LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

226 Nataˇsa Jonoska, Shiping Liao, and Nadrian C. Seeman
3 Primitive Recursive Functions from Transducers
In this section we show how the basic idea of modeling finite transducers by Wang
tiles, with a small modification to allow composition of such computations, can
perform computations of any primitive recursive function over the set of natural
numbers. There is extensive literature on (primitive) recursive functions, and
some introduction to the topic can be found in [5].
The primitive recursive functions are defined inductively as follows:
Definition 31
(i) Three initial functions: “zero” function z(x) =0, “add one” function s(x) =
x +1 and the “i-th projection” function π n i (x1,...,xn) =xi are primitive
recursive.
(ii) Let x denote (x1,...,xn). Iff(x1,...,xk) and g1(x),...,gk(x) are primitive
recursive, then the composition h(x) =f(g1(x),...,gk(x)) is primitive
recursive too.
(iii) Let f(x1,...,xn) and g(y, x1,...,xn,z) be two primitive recursive functions,
then the function h defined recursively (recursion):
h(x1,...,xn, 0) = f(x1,...,xn)
h(x1,...,xn,t+1)=g(h(x1,...,xn,t),x1,...,xn,t)
is also primitive recursive.
Primitive recursive functions are total functions (defined for every input of
natural numbers) and their range belongs to the class of recursive sets of numbers
(see Section [5] and 4). Some of the primitive recursive functions include: x + y,
x · y, x!, x y ... In fact, in practice, most of the total functions (defined for all
initial values) are primitive recursive. All computable functions are recursive
(see Section 4), and there are examples of such functions that are not primitive
recursive (see for ex. [5]).
In order to show that the previous tiling model can simulate every primitive
recursive function we need to show that the initial functions can be modeled in
this manner, as well as their composition and recursion.
3.1 Initial Functions
The input and the output alphabets for the transducers are equal: Σ = Σ ′ =
{0, 1}. An input representing the number n is given with a non-empty list of 0’s
followed with n + 1 ones and a non-empty list of 0’s, i.e., 0 ...01 n+1 0 ...0. That
means that the string 01110 represents the number 2. The input as an n-tuple
x =(x1,...,xn) is represented with
0 ···01 x1+1 0 s1 1 x2+1 0 s2 ···0 sn 1 xn+1 0 ···0
where si > 0. This is known as the unary representation of x. InFigure5,transducers
corresponding to the initial primitive recursive functions z(x),s(x),U n i (x)