1. Pairing Functions and Gödel Numbers
x, y + 1 = 2x(2y + 1).
Examples:
x, y = 41
x = 1
y = 10
x, y = 31
x = 5
y = 0
x, y = 36
x = 0
y = 18
x, y = 0
y = 0
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

2. Pairing Functions and Gödel Numbers
This way the equation x, y = z defines functions x = l(z) and y = r(z).
x, y = z also implies that x, y < z + 1, and therefore l(z)  z, r(z)  z.
Then we can write:
l(z) = minxz[(y)z(z = x, y)],
r(z) = minyz[(x)z(z = x, y)],
showing that l(z) and r(z) are primitive recursive functions.
It is also true that x, y = z  x = l(z) & y = r(z).
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

3. Pairing Functions and Gödel Numbers
Theorem 8.1 (Pairing Function Theorem):
The functions x, y, l(z) and r(z) have the following properties:
1. they are primitive recursive;
2. l(x, y) = x; r(x, y) = y;
3. l(z), r(z) = z;
4. l(z), r(z)  z.
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

4. Pairing Functions and Gödel Numbers
We now want to develop primitive recursive functions that encode and decode arbitrary finite sequences of numbers.
Our method (actually invented by Gödel) will be based on the prime power decomposition of integers.
We define the Gödel number of the sequence (a1, …, an) to be the number
For example, the Gödel number of the sequence (7, 6, 4, 4, 3) is
27  36  54  74  113.
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

5. Pairing Functions and Gödel Numbers
For each n, the function [a1, …, an] is clearly primitive recursive.
Gödel numbering satisfies the following uniqueness property:
Theorem 8.2:
If [a1, …, an] = [b1, …, bn] then ai = bi for i = 1, …, n.
This follows immediately from the fundamental theorem of arithmetic, i.e., the uniqueness of the factorization of integers into primes.
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

6. Pairing Functions and Gödel Numbers
However, it is important to note that
[a1, …, an] = [a1, …, an, 0],
because for any n+1, p0n+1 = 1.
Actually, we could add any number of 0s to the right end of a sequence without changing its Gödel number.
Since we have 1 = 20 = 2030 = = … ,
it is useful to define 1 as the Gödel number of the empty sequence of length 0.
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

7. Pairing Functions and Gödel Numbers
Obviously, adding a zero to the left of the sequence will lead to a Gödel number different from the initial one.
Examples:
[1, 4] = 21  34 = 162
[1, 4, 0] = 21  34  50 = 162
[0, 1, 4] = 20  31  54 = 1875
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

8. Pairing Functions and Gödel Numbers
We will now define a primitive recursive function (x)i so that if x = [a1, …, an], then (x)i = ai.
We set
(x)i = mintx(pit+1 | x).
Then we define the length Lt(x) of the sequence for the Gödel number x:
Lt(x) = minix((x)i  0 & (j)x(j  i  (x)j = 0)).
Example: If x = 20 = 22  51 = [2, 0, 1],
then (x)1 = 2, (x)2 = 0, (x)3 = 1, (x)4 = (x)5 = … 0,
Lt(x) = 3.
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

9. Pairing Functions and Gödel Numbers
If x > 1 and Lt(x) = n, then pn divides x but no prime greater than pn divides x.
Note that Lt([a1, …, an]) = n if and only if an  0.
Theorem 8.3 (Sequence Number Theorem):
([a1, …, an])i = ai if 1  i  n
([(x)1, …, (x)n]) = x if n  Lt(x)
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

10. Coding Programs by Numbers
After having developed appropriate coding techniques, it will be our goal to enumerate all programs of the language L .
In other words, each program P of L will receive a number #(P ) so that the program can be retrieved from its number.
Let us first arrange the variables in the following order:
Y X1 Z1 X2 Z2 X3 Z3 …
And also the labels:
A1 B1 C1 D1 E1 A2 B2 C2 D2 E2 A3 …
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

11. Coding Programs by Numbers
We write #(V), #(L) for the position of a given variable or label in the appropriate ordering.
For example, #(X2) = 4, #(Z) = 3, #(C2) = 8.
Now let I be an instruction (labeled or unlabeled) of the language L .
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

12. Coding Programs by Numbers
Then we write
#(I) = a, b, c ,
where
if I is unlabeled, then a = 0; if I is labeled L, then a = #(L);
if the variable V is mentioned in I, then c = #(V) – 1;
if the statement in I is V  V, V  V+1, or V  V-1, then b = 0, 1, or 2, respectively;
if the statement in I is IF V0 GOTO L’ then b = #(L’) + 2.
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

13. Coding Programs by Numbers
Examples:
The number of the unlabeled instruction X  X-1 is
0, 2, 1 = 0, 11 = 22.
The number of the instruction [A] X  X-1 is
1, 2, 1 = 1, 11 = 45.
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

14. Coding Programs by Numbers
Note that for any given number q there is a unique instruction I with #(I) = q.
We first calculate l(q).
If l(q) = 0, I is unlabeled; otherwise I has the l(q)-th label in our list.
To find the variable mentioned in I, we compute i = r(r(q)) + 1 and locate the i-th variable V in our list.
Then the statement will be V  V, V  V+1, or V  V-1, if l(r(q)) = 0, 1, or 2, respectively; otherwise, it will be the statement IF V0 GOTO L, where L is the j-th label in our list and j = l(r(q)) –2.
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

15. Coding Programs by Numbers
Finally, for a program P that consists of the instructions I1, I2, …, Ik, we set
#(P ) = [#(I1), #(I2), …, #(Ik)] – 1.
This way we associated every possible program in L with a unique number.
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

16. Coding Programs by Numbers
Gödel numbers are usually very large, even for small programs.
Let us look at the following example:
[A] X  X+1
IF X0 GOTO A
#(I1) = 1, 1, 1 = 1, 5 = 21
#(I2) = 0, 3, 1 = 0, 23 = 46
So the number of our small program is
221  346 – 1.
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

17. Coding Programs by Numbers
Note that the number of the unlabeled instruction Y  Y is 0, 0, 0 = 0, 0 = 0.
Thus, the number of a program will be unchanged if an unlabeled instruction Y  Y is appended to it.
Although this ambiguity is harmless, we avoid it by adding a sentence to our definition of programs of L :
The final instruction in a program is not permitted to be the unlabeled statement Y  Y.
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II

18. Coding Programs by Numbers
Then each number determines a unique program. As an example, let us determine the program whose number is 199:
= 200 = 23  30  52 = [3, 0, 2].
So if #(P ) = 199, P consists of 3 instructions, the second of which is the unlabeled statement Y  Y.
3 = 2, 0 = 2, 0, 0
2 = 0, 1 = 0, 1, 0
Thus, the program is: [B] Y  Y
Y  Y
Y  Y+1
March 7, 2019
Theory of Computation Lecture 10: A Universal Program II