LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

222 Nataˇsa Jonoska, Shiping Liao, and Nadrian C. Seeman
Examples:
1. The transducer T1 presented in Figure 1 (a) has initial and terminal state s0.
The input alphabet is {0, 1} and the output alphabet is Σ ′ = ∅. It recognizes
the set of binary strings that represent numbers divisible by 3. The states
s0,s1,s2 represent the remainder of the division of the input string with 3.
0 1
0
s0 s1 s2 1
0
1
0 0
s 0
1 0
1 1
(a) (b)
Fig. 1. A finite state machine that accepts binary strings that are divisible by 3.
The machine in figure (b) outputs the result of dividing the binary string with
3inbinary
2. The transducer T2 presented in Figure 1 (b) is essentially the same as T1
except that now the output alphabet is also {0, 1}. The output in this case
is the result of the division of the binary string with three. On input 10101
(21 in decimal) the transducer gives the output 00111 (7 in decimal).
3. Our next example refers to encoders. Due to manufacturing constraints of
magnetic storage devices, the binary data cannot be stored verbatim on the
disk drive. One method of storing binary data on a disk drive is by using the
modified frequency modulation (MFM) scheme currently used on many disk
drives. The MFM scheme inserts a 0 between each pair of data bits, unless
both data bits are 0, in which case it inserts a 1. The finite state machine,
transducer, that provides this simple scheme is represented in Figure 2 (a).
In this case the output alphabet is Σ ′ = {00, 01, 10}. If we consider rewriting
of the symbols with 00 ↦→ α, 01↦→ β and 10 ↦→ γ we have the transducer in
Figure 2 (b).
4. A transducer that performs binary addition is presented in Figure 2 (c).
The input alphabet is Σ = {00, 01, 10, 11} representing a pair of digits to
be added, i.e., if x = x1 ···xk and y = y1 ···yk are two numbers written
in binary (xi,yi = {0, 1}), the input for the transducer is written in form
[xkyk] [xk−1yk−1] ··· [x1y1]. The output of the transducer is the sum of
those numbers. The state s1 is the “carry”, s0 is the initial state and all
states are terminal. In [16] essentially the same transducer was simultaed by
gradually connecting TX molecules.
For a given T =(Σ,Σ ′ ,Q,δ,s0,F) the transition (q, a) δ
↦→ (a ′ ,q ′ )schematically
can be represented with a square as shown in Figure 3. Such a square can
be considered as a Wang tile with colored edges, such that left and right we have
the state colors encoding the input and output states of the transition and down
s 1
0 0
0 1
s 2
1 1