CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 14

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CD5560

FABER

Formal Languages, Automata and Models of Computation

Lecture 14

Mälardalen University

2010

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Content

Recursive Functions

Primitive Recursion

Ackermann function & -recursive functions

Relations Among Function Classes

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Recursion

In computer programming, recursion is related to performing computations in a loop.

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Recursion in Problem Modelling

Reducing the complexity by

• breaking up computational sequences into its simplest forms.

• synthesizing components into more complex objects by replicating simple component sequences over and over again.

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"A reduction is a way of converting one problem into another problem in such a way that a solution to the second problem can be used to solve the first problem."

Michael Sipser, Introduction to the Theory of Computation

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Recursion can be seen as concept of well-defined self-reference.

We use recursion frequently. Consider, for example, the following hypothetical “definition of a Jew”. I found it on web, as a joke.

“Somebody is a Jew if she is Abraham's wife Sarah, or if his or her mother is a Jew.”

(My digression: I wonder what about Abraham?)

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So if I want to know if I am a Jew, I look at this definition. I'm not Sarah, so I need to know whether my mother is a Jew.

How do I know about my mother? I look at the definition again. She isn't Sarah either, so I ask about her mother.

I keep going back through the generations - recursively.

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Self-referential definitions can be dangerous if we're not careful to avoid circularity.

The definition ''A rose is a rose'‘* just doesn't cut it.

This is why our definition of recursion includes the word well-defined.

*Know Gertrude Stein? '' A rose is a rose is a rose''

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We can write pseudocode to determine whether somebody is an immigrant:

FUNCTION isAnImmigrant(person): IF person immigrated herself, THEN:

return true ELSE:

return isAnImmigrant(person's parent) END IF

This is a recursive function, since it uses itself to compute its own value.

[According to some authors (Rudbeckius) Adam and Eve were Swedish.]

Yet another recursive definition: an immigrant…

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Functions

From math classes, we have seen many ways of defining and combining numerical functions.– Inverse f-1

– Composition f ◦ g– Derivatives f´(x), f´´(x), …– Iteration f1(x), f2(x), f3(x), f4(x), …

– …

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Functions

Look at what happens when we use only some of these.

– How can we define standard interesting functions?

– How do these relate to e.g. TM computations? We have seen TMs as functions. They are cumbersome!

As alternative, look at a more intuitive definition of functions.

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Notation

For brevity, limit to functions on natural numbers

N = {0,1,2,…}

Notation will also use n-tuples of numbers

(m1, …, mn)

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Natural Numbers

Start with standard recursive definition of natural numbers (remember Peano?):

A natural number is either

• 0, or• successor(n), where n is a natural number.

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What is a recurrence?

A recurrence is a well-defined mathematical function written in terms of itself.

It is a mathematical function defined recursively.

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Fibonacci sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...

The first two numbers of the sequence are both 1, while each succeeding number is the sum of the two numbers before it.

(We arrived at 55 as the tenth number, since it is the sum of 21 and 34, the eighth and ninth numbers.)

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F is called a recurrence, since it is defined in terms of itself evaluated at other values.

F(0) = 1 F(1) = 1 (base cases)

F(n) = F(n - 1) + F(n - 2)

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A recursive process is one in which objects are defined in terms of other objects of the same type.

Using some sort of recurrence relation*, the entire class of objects can then be built up from a few initial values and a small number of rules.

Recursion & Recurrence

(*Recurrence is a mathematical function defined recursively.)

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Computable Function

Any computable function can be programmed using while-loops (i.e., "while something is true, do something else").

For-loops (which have a fixed iteration limit) are a special case of while-loops.

Computable functions could also be coded using a combination of for- and while-loops.

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Total Function

A function defined for all possible input values.

Primitive Recursive Function

A function which can be implemented using only for-loops.

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103

An example function

1)( 2 nnfDomain Range

10)3( f

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We need a way to define functions.

We need a set of basic functions.

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Zero function: 0)( xzero

Successor function: 1)( xxsucc

Projection functions: 1211 ),( xxxp

2212 ),( xxxp

Basic Primitive Recursive Functions

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Building functions

Composition

)),(),,((),( 21 yxgyxghyxf

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Composition, Generally

Giveng1 : Nk N . . . gm : Nk N

f : Nm N

h(n1,…,nk) = f(g1(n1,…,nk), …, gm(n1,…,nk))

h = f ◦ (g1,…,gm) Alternate notation.

Create h : Nk N

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Primitive Recursion “Template”

)),(),,(()1,( 2 yxfyxghyxf

)()0,( 1 xgxf

N.B. For primitive recursive functions recursion in only one argument.

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Any function built from

the basic primitive recursive functions

is called Primitive Recursive Function.

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0)( xzero

)())(( xzeroxsucczero

Basic Primitive Zero function

(a constant function)

0)0()1()2()3( zerozerozerozero

Example

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Basic Primitive Identity function

...

xxidentity

xx

210)(

210

))(())((

0)0(

xidentsuccxsuccidentity

identity

Recursive definition

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Basic Primitive Successor function

...

1321)(

210

xxsucc

xx

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))(()( xzerosuccxone

Using Basic Primitive Zero function

and a Successor function we can construct Constant functions

etc..

))(()( xonesuccxtwo

))(()( xtwosuccxthree

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)2(

))1((

)))0(((

))))((((

)))(((

))(()(

succ

succsucc

succsuccsucc

xzerosuccsuccsucc

xonesuccsucc

xtwosuccxthree

Example

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A Primitive Recursive Function ),( yxadd

xxadd )0,( (projection)

)),(()1,( yxaddsuccyxadd (successor function)

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5

)4(

))3((

)))0,3(((

))1,3(()2,3(

succ

succsucc

addsuccsucc

addsuccadd

Example

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5

14

1)13(

1)1)0,3((

1))1,3(()2,3(

add

addadd

Example

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Basic Primitive Predecessor function

...

1100)(

210

xxpred

xx

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Predecessor

xxsuccpred

pred

))((

0)0(

1)( xxpred

)())((

0)0(

xGxsuccpred

pred

Predecessor is a primitive recursive function with no direct self-reference.

x) identity(G(x) templaterecursive primitive

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Subtraction

)),(())(,(

)0,(

xysubpredxsuccysub

yysub

xyxysub ),(

)1)()1(( xyxy

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1

)2(

))3((

)))0,3(((

))1,3(()2,3(

pred

predpred

subpredpred

subpredsub

Example

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0)0,( xmult

)),(,()1,( yxmultxaddyxmult

),( yxmultA Primitive Recursive Function

))()1(( xxyyx

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x

xxadd

xxaddxadd

xxaddxaddxadd

xaddxaddxaddxadd

xmultxaddxaddxaddxadd

xmultxaddxaddxadd

xmultxaddxadd

xmultxaddxmult

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)3,(

))2,(,(

))),(,(,(

))))0,(,(,(,(

)))))0,(,(,(,(,(

))))1,(,(,(,(

)))2,(,(,(

))3,(,()4,(

Example

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1),0( xexp

)),,((),1( yyxexpmultyxexp

),( yxexpA Primitive Recursive Function

)( 1 yyy xx

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Example

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)),(

)),,((

)),),,(((

)),),),,1((((

)),),),),,0(((((

)),),),,1((((

)),),,2(((

)),,3((),4(

yyyyy

yyyymult

yyyymultmult

yyyymultmultmult

yyyymultmultmultmult

yyyyyexpmultmultmultmult

yyyyexpmultmultmult

yyyexpmultmult

yyexpmultyexp

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Primitive Recursion: Logic

A predicate (Boolean function) with output in the set {0,1} which is interpreted as {yes, no}, can be used to define standard functions.– Logical connectives , ,, , …– Numeric comparisons =, < ,, …– Bounded existential quantification in, f(i)– Bounded universal quantification in, f(i)– Bounded minimization min i in, f(i)

where result = 0 if f(i) never true within bounds.

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Recursive Predicates and?zero ?_ zeronon

1110))?(?(?_

0001)),(()?(

3210

xzerozerozeronon

xxonesubxzero

x

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),( yxand ),( yxor ),( yxless )(xnon returns

1

0

00 yx 00 yx

00 yx00 yx

yx

yx

0x

0x

More Recursive Predicates

))),(?((),(

))),(?((),(

))),(?((),(

yxsubzerononyxless

yxaddzerononyxor

yxmultzerononyxand

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)),((),(_ yxequalnonyxequalnon

))),(()),,(((),( xylessnonyxlessnonandyxequal

More Recursive Predicates...

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Example

Recursive predicates can combine into powerful functions.

What does this compute?

Tests primality.

???(n) = in, jn, ((i=1 j=n) (j=1 i=n) ijn)

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prime(n) = n2 i<n, (i1 mod(n,i) > 0)

mod(m,n) = if n>0 then (min i im, div(m,n)n+i=m) else 0

div(m,n) = if n>0 then (min i im, (i+1)n>m) else 0

ExampleAnother version of prime(n)

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Function

0

0),,(

xify

xifzzyxif

if

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yzyxsuccif

zzyif

),),((

),,0(

)(),),((

)(),,0(

yGzyxsuccif

zBzyif

identityG Bwith

our construction

primitive recursive template

)),(),,(()1,( 2 yxfyxghyxf

)()0,( 1 xgxf

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Division example: x/4

rdqx quotient remainderx

0

1

2

3

4

5

6

7

8

0400

1401

2402

3403

0414

1415

2416

3417

0428

0

0

0

0

1

1

1

1

2

0

1

2

3

0

1

2

3

0

quotientq remainderr 4d

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Division as Primitive Recursion

))),,((

,

),,((),(

ddxsubremain

x

dxlessifdxremain

)))),,(((

,0

),,((),(

ddxsubquotsucc

dxlessifdxquot

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Division example: x/4

))),,((

,

),,((),(

ddxsubremain

x

dxlessifdxremain

rdqx quotient remainderx

0

1

2

3

4

5

6

7

8

0400

1401

2402

3403

0414

1415

2416

3417

0428

0

0

0

0

1

1

1

1

2

0

1

2

3

0

1

2

3

0

quotientq

remainderr

4d

)))),,(((

,0

),,((),(

ddxsubquotsucc

dxlessifdxquot

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Division as Primitive Recursion

)0

)),,(((

),)),,(((()),((

0),0(

dxsubremainsucc

ddxremainsucclessifdxsuccremain

dremain

)),()),),((?(()),((

0),0(

dxquotdxsuccremainzeroadddxsuccquot

dquot

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)),(?(),( dxremainzerodxdivisible

Recursive Predicate divisible

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)),((),(_ yxequalnonyxequalnon

Recursive Predicate

)),(?(),( dxremindzerodxdivisible

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Theorem

The set of primitive recursive functions

is countable.

Proof

Each primitive recursive function

can be encoded as a string.

Enumerate all strings in proper order.

Check if a string is a function.

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There is a function that

is not primitive recursive.

Proof

Enumerate the primitive recursive functions,,, 321 fff

Theorem

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Define function

1)()( ifig i

g differs from every if

g is not primitive recursive

END OF PROOF

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A specific function that is not

primitive recursive:

Ackermann’s function:

)),(,1()1,(

)1,1()0,(

1),0(

yxAxAyxA

xAxA

yyA

Grows very fast,

faster than any primitive recursive function

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The Ackermann function is the simplest example of a well defined total function which is computable but not primitive recursive, providing a counterexample to the belief in the early 1900s that every computable function was also primitive recursive.

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Recursive Functions

0),(such that smallest )),(( yxgyyxgy

Ackermann’s function is a

Recursive Function

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Primitive recursive functions

Recursive Functions

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Primitive Recursion: Extended Example

Needs following building blocks:– constants– addition– multiplication– exponentiation– subtraction

A polynomial function:

f(x,y) = 3x7+ xy – 7y2.

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Additionadd(m,n) = m+n

add(0,n) =add(m+1,n) =

nsucc(add(m,n))

Multiplication:mult(m,n) = mn

mult(0,n) =mult(m+1,n) =

0add(mult(m,n),n)

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Exponentiation:exp(m,n) = nm

exp(0,n) =exp(m+1,n) =

1mult(exp(m,n),n)

= one(n)

Subtraction sub(m,n) = m-n

sub(0,n) =sub(m+1,n) =

0 = zero(n)succ(sub(m,n))

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Primitive Recursion: Extended Example

f(x,y) = (3x7+ xy) - 7y2

f = sub◦ (add ◦ (f1,f2), f3)

f1(x,y) = mult(3,exp(7,x)) f1 = mult ◦ (three, exp ◦ (seven))

f2(x,y) = mult(x,y) f2 = mult

f3(x,y) = mult(7,exp(2,y)) f3 = mult ◦ (seven, exp ◦ (two))

f(x,y) = sub(add(f1(x,y),f2(x,y)),f3(x,y))

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Primitive Recursion

All primitive recursive functions are total.I.e., they are defined for all values.

Primitive recursion lack some interesting functions.“True” subtraction – when using natural numbers.“True” division – undefined when divisor is 0.Trigonometric functions – undefined for some values.…

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Partial Recursive

A function is partial recursive it can be defined by the previous constructions.

A function is recursive it is partial recursive and total.

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Division:div(m,n) = m n

div(m,n) = min i, sub(succ(m),add(mult(i,n),n)) = 0

div(m,n) = minimum i such thati mnin m-(n-1)in+n m+1(m+1) – (in+n) 0(m+1) (in+n) = 0

Example

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Relations Among Function Classes

Functions TMs– Define TMs in terms of the

function formers.– Straightforward, but long.

TMs Functions– Define functions where

subcomputations encode TM behavior.

– Simple encoding scheme.– Straightforward, but very

messy.

partial recursive= recognizable

recursive= decidable

primitiverecursive

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otherwise 0,

)(

even isn if,1

neven

))(,1()1(

1)0(

kevensubkeven

even

More Examples of Primitive Recursion

A recursive function is a function that calls itself (by using its own name within its function body).

Even

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))1(),(()1(

1)0(

xxfactmultxfact

fact

Factorials

))1)1(! nnn

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),(),)),(((()),((

),0(),0(

),()?(

),()(

yxisSquareyxsuccsquareequaloryxsuccisSquare

yequalyisSquare

xxisSquarexsquare

xxmultxsquare

)),(,()),(,(),),,(((),(

)0,()?(

yymultxequalysuccxhxyymultlessifyxh

xhxsquare

Is a number a square?

Forward recursion (-recursion)

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number. naturalany of square anot is 5

0))5,0(

))5,0(.......................................................................

))5,1(..........................................................................

))5,2(..........................................................................

))))5,3(),5,4(((),5,5(((

)5,4(),5,5((()5,5(

)5,5()5?(

isSquare

isSquareetc

isSquareetc

isSquareetc

isSquaresquareequalorsquareequalor

isSquaresquareequalorisSquare

isSquaresquare

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etc

multequalhmultlessifh

hsquare

...

))0,0(,5(),1,5(),5),0,0((()0,5(

)0,5()5?(