Question

Why development is called a universal process



b.Ed answer?

Answer 1

he classic recursion theory [Rog87, Soa87] is based on two fundamental observations. The first is

that there is an effective function φ

k

that enumerates all the k-ary recursive functions. By fixing an

enumeration function we can write φ

k

i

for φ

k

(i), the i-th k-ary recursive function. The number i is

called the G ¨odel number, or the G ¨odel index of the recursive function. The effectiveness of φ

k

i

comes

in both directions. One can effectively calculate a unique number from a given recursive function.

One can also effectively recover a unique recursive function from a given number. The S-m-n

Theorem states that for all k0, k1 there is a total (k0+1)-ary recursive function s

k0

k1

(z, x1, . . . , xk0

) such

that φ

k0+k1

k

(i1, . . . , ik0

, j1, . . . , jk1

) ' φ

k1

s

k0

k1

(k,i1,...,ik0

)

(j1, . . . , jk1

) for all numbers k, i1, . . . , ik0

, j1, . . . , jk1

.

The equality ' means that either both sides are defined and they are equal or neither side is defined.

The second important observation is that there exists a (k+1)-ary universal function Uk

that, upon

receiving an index j of a k-ary recursive function f and k numbers i1, . . . , ik, evaluates f(i1, . . . , ik).

In other words, Uk

(j, i1, . . . , ik) ' φ

k

j

(i1, . . . , ik). The existence of such a universal function depends

crucially on Godelization. It is by G ¨ odelization that we can see a number both as a datum and a ¨

program. The S-m-n Theorem and the universal functions are the foundational tools in recursion

theory. The practical counterpart of a universal function is a general purpose computer. The central

idea of the von Neumann structure of such a computer is that of the stored program, which is

essentially the same thing as Godelization. From the point of view of programming, a universal ¨

function is an interpreter that works by interpreting a datum as a program. Again this is the idea of

Godelization. ¨

1998 ACM Subject Classification: [Theory of computation]: Models of computation—Concurrency.