Download Primitive Recursive Functions (Chapter 3)

Primitive Recursive Functions
(Chapter 3)
1
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.
2
Composition

Let f : A → B and g : B → 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)
3
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)
4
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)
5
Input: f ( 3, 2 ) => f ( 3 , 1 ) + 1 => ( f ( 3 , 0 ) + 1 ) + 1 => ( 3 + 1 ) + 1 => 5
PRC: Initial functions

Primitive Recursively Closed (PRC) class of functions.

Initial functions:
s(x) = x + 1
n(x) = 0
ui (x1 , … , xn) = xi

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.
6
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.
7
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.
8
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)))
9
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))))
10
Primitive recursive functions:
power and predecessor
Power function
x0 = 1
x y+1 = x y * x
Predecessor
function
p (0) = 0
p(t+1)=t
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) ;
In LRR the predecessor is as follows:
pred(1) => 0 ;
pred(succ(x)) => x ;
11
Primitive recursive functions:
∸, | x – y | and α
x∸0=x
x ∸ ( t + 1) = p( x ∸ t )
|x–y|=(x∸y)+(y∸x)
α(x) = 1 ∸ x
if
x0
1
 ( x)  
0 otherwise
dotsub(x,x) => 0 ;
dotsub(x,succ(y)) => pred(dotsub(x,y)) ;
What would be the output?
 dotsub(succ(succ(succ(0))),succ(0))
abs(x,y) => sum(dotsub(x,y),dotsub(y,x)) ;
α(x) => dotsub(1,x) ;
Output for the following?
 a(succ(succ(0)))
 a(null(0))
12
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
x∸0=x
x ∸ ( t + 1) = p( x ∸ t )
if x ≥ y then x ∸ y = x – y; else x ∸ y = 0
|x–y|
α(x)
|x–y|=(x∸y)+(y∸x)
α(x) = 1 ∸ x
13
Bounded quantifiers

Theorem: Let C be a PRC class. If f( t , x1 , … ,
xn) belongs to C then so do the functions
y
g( y , x1 , ... , xn ) =  f( t , x1 , …, xn )
t 0 y
g( y , x1 , ... , xn ) =  f( t , x1 , …, xn )
t 0
14
Bounded quantifiers

Theorem: Let C be a PRC class. If f( t , x1 , … ,
xn) belongs to C then so do the functions
y
g( y , x1 , ... , xn ) =  f( t , x1 , …, xn )
t 0
y
g( y , x1 , ... , xn ) =  f( t , x1 , …, xn )
t 0

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 )
15
Primitive recursive predicates
x=y
d( x , y ) = α( | x – y | )
x≤y
α(x∸y)
~P
α( P )
P&Q
P*Q
PvQ
~ ( ~P & ~Q )
y|x
Prime(x)
y | x = (∃t)≤x { y * t = 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
16
Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
g t, x1,..., xn  =  Pt, x1,..., xn 
y
u
u=0 t=0
17
Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
g t, x1,..., xn  =  Pt, x1,..., xn 
y
u
u=0 t=0
,where
P (t,x 1, . . . ,xn )= 0 for t<t 0
P (t 0, x 1, . .. ,x n )= 1
18
Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
g t, x1,..., xn  =  Pt, x1,..., xn 
y
u
u=0 t=0

,where
P (t,x 1, . . . ,xn )= 0 for t<t 0
P (t 0, x 1, . .. ,x n )= 1
Function g also belongs to C as it is attained from composition of primitive
recursive functions.
19
Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
g t, x1,..., xn  =  Pt, x1,..., xn 
y
u
u=0 t=0


,where
P (t,x 1, . . . ,xn )= 0 for t<t 0
P (t 0, x 1, . .. ,x n )= 1
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).
20
Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
g t, x1,..., xn  =  Pt, x1,..., xn 
y
u
u=0 t=0


,where
P (t,x 1, . . . ,xn )= 0 for t<t 0
P (t 0, x 1, . .. ,x n )= 1
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).
t < t0 :  P t, x1,..., xn  = 1  True
t  t0 :  P t, x1,..., xn  = 0  False
21
Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
g t, x1,..., xn  =  Pt, x1,..., xn 
y
u
,where
u=0 t=0


P (t,x 1, . . . ,xn )= 0 fort<t 0
P (t 0, x1, . . . ,x n )= 1
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).
  Pt, x
u
t < t0 :  P t, x1,..., xn  = 1  True
t  t0 :  P t, x1,..., xn  = 0  False
1,.
.., xn  = 1if u < t0
t =0
  Pt, x
u
1,.
.., xn  = 0if u  t0
t =0
22
Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
g t, x1,..., xn  =  Pt, x1,..., xn 
y
u
,where
u=0 t=0


P (t,x 1, . . . ,xn )= 0 for t<t 0
P (t 0, x 1, . .. ,x n )= 1
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).
  Pt, x
u
t < t0 :  P t, x1,..., xn  = 1  True
t  t0 :  P t, x1,..., xn  = 0  False
1,.
.., xn  = 1if u < t0
t =0
  Pt, x
u
1,.
g (y,x 1, . . . ,xn )= ∑ 1= 1=t 0 for u<t 0
.., xn  = 0if u  t0
t =0
23
Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
g t, x1,..., xn  =  Pt, x1,..., xn 
y
u
,where
u=0 t=0


P (t,x 1, . . . ,xn )= 0 for t<t 0
P (t 0, x 1,. .. ,x n )= 1
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).
  Pt, x
u
t < t0 :  P t, x1,..., xn  = 1  True
t  t0 :  P t, x1,..., xn  = 0  False
1,.
.., xn  = 1if u < t0
t= 0
  Pt, x
u
1,.
g (y,x 1, . . . ,xn )= ∑ 1= 1=t 0 for u<t 0
.., xn  = 0if u  t0
t= 0

g( y , x1 , ... , xn ) produces the least value for which P is true. Finally the
definition for bounded minimalization can be given as:
min
t y
Pt, x1,. .., xn = g  y, x1,. .., xn  if t t  y Pt, x1,. .., xn 
min
t y
Pt, x1,. .., xn = 0otherwise
24
Bounded minimalization
Let P(t, x1, … ,xn) be in some PRC class C and we can define a function g as follows:
g t, x1,..., xn  =  Pt, x1,..., xn 
y
u
,where
u=0 t=0
P (t,x 1, . . . ,xn )= 0 for t<t 0
P (t 0, x 1, . .. ,x n )= 1
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).


  Pt, x
u
t < t0 :  P t, x1,..., xn  = 1  True
t  t0 :  P t, x1,..., xn  = 0  False
1,.
.., xn  = 1if u < t0
t= 0
  Pt, x
u
1,.
g (y,x 1, . . . ,xn )= ∑ 1= 1=t 0 for u<t 0
.., xn  = 0if u  t0
t= 0
g( y , x1 , ... , xn ) produces the least value for which P is true. Finally the
definition for bounded minimalization can be given as:

min
t y
Pt, x1,. .., xn = g  y, x1,. .., xn  if t t  y Pt, x1,. .., xn 
min
t y

Pt, x1,. .., xn = 0otherwise
Theorem: If P(t,x1, … ,xn) belongs to some PRC class C and there is function g 25
that does the bounded minimalization for P, then f belongs to C.
Unbounded minimalization
min P ( x1,. .. , x n , y )
y

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.
26
Unbounded minimalization
min P ( x1,. .. , x n , y )
y


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:
x− y= min [ y+ z= x ]
z
27
Unbounded minimalization
min P ( x1,. .. , x n , y )
y


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:
x− y= min [ y+ z= x ]
z

Theorem: If P(x1, … , xn, y) is a computable predicate and if
g ( x1,. .. , xn )= min P (x1,. .. , x n , y)
y
then g is a partially computable function.
28
(Proof by construction)
Additional primitive recursive functions

[ x / y ] , the whole part of the division i.e. [10/4]=2
x / y= mt inx t +1* y > x

R(x,y) , remainder of the division of x by y.
Rx, y  = x -  y * x / y

pn , nth prime number i.e p1=2 , p2=3 etc.
p0 = 0,
pn+ 1= min [ Prime(t)& t> pn ]
t < p n! + 1
29
Pairing functions
Let us consider the following primitive recursive function that provides a
coding for two numbers x and y.

< x, y >= 2 x 2y +1 - 1
30
Pairing functions
Let us consider the following primitive recursive function that provides a
coding for two numbers x and y.

< x, y >= 2 x 2y +1 -1
Example: <1,1> = 2 * ( 2 + 1 ) ∸ 1 = 6 ∸ 1 = 5
<1,2> = 2 * ( 2*2 + 1) ∸ 1 = 10 ∸ 1 = 9
31
Pairing functions
Let us consider the following primitive recursive function that provides a
coding for two numbers x and y.

< x, y >= 2 x 2y +1 -1
Example: <1,1> = 2 * ( 2 + 1 ) ∸ 1 = 6 ∸ 1 = 5
<1,2> = 2 * ( 2*2 + 1) ∸ 1 = 10 ∸ 1 = 9
Note that 2 x 2y +1  0 , so
we can rearrange
as
follows : < x, y > +1= 2 x 2y +1
32
Pairing functions
Let us consider the following primitive recursive function that provides a
coding for two numbers x and y.

< x, y >= 2 x 2y +1 -1
Example: <1,1> = 2 * ( 2 + 1 ) ∸ 1 = 6 ∸ 1 = 5
<1,2> = 2 * ( 2*2 + 1) ∸ 1 = 10 ∸ 1 = 9
Note that 2 x 2y +1  0 , so we can rearrange

as
follows : < x, y > +1= 2 x 2y +1
Define z to be as: < x , y > = z
Then for any z there is always a unique solution x and y.
33
Pairing functions
Let us consider the following primitive recursive function that provides a
coding for two numbers x and y.

< x, y >= 2 x 2y +1 -1
Example: <1,1> = 2 * ( 2 + 1 ) ∸ 1 = 6 ∸ 1 = 5
<1,2> = 2 * ( 2*2 + 1) ∸ 1 = 10 ∸ 1 = 9
Note that 2 x 2y +1  0 , so

we can rearrange
as
follows : < x, y > +1= 2 x 2y +1
Define z to be as: < x , y > = z
Then for any z there is always a unique solution x and y.
Given that
z +1= 2 x 2y +1 , then 2 x is the divisor of
z +1
Then 2y +1= z +1 / 2 x
34
Pairing functions
Let us consider the following primitive recursive function that provides a
coding for two numbers x and y.

< x, y >= 2 x 2y +1 -1
Example: <1,1> = 2 * ( 2 + 1 ) ∸ 1 = 6 ∸ 1 = 5
<1,2> = 2 * ( 2*2 + 1) ∸ 1 = 10 ∸ 1 = 9
Note that 2 x 2y +1  0 , so

we can rearrange
as
follows : < x, y > +1= 2 x 2y +1
Define z to be as: < x , y > = z
Then for any z there is always a unique solution x and y.
Given that
z +1= 2 x 2y +1 , then 2 x is the divisor of
z +1
Then 2y +1= z +1 / 2 x

Thus we have the solutions for x and y which can then be defined using
the following functions:
x = l z   z
y = r z   z
35
Pairing functions
x = l z   z
y = r z   z

More formally this can written as:
l z  = xminz [y 
r z  = my inz [x 

z =< x, y > ]
z  z =< x, y > ]
 z

Pairing Function Theorem: functions <x,y>, l(z), r(z) have the
following properties:

are primitive recursive

l(<x,y>) = x

< l(z) , r(z) > = z

l(z) , r(z)≤ z
and r(<x,y>) = y
36
Gödel numbers

Let (a1, … , an) be any sequence, then the Gödel
number is computed as follows:
[a ... , a ]= ∏ p a
n
1,
n
i
i
i= 1
37
Gödel numbers

Let (a1, … , an) be any sequence, then the Gödel
number is computed as follows:
[a ... , a ]= ∏ p a
n
1,
n
i
i
i= 1

Example: Take a sequence (1,2,3,4), the Gödel number
1
2
3
4
will be computed as follows: 1,2,3,4= 2  3  5  7
38
Gödel numbers

Let (a1, … , an) be any sequence, then the Gödel
number is computed as follows:
[a ... , a ]= ∏ p a
n
1,
n
i
i
i= 1


Example: Take a sequence (1,2,3,4), the Gödel number
1
2
3
4
will be computed as follows: 1,2,3,4= 2  3  5  7
Gödel numbering has a special uniqueness property:
If [a1, … , an ] = [ b1, … , bn ] then
ai = bi , where i = 1, … , n
39
Gödel numbers

Let (a1, … , an) be any sequence, then the Gödel
number is computed as follows:
[a ... , a ]= ∏ p a
n
1,
n
i
i
i= 1


Example: Take a sequence (1,2,3,4), the Gödel number
1
2
3
4
will be computed as follows: 1,2,3,4= 2  3  5  7
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 ]
40
Gödel numbers

Given that x = [a1, … , an ], we can now define two important functions:
x i = ai = mt inx ~ pi t 1 | x 
Lt x  = miinx x i  0j  x  j  i  x  j = 0
41
Gödel numbers

Given that x = [a1, … , an ], we can now define two important functions:
x i = ai = mt inx ~ pi t 1 | x 
Lt x  = miinx x i  0j  x  j  i  x  j = 0

Example: Let x = [ 4 , 3 , 2 , 1 ], then (x)2 = 3 and (x)4=1 and (x)0 = 0
42
Gödel numbers

Given that x = [a1, … , an ], we can now define two important functions:
x i = ai = mt inx ~ pi t 1 | x 
Lt x  = miinx x i  0j  x  j  i  x  j = 0

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
43
Gödel numbers

Given that x = [a1, … , an ], we can now define two important functions:
x i = ai = mt inx ~ pi t 1 | x 
Lt x  = miinx x i  0j  x  j  i  x  j = 0

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
44
Gödel numbers

Given that x = [a1, … , an ], we can now define two important functions:
x i = ai = mt inx ~ pi t 1 | x 
Lt x  = miinx x i  0j  x  j  i  x  j = 0

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)
 ai if 1  i  n


...
,
a
=


1,
n
 0 otherwise 
a
45
Gödel numbers

Given that x = [a1, … , an ], we can now define two important functions:
x i = ai = mt inx ~ pi t 1 | x 
Lt x  = miinx x i  0j  x  j  i  x  j = 0

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:
 ai if 1  i  n


...
,
a
=


1,
n
 0 otherwise 
(1)
a
(2)
x  ...,x  = x
1,
n
if
n  Lt x 
46