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The domain of apartial functionon set A contains the subset of A.

The domain of a total functionon set A contains the entire set A.

Apartial functionfis called partially computable if thereis some program that computes it. Another term for suchfunctions partial recursive.

Similarly, a function fis called computable if it is bothtotalandpartially computable. Another term for suchfunction is recursive.

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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)

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Definition: Let gbe a function containing kvariables and f1 ...fkbefunctions of nvariables, 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) )

his obtained from gand f1... fkby composition.

If gand f1...fkare (partially) computable, then his(partially) computable. (Proof by construction)

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From programming experience we know that recursionrefers to a function calling upon itself within its owndefinition.

Definition: Let gbe a function containing kvariablesthen his obtained through recursion as follows:

h(x1 , , xn) = g( , h(x1 , , xn) )

Example: x + yf( 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 =>55

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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 Cis called PRC class if:

The initial functions belong to C. Function obtained from functions belonging to Cby

either composition or recursion belongs to C.

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s(x) = x + 1

n(x) = 0ui(x1 , , xn) = xi

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There exists a class of computable functionsthat is a PRC class.

Definition: Function is considered primitive recursive if it

can be obtained from initial functions and through finitenumber of composition and recursion steps.

Theorem: A function is primitive recursive iffit belongs tothe PRC class. (see proof in chapter 3)

Corollary: Every primitive recursive function is computable.

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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

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NB: To prove that a function is primitive recursive you need show that itcan be obtained from the initial functions using only concatenation andrecursion.

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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)))

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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))))

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xx=x

=x

y+y *

1

1

0

In LRR the power function canbe 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

Predecessorfunction

In LRR the predecessor is as follows:

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

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x 0 = x

x ( t + 1) = p( x t )

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| xy | = ( x y ) + ( y x )

(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)) ;

(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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x + y f( x , 0 ) = xf( x , y + 1 ) = f( x , y ) + 1

x * y h( x , 0 ) = 0h( x , y + 1 ) = h( x , y ) + x

x! 0! = 1

( x + 1 )! = x! * s(x)

x^y x^0 = 1x^( y + 1 ) = x^y * x

p(x) p( 0 ) = 0p( x + 1 ) = x

x yif x y then x y = x y; else x y = 0

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

| xy | | xy | = ( x y ) + ( y x )

(x) (x) = 1 x