LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Transducers with Programmable Input by DNA Self-assembly 231
Check for n −tuple
input with
2n+1 states
α 1
λ 0 λ α
Fig. 7. The transducer T n add(0) that increases the input from n-tuple x to (n+1)tuple
(x, 0).
(i1, 0) ↦→ (ν, s0) where i0,i1 are new states and s0 is the starting state for
Id(x, 0). With this, our entry is in the desired form of (y, x, 0) = (t +1| x, 0).
Execution of the recursion. Each transducer T that is used for computation
of functions f and g is adjusted to T ′ which skips the input until symbol γ is
obtained. This is obtained with one additional state (cT ,i) ↦→ (i, cT )fori =0, 1
and (cT ,ν) ↦→ (ν, s0) wheres0is the starting state for T .Asinthecaseof
composition we further adjust the prototiles for functions f and g into prototiles
¯P (f) ′ and ¯ P (g) ′ which read/write pairs of symbols, but the computation of f and
g is performed on the first coordinate, i.e., every prototile of the form [q, a, a ′ ,q ′ ]
in the set of prototiles for f and g is substituted with [q, (a, a2), (a ′ ,a2),q ′ ]where
a2 are in {0, 1}. Second coordinates are kept to “remember” the input for f and g
that is used in the current computation and will be used in the next computation
as well.
In this case the recursion is obtained by the following procedure:
– Translate input (t+1| x, 0) into (t+1| ¯x, ¯0) using the translation transducer
T2 as presented in Figure 6 and adjusted with the additional state to skip
the initial t +1.Noweachsymbolain the input portion (x, 0) is translated
into a pair (a, a).
– Apply ¯ P (f) ′ , hence the result f(x) can be read from the first coordinates of
the input symbols and the input (x, 0) is read in the second coordinates.
– Mark with M ′ 1 the end of input from the first coordinate. This is the same
transducer M1 as used in the composition, except in this case there is an
extra state cM1 that skips the counter t + 1 at the beginning.
– Shift coordinates using σ ′ 2 , i.e., the same transducer as σ2 for the composition
with the additional state cσ2 to skip the counter.
– Translate back from pairs into {0, 1} with T ′ R . Now the result reads as (t +
1 | f(x), x, 0), i.e., h(x, 0) = f(x) isreadrightaftersymbolν.
– For an input (t +1| f(x), x, 0) reduce t + 1 for one. This is done with transducer
presented in Figure 8 (a). The new input reads (t | f(x), x, 0).
– Check for end of computation with transducer that accepts 0 ···0νw for any
word w. Note that the language 0∗ν(0 + 1) ∗ is regular and so accepted by
a finite state transducer that has each transition with output symbols same
as input symbols.
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