1 1 cdt314 faber formal languages, automata and models of computation lecture 14 school of...

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CDT314

FABER

Formal Languages, Automata and Models of Computation

Lecture 14

School of Innovation, Design and Engineering Mälardalen University

2012

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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.

Visualisation of a recursive structure

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A Sierpinksi carpet is a famous fractal objects in mathematics. Creating one is an iterative procedure. Start with a square, divide it into nine equal squares and remove the central one. That leaves eight squares around a central square hole.  In the next iteration, repeat this process with each of the eight remaining squares and so on (see above). 

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Recursion used in Problem Modelling

Reducing the complexity by

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

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

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

*Gertrude Stein: ''A rose is a rose is a rose''

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We can write a 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.

Recursive definition: an immigrant…

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Functions

From math classes, we have seen many ways of defining and combining numerical functions.

Inverse

Composition

Derivatives

Iteration

(iterated function is composed with itself)

…. etc.

1f

gf

),(),(),( xfxfxf

),(),(),( 321 xfxfxf

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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 awkward!

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

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Notation

For brevity, we will limit ourselves to functions on natural numbers

Notation will also use n-tuples of numbers

},2,1,0{ N

),,( 1 nmm

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Natural Numbers

Start with standard recursive definition of natural numbers (Peano axioms)

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 following 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.

)2()1()(

)(1)1(1)0(

nFnFnF

casesbaseFF

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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 Functions

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 of course 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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Building functions from basic ones

Based on C Busch, RPI, Models of Computation

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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

Given

NNh

NNf

NNg

NNg

k

m

km

k

:

:

:

...

:1

),,( 1 mggfh

)),,(,),,,((),,( 1111 kmkk nngnngfnnh

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Primitive Recursion “Template”

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

)()0,( 1 xgxf

For primitive recursive functions recursion is 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

303

)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

),( yxmult

Multiplication

))()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,exponentiation

)( 1 yyy xx

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Example

4

)),(

)),,((

)),),,(((

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

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

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

)),),,2(((

)),,3((),4(

yyyyy

yyyymult

yyyymultmult

yyyymultmultmult

yyyymultmultmultmult

yyyyyexpmultmultmultmult

yyyyexpmultmultmult

yyyexpmultmult

yyexpmultyexp

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Primitive Recursion in 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!

))1()1((,,)(? njinijnjinjnin

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

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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 (by Cantor diagonal argument)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:27 73),( yxyxyxf

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Addition

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

nsucc(add(m,n))

Multiplication:

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

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

nmnmadd ),(

nmnmmult ),(

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Exponentiation:

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

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

= one(n)

Subtraction

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

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

nmnmsub ),(

mnnm ),(exp

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Primitive Recursion: Extended Example

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))

27 73),( yxyxyxf

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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 vs. 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

Additional 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?(