial
1 (1,...1) 0 (0,...0) 1 (1,...1) 0 (0,...0)
T k
1 (1,...1)
1 (1,...1)
s n
0 (0,...0)
s n
’
terminal
0 (0,...0)
Fig. 6. The transducer Tk that translates symbols into k-tuples
Step 3 Each computational prototile for gi of the form [q, a, a ′ ,q ′ ] is substituted
with a set of prototiles [q, (a1,...,ai−1,a,...,ak), (a1,...,ai−1,a ′ ,...,ak),q ′ ]
where ai is any symbol in the alphabet Σgi used by transducers for gi. The
end tile !α =[qi,βb,α,βr] that increases the computational space is substituted
with ! (α,...,α) =[qi,βb, (α,...,α),βr]. We call this set of prototiles ¯ P (gi). The
idea is to use the tiles ¯ P (gi) to compute the function gi on coordinate i and leave
the rest of the coordinates unchanged.
Step 4 After the application of the tiles ¯ P (g1) to ¯ P (gk), the k-tuples have the
property that for each i the sequence of symbols that appears in the i-th coordinate
represents a number mi = g(x) oftheformwi =0···01 mi 0 ···0α si
for some si ≥ 1. The end of input is obtained with the symbol (α,...,α). In
order to prepare an input for the function f, all1’sinthei-th coordinate have
to appear to the right of the first 0 after the 1’s in the i − 1 coordinate for all
i =2,...,k. Hence, first we mark the end of 1’s in each coordinate with an
additional symbol γ. The transducer Mi substitutes γ instead of the first 0 after
the sequence of 1’s in the i-th coordinate. It has the following transitions (all