1. Primitive Recursive Functions (Chapter 3)

2. Preliminaries: partial and total functions
The domain of a partial function on set A contains the subset of A.
The domain of a total function on set A contains the entire set A.
A partial function f is called partially computable if there is some program that computes it. Another term for such functions partial recursive.
Similarly, a function f is called computable if it is both total and partially computable. Another term for such function is recursive.

3. Composition Let f : A → B and g : B → C g ͦ f : A → C
Composition of f and g can then be expressed as:
g ͦ f : A → C
(g ͦ f)(x) = g(f(x))
h(x) = g(f(x))
NB: In general composition is not commutative:
( g ͦ f )(x) ≠ ( f ͦ g )(x)

4. Composition
Definition: Let g be a function containing k variables and f1 ... fk be functions of n variables, so the composition of g and f is defined as:
h(0) = k Base step
h( x1, ... , xn) = g( f1(x1 ,..., xn), … , fk(x1 ,..., xn) ) Inductive step
Example: h(x , y) = g( f1(x , y), f2(x , y), f3(x , y) )
h is obtained from g and f1... fk by composition.
If g and f1...fk are (partially) computable, then h is (partially) computable. (Proof by construction)

5. Recursion
From programming experience we know that recursion refers to a function calling upon itself within its own definition.
Definition: Let g be a function containing k variables then h is obtained through recursion as follows:
h(x1 , … , xn) = g( … , h(x1 , … , xn) )
Example: x + y
f( x , 0 ) = x (1)
f(x , y+1 ) = f( x , y ) + 1 (2)
Input: f ( 3, 2 ) => f ( 3 , 1 ) + 1 => ( f ( 3 , 0 ) + 1 ) + 1 => ( ) + 1 => 5

6. PRC: Initial functions
Primitive Recursively Closed (PRC) class of functions.
Initial functions:
Example of a projection function: u2 ( x1 , x2 , x3 , x4 , x5 ) = x2
Definition: A class of total functions C is called PRC² class if:
The initial functions belong to C.
Function obtained from functions belonging to C by either composition or recursion belongs to C.
s(x) = x + 1
n(x) = 0
ui (x1 , … , xn) = xi

7. PRC: primitive recursive functions
There exists a class of computable functions that is a PRC class.
Definition: Function is considered primitive recursive if it can be obtained from initial functions and through finite number of composition and recursion steps.
Theorem: A function is primitive recursive iff it belongs to the PRC class. (see proof in chapter 3)
Corollary: Every primitive recursive function is computable.

8. Primitive recursive functions: sum
We have already seen the addition function, which can be rewritten in LRR as follows:
sum( x, succ(y) ) => succ( sum( x , y)) ;
sum( x , 0 ) => x ;
Example: sum(succ(0),succ(succ(succ(0)))) => succ(sum(succ(0),succ(succ(0)))) => succ(succ(sum(succ(0),succ(0)))) => succ(succ(succ(sum(succ(0),0) => succ(succ(succ(succ(0))) => succ(succ(succ(1))) => succ(succ(2)) => succ(3) => 4
NB: To prove that a function is primitive recursive you need show that it can be obtained from the initial functions using only concatenation and recursion.

9. Primitive recursive functions: multiplication
h( x , 0 ) = 0
h( x , y + 1) = h( x , y ) + x
In LRR this can be written as:
mult(x,0) => 0 ;
mult(x,succ(y)) => sum(mult(x,y),x) ;
What would happen on the following input?
mult(succ(succ(0)),succ(succ(0)))

10. Primitive recursive functions: factorial
0! = 1
( x + 1 ) ! = x ! * s( x )
LRR implementation would be as follows:
fact(0) => succ(null(0)) ;
fact(succ(x)) => mult(fact(x),succ(x)) ;
Output for the following? fact(succ(succ(null(0))))

11. Primitive recursive functions: power and predecessor
Power function
In LRR the power function can
be expressed as follows:
pow(x,0) => succ(null(0)) ;
pow(x,succ(y)) => mult(pow(x,y),x) ;
Predecessor function
In LRR the predecessor is as follows:
pred(1) => 0 ;
pred(succ(x)) => x ;
p (0) = 0
p ( t + 1 ) = t

12. Primitive recursive functions: ∸, | x – y | and α
dotsub(x,x) => 0 ;
dotsub(x,succ(y)) => pred(dotsub(x,y)) ;
x ∸ 0 = x
x ∸ ( t + 1) = p( x ∸ t )
What would be the output?
dotsub(succ(succ(succ(0))),succ(0))
| x – y | = ( x ∸ y ) + ( y ∸ x )
abs(x,y) => sum(dotsub(x,y),dotsub(y,x)) ;
α(x) = 1 ∸ x
α(x) => dotsub(1,x) ;
Output for the following?
a(succ(succ(0)))
a(null(0))

13. Primitive recursive functions
x + y
f( x , 0 ) = x
f( x , y + 1 ) = f( x , y ) + 1
x * y
h( x , 0 ) = 0
h( x , y + 1 ) = h( x , y ) + x x!
0! = 1
( x + 1 )! = x! * s(x)
x^y
x^0 = 1
x^( y + 1 ) = x^y * x
p(x)
p( 0 ) = 0
p( x + 1 ) = x
x ∸ y
if x ≥ y then x ∸ y = x – y; else x ∸ y = 0
x ∸ 0 = x
x ∸ ( t + 1) = p( x ∸ t )
| x – y |
| x – y | = ( x ∸ y ) + ( y ∸ x )
α(x)
α(x) = 1 ∸ x

14. Bounded quantifiers
Theorem: Let C be a PRC class. If f( t , x1 , … , xn) belongs to C then so do the functions
g( y , x1 , ... , xn ) = f( t , x1 , …, xn )
g( y , x1 , ... , xn ) = f( t , x1 , …, xn )

15. Bounded quantifiers
Theorem: Let C be a PRC class. If f( t , x1 , … , xn) belongs to C then so do the functions
g( y , x1 , ... , xn ) = f( t , x1 , …, xn )
g( y , x1 , ... , xn ) = f( t , x1 , …, xn )
Theorem: If the predicate P( t, x1 , … , xn ) belongs to some PRC class C, then so do the predicates:
( t)≤y P(t, x1, … , xn )
(∃t)≤y P(t, x1, … , xn )

16. Primitive recursive predicates
x = y
d( x , y ) = α( | x – y | )
x ≤ y
α ( x ∸ y ) ~P
α( P )
P & Q
P * Q
P v Q
~ ( ~P & ~Q )
y | x
y | x = (∃t)≤x { y * t = x }
Prime(x)
Prime(x) = x > 1 & ( t)≤x { t = 1 v t = x v ~( t | x ) }
Exercises for Chapter 3: page 62 Questions 3,4 and 5. Fibonacci function

17. Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:

18. Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
,where

19. Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
,where
Function g also belongs to C as it is attained from composition of primitive recursive functions.

20. Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
,where
Function g also belongs to C as it is attained from composition of primitive recursive functions.
t0 is the least value for for which the predicate P is true (1).

21. Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
,where
Function g also belongs to C as it is attained from composition of primitive recursive functions.
t0 is the least value for for which the predicate P is true (1).

22. Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
,where
Function g also belongs to C as it is attained from composition of primitive recursive functions.
t0 is the least value for for which the predicate P is true (1).

23. Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
,where
Function g also belongs to C as it is attained from composition of primitive recursive functions.
t0 is the least value for for which the predicate P is true (1).

24. Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
,where
Function g also belongs to C as it is attained from composition of primitive recursive functions.
t0 is the least value for for which the predicate P is true (1).
g( y , x1 , ... , xn ) produces the least value for which P is true. Finally the definition for bounded minimalization can be given as:

25. Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
,where
Function g also belongs to C as it is attained from composition of primitive recursive functions.
t0 is the least value for for which the predicate P is true (1).
g( y , x1 , ... , xn ) produces the least value for which P is true. Finally the definition for bounded minimalization can be given as:
Theorem: If P(t,x1, … ,xn) belongs to some PRC class C and there is function g that does the bounded minimalization for P, then f belongs to C.

26. Unbounded minimalization
Definition: y is the least value for which predicate P is true if it exists. If there is no value of y for which P is true, the unbounded minimalization is undefined.

27. Unbounded minimalization
Definition: y is the least value for which predicate P is true if it exists. If there is no value of y for which P is true, the unbounded minimalization is undefined.
We can then define this as a non-total function in the following way:

28. Unbounded minimalization
Definition: y is the least value for which predicate P is true if it exists. If there is no value of y for which P is true, the unbounded minimalization is undefined.
We can then define this as a non-total function in the following way:
Theorem: If P(x1, … , xn, y) is a computable predicate and if
then g is a partially computable function.
(Proof by construction)

29. Additional primitive recursive functions
[ x / y ] , the whole part of the division i.e. [10/4]=2
R(x,y) , remainder of the division of x by y.
pn , nth prime number i.e p1=2 , p2=3 etc.

30. Pairing functions
Let us consider the following primitive recursive function that provides a
coding for two numbers x and y.

31. Pairing functions
Let us consider the following primitive recursive function that provides a
coding for two numbers x and y.
Example: <1,1> = 2 * ( ) ∸ 1 = 6 ∸ 1 = 5
<1,2> = 2 * ( 2*2 + 1) ∸ 1 = 10 ∸ 1 = 9

32. Pairing functions
Let us consider the following primitive recursive function that provides a
coding for two numbers x and y.
Example: <1,1> = 2 * ( ) ∸ 1 = 6 ∸ 1 = 5
<1,2> = 2 * ( 2*2 + 1) ∸ 1 = 10 ∸ 1 = 9

33. Pairing functions
Let us consider the following primitive recursive function that provides a
coding for two numbers x and y.
Example: <1,1> = 2 * ( ) ∸ 1 = 6 ∸ 1 = 5
<1,2> = 2 * ( 2*2 + 1) ∸ 1 = 10 ∸ 1 = 9
Define z to be as: < x , y > = z
Then for any z there is always a unique solution x and y.

34. Pairing functions
Let us consider the following primitive recursive function that provides a
coding for two numbers x and y.
Example: <1,1> = 2 * ( ) ∸ 1 = 6 ∸ 1 = 5
<1,2> = 2 * ( 2*2 + 1) ∸ 1 = 10 ∸ 1 = 9
Define z to be as: < x , y > = z
Then for any z there is always a unique solution x and y.

35. Pairing functions
Let us consider the following primitive recursive function that provides a
coding for two numbers x and y.
Example: <1,1> = 2 * ( ) ∸ 1 = 6 ∸ 1 = 5
<1,2> = 2 * ( 2*2 + 1) ∸ 1 = 10 ∸ 1 = 9
Define z to be as: < x , y > = z
Then for any z there is always a unique solution x and y.
Thus we have the solutions for x and y which can then be defined using the following functions:

36. Pairing functions More formally this can written as:
Pairing Function Theorem: functions <x,y>, l(z), r(z) have the following properties:
are primitive recursive
l(<x,y>) = x and r(<x,y>) = y
< l(z) , r(z) > = z
l(z) , r(z)≤ z

37. Gödel numbers
Let (a1, … , an) be any sequence, then the Gödel number is computed as follows:

38. Gödel numbers
Let (a1, … , an) be any sequence, then the Gödel number is computed as follows:
Example: Take a sequence (1,2,3,4), the Gödel number will be computed as follows:

39. Gödel numbers
Let (a1, … , an) be any sequence, then the Gödel number is computed as follows:
Example: Take a sequence (1,2,3,4), the Gödel number will be computed as follows:
Gödel numbering has a special uniqueness property:
If [a1, … , an ] = [ b1, … , bn ] then
ai = bi , where i = 1, … , n

40. Gödel numbers
Let (a1, … , an) be any sequence, then the Gödel number is computed as follows:
Example: Take a sequence (1,2,3,4), the Gödel number will be computed as follows:
Gödel numbering has a special uniqueness property:
If [a1, … , an ] = [ b1, … , bn ] then
ai = bi , where i = 1, … , n
Also notice: [ a1, … , an ] = [ a1, … , an, 0 ]

41. Gödel numbers
Given that x = [a1, … , an ], we can now define two important functions:

42. Gödel numbers
Given that x = [a1, … , an ], we can now define two important functions:
Example: Let x = [ 4 , 3 , 2 , 1 ], then (x)2 = 3 and (x)4=1 and (x)0 = 0

43. Gödel numbers
Given that x = [a1, … , an ], we can now define two important functions:
Example: Let x = [ 4 , 3 , 2 , 1 ], then (x)2 = 3 and (x)4=1 and (x)0 = 0
Lt(10) = will be the length of the sequence derived using Gödel numbering

44. Gödel numbers
Given that x = [a1, … , an ], we can now define two important functions:
Example: Let x = [ 4 , 3 , 2 , 1 ], then (x)2 = 3 and (x)4=1 and (x)0 = 0
Lt(10) = will be the length of the sequence derived using Gödel numbering
So: 10 = 2^1 * 3^0 * 5^1 = [ 1, 0, 1 ] => Lt(10) = 3

45. Gödel numbers
Given that x = [a1, … , an ], we can now define two important functions:
Example: Let x = [ 4 , 3 , 2 , 1 ], then (x)2 = 3 and (x)4=1 and (x)0 = 0
Lt(10) = will be the length of the sequence derived using Gödel numbering
So: 10 = 2^1 * 3^0 * 5^1 = [ 1, 0, 1 ] => Lt(10) = 3
Sequence Number Theorem:
(1)

46. Gödel numbers
Given that x = [a1, … , an ], we can now define two important functions:
Example: Let x = [ 4 , 3 , 2 , 1 ], then (x)2 = 3 and (x)4=1 and (x)0 = 0
Lt(10) = will be the length of the sequence derived using Gödel numbering
So: 10 = 2^1 * 3^0 * 5^1 = [ 1, 0, 1 ] => Lt(10) = 3
Sequence Number Theorem:
(1)
(2)