LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Transducers with Programmable Input by DNA Self-assembly 233
4 Recursive Functions as Composition of Transducers
Recursive functions are defined through the so called minimization operator.
Given a function g(x1,...,xn,y)=g(x,y) the minimization f(x) =µyg(x,y)is
defined as:
⎧
⎨ y if y is the least natural number such that
f(x) =µyg(x,y)=
g(x,y)=0
⎩
undefined if g(x,y) �= 0forally
A function f(x) is called total if it is defined for all x. Theclassofrecursive
or computable functions is the class of total functions that can be obtained
from the initial functions z(x),s(x),Ui n (x) with a finite number of applications
of composition, recursion and minimization. The Kleene normal form theorem
shows that every recursive function can be obtained with only one application
of minimization (see for example [5]).
In order to show that all recursive (computable) functions can be obtained
as a composition of transducers we are left to observe that the minimization
operation can be obtained in this same way. As in the case of recursion we
adjust the prototiles for function g into prototiles ¯P (g) which contain pairs as
symbols, but the computation of g is performed on the first coordinate i.e.,
every prototile of the form [q, a, a ′ ,q ′ ] in the set of prototiles for g is substituted
with [q, (a, a2), (a ′ ,a2),q ′ ]wherea2are in {0, 1}. Second coordinates are kept to
“remember” the input for g. Now the minimization is obtained in the following
way (all representations of x are in unary).
– Fix the input x into (x, 0) by the transducer T n add(0) .
– Translate input (x, 0) into (¯x, ¯0) using the translation transducer T2 as presented
in Figure 6. Now each symbol a in the input portion (x, 0) is translated
into a pair (a, a).
– Apply ¯ P (g), hence the result g(x) can be read from the first coordinates of
the input symbols. The second coordinates read (x, 0).
– Until the first coordinate reads 0 continue:
• (*) Check whether g(x) = 0 in the first coordinate. This can be done
with a transducer which “reads” only the first coordinate of the input
pair and accepts strings of the form 0 + 10 + and rejects 0 + 11 + 0 + .
• If the transducer is in “accept” form, then
∗ Change 1 from the first coordinate in the input into 0.
∗ Translate back from pairs of symbols into a single symbol with TR.
∗ Apply U n+1
n+1 ,stop.
• If the transducer is in “reject” form,
∗ Change all 1’s from the first coordinate with 0’s. (I.e., apply z(x) on
the first coordinate.)
∗ Apply T n+1
add(0) to the second coordinate. Hence an input that reads
from the second coordinate (x,y) is changed into (x,y+1).
∗ Copy the second coordinates to the first with transitions: (s, (a1,a2))
↦→ ((a2,a2),s).