1 primitive recursive functions (chapter 3). 2 preliminaries: partial and total functions the domain...

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 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 => ( 3 + 1 ) + 1 => 5

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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

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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

xx=x

=xy+y *

11

0

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) ;

p (0) = 0p ( t + 1 ) = t

Power function

Predecessor function In LRR the predecessor is as follows:

pred(1) => 0 ;pred(succ(x)) => x ;

12

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∸

dotsub(x,x) => 0 ;dotsub(x,succ(y)) => pred(dotsub(x,y)) ;

abs(x,y) => sum(dotsub(x,y),dotsub(y,x)) ;

α(x) => dotsub(1,x) ;

What would be the output? dotsub(succ(succ(succ(0))),succ(0))

Output for the following?a(succ(succ(0)))a(null(0))

0

0

1)(

x

otherwise

ifx

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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∸

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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 )

y

t 0

y

t 0

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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 )

y

t 0

y

t 0

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Primitive recursive predicatesx = 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

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

n1,.

y

u=

u

=tn1, x,xt,P=x,xt,g .....

0 0

18


n1,.

y

u=

u


0 0 P (t,x1, . . . ,xn )=0 for t<t0

P (t 0, x1, . .. ,xn )=1,where

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n1,.

y

u=

u


0 0 P (t,x1, . . . ,xn )=0 for t<t0

P (t 0, x1, . .. ,xn )=1,where

Function g also belongs to C as it is attained from composition of primitive recursive functions.

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n1,.

y

u=

u


0 0 P (t,x1, . . . ,xn )=0 for t<t0

P (t 0, x1, . .. ,xn )=1,where


t0 is the least value for for which the predicate P is true (1).

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n1,.

y

u=

u


0 0 P (t,x1, . . . ,xn )=0 for t<t0

P (t 0, x1, . .. ,xn )=1,where



False=x,xt,Ptt

True=x,xt,Pt<t

n1,

n1,0

0...:

1...:

0

22


n1,.

y

u=

u


0 0 P (t,x1, . . . ,xn )=0 fort<t0

P (t 0, x1, . . . ,xn)=1,where



False=x,xt,Ptt

True=x,xt,Pt<t

n1,

n1,0

0...:

1...:

0

00

0

0..

1..

tu if=x,xt,P

t<u if=x,xt,P

n1,.

u

= t

0n1,.

u

= t

23


n1,.

y

u=

u


0 0 P (t,x1, . . . ,xn )=0 for t<t0

P (t 0, x1, . .. ,xn )=1,where



False=x,xt,Ptt

True=x,xt,Pt<t

n1,

n1,0

0...:

1...:

0

00

0

0..

1..

tu if=x,xt,P

t<u if=x,xt,P

n1,.

u

= t

0n1,.

u

= t

g ( y,x1, . . . ,xn )=∑ 1=1=t 0 for u<t0

24


n1,.

y

u=

u


0 0 P (t,x1, . . . ,xn )=0 for t<t0

P (t 0, x1,. .. ,xn )=1,where



False=x,xt,Ptt

True=x,xt,Pt<t

n1,

n1,0

0...:

1...:

0

00

0

0..

1..

tu if=x,xt,P

t<u if=x,xt,P

n1,.

u

=t

0n1,.

u

=t

g ( y,x1, . . . ,xn )=∑ 1=1=t 0 for u<t0

g( y , x1 , ... , xn ) produces the least value for which P is true. Finally the definition for bounded minimalization can be given as:

otherwise=x,xt,P

x,xt,Ptifx,xy,g=x,xt,P

nnimyt

nytnnnimyt

0..

......

1,.

1,.1,.1,.

25


n1,.

y

u=

u


0 0 P (t,x1, . . . ,xn )=0 for t<t0

P (t 0, x1, . .. ,xn )=1,where



False=x,xt,Ptt

True=x,xt,Pt<t

n1,

n1,0

0...:

1...:

0

00

0

0..

1..

tu if=x,xt,P

t<u if=x,xt,P

n1,.

u

=t

0n1,.

u

=t

g ( y,x1, . . . ,xn )=∑ 1=1=t 0 for u<t0

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.

otherwise=x,xt,P

x,xt,Ptifx,xy,g=x,xt,P

nnimyt

nytnnnimyt

0..

......

1,.

1,.1,.1,.

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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.

miny

P ( x1,. .. , x n , y )

27



We can then define this as a non-total function in the following way:

miny

P ( x1,. .. , x n , y )

x− y=minz

[ y+ z= x ]

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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)

miny

P ( x1,. .. , x n , y )

x− y=minz

[ y+ z= x ]

g ( x1,. .. , xn )= miny

P (x1,. .. , x n , y )

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.

yxyx=yx,R /*-

x>y+t=yx nimxt

*1/

p0=0,pn+ 1= min

t < p n!+ 1[ Prime( t )& t> pn]

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

1 - 12y2>< +=yx, x

31


Example: <1,1> = 2 * ( 2 + 1 ) 1 = 6 1 = 5∸ ∸ <1,2> = 2 * ( 2*2 + 1) 1 = 10 1 = 9∸ ∸

1-12y2>< +=yx, x

32


Example: <1,1> = 2 * ( 2 + 1 ) 1 = 6 1 = 5∸ ∸ <1,2> = 2 * ( 2*2 + 1) 1 = 10 1 = 9∸ ∸

12y21:012y2 +=+>yx,<followsasrearrangecanweso,+thatNote xx

1-12y2>< +=yx, x

33


Example: <1,1> = 2 * ( 2 + 1 ) 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.

1-12y2>< +=yx, x

34


Example: <1,1> = 2 * ( 2 + 1 ) 1 = 6 1 = 5∸ ∸ <1,2> = 2 * ( 2*2 + 1) 1 = 10 1 = 9∸ ∸



1212y21 +zof divisor the isthen,+=+zthatGiven xx

x+z=+Then 2/112y

1-12y2>< +=yx, x

35


Example: <1,1> = 2 * ( 2 + 1 ) 1 = 6 1 = 5∸ ∸ <1,2> = 2 * ( 2*2 + 1) 1 = 10 1 = 9∸ ∸



1212y21 +zof divisor the isthen,+=+zthatGiven xx

x+z=+Then 2/112y

Thus we have the solutions for x and y which can then be defined using the following functions:

zzr=y

zzl=x

1-12y2>< +=yx, x

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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

zzr=y

zzl=x

]><[

]><[

yx,=zx=zr

yx,=zy=zl

znimzy

znimzx

37

Gödel numbers

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

[a1, ... , an]=∏i=1

n

pia i

38

Gödel numbers


Example: Take a sequence (1,2,3,4), the Gödel number will be computed as follows: 4321 75321,2,3,4 =

[a1, ... , an]=∏i=1

n

pia i

39

Gödel numbers


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

4321 75321,2,3,4 =

[a1, ... , an]=∏i=1

n

pia i

40

Gödel numbers


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 ]

4321 75321,2,3,4 =

[a1, ... , an]=∏i=1

n

pia i

41

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

00Lt

|~ 1t

=xijjx=x

xp=a=x

jxinimxi

inimxtii

42


Example: Let x = [ 4 , 3 , 2 , 1 ], then (x)2 = 3 and (x)4=1 and (x)0 = 0

00Lt

|~ 1t

=xijjx=x

xp=a=x

jxinimxi

inimxtii

43



Lt(10) = will be the length of the sequence derived using Gödel numbering

00Lt

|~ 1t

=xijjx=x

xp=a=x

jxinimxi

inimxtii

44




So: 10 = 2^1 * 3^0 * 5^1 = [ 1, 0, 1 ] => Lt(10) = 3

00Lt

|~ 1t

=xijjx=x

xp=a=x

jxinimxi

inimxtii

45




So: 10 = 2^1 * 3^0 * 5^1 = [ 1, 0, 1 ] => Lt(10) = 3

Sequence Number Theorem:

otherwise

niifa=a,a i

n 0

1...1,(1)

00Lt

|~ 1t

=xijjx=x

xp=a=x

jxinimxi

inimxtii

46




So: 10 = 2^1 * 3^0 * 5^1 = [ 1, 0, 1 ] => Lt(10) = 3

Sequence Number Theorem:

otherwise

niifa=a,a i

n 0

1...1,(1)

(2) xnifx=x,x n Lt...1,

00Lt

|~ 1t

=xijjx=x

xp=a=x

jxinimxi

inimxtii

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