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CD5560

FABER

Formal Languages, Automata and Models of Computation

Lecture 14

Mälardalen University

2006

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Rices sats

Om är en mängd av Turing-accepterbara språk som innehåller något men inte alla sådana språk, så kan ingen TM avgöra för ett godtyckligt Turing-accepterbart språk L om L tillhör eller ej.

(Varje icke-trivial egenskap av Turing-accepterbara språk är oavgörbar.)

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Avgörbart? Motivera!

a) ”Är L(M) oändlig?” Givet att M är en godtycklig DFA.

b) ”Är L(M) oändlig?” Givet att M är en godtycklig TM.

Svar

a) AVGÖRBART! Man behöver bara kontrollera om M innehåller någon slinga på väg till acceptans, vilket kan göras i ändligt många steg. Se Sallings bok uppgift 7.2.

b) OAVGÖRBART! Följer av Rices sats, om man väljer som mängden av alla oändliga Turingaccepterbara språk, eftersom denna mängd är icketrivial.

Exempel

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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 FunctionA 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, GenerallyGiven

g1 : 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 fromthe basic primitive recursive functionsis 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

...

xxidentityxx

210)(210

))(())((0)0(

xidentsuccxsuccidentityidentity

Recursive definition

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Basic Primitive Successor function

...

1321)(210

xxsuccxx

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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((())))((((

)))((())(()(

succsuccsucc

succsuccsuccxzerosuccsuccsucc

xonesuccsuccxtwosuccxthree

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(

succsuccsucc

addsuccsuccaddsuccadd

Example

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5 14

1)13( 1)1)0,3((

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

addaddadd

Example

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Basic Primitive Predecessor function

...

1100)(210

xxpredxx

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Predecessor

xxsuccpredpred

))((0)0(

1)( xxpred

)())((0)0(

xGxsuccpredpred

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

x) identity(G(x) templaterecursive primitive

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Subtraction

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

xysubpredxsuccysubyysub

xyxysub ),(

)1)()1(( xyxy

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1 )2(

))3(( )))0,3(((

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

predpredpred

subpredpredsubpredsub

Example

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0)0,( xmult

)),(,()1,( yxmultxaddyxmult

),( yxmultA Primitive Recursive Function

))()1(( xxyyx

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xxxadd

xxaddxaddxxaddxaddxadd

xaddxaddxaddxaddxmultxaddxaddxaddxadd

xmultxaddxaddxaddxmultxaddxadd

xmultxaddxmult

4)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

yyyymultyyyymultmult

yyyymultmultmultyyyymultmultmultmult

yyyyyexpmultmultmultmultyyyyexpmultmultmult

yyyexpmultmultyyexpmultyexp

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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

xzerozerozerononxxonesubxzero

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

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

yxsubzerononyxlessyxaddzerononyxoryxmultzerononyxand

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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

00

),,(

xifyxifz

zyxif

if

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yzyxsuccifzzyif

),),((),,0(

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

yGzyxsuccifzBzyif

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

))),,((,

),,((),(

ddxsubremainx

dxlessifdxremain

)))),,(((,0

),,((),(

ddxsubquotsucc

dxlessifdxquot

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Division example: x/4

))),,((,

),,((),(

ddxsubremainx

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(

dxsubremainsuccddxremainsucclessifdxsuccremain

dremain

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

dxquotdxsuccremainzeroadddxsuccquotdquot

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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.

ProofEnumerate the primitive recursive functions

,,, 321 fff

Theorem

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Define function

1)()( ifig i

g differs from every ifg is not primitive recursive

END OF PROOF

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A specific function that is notprimitive recursive:

Ackermann’s function:

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

1),0(

yxAxAyxAxAxA

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,1neven

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

kevensubkeveneven

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(

xxfactmultxfactfact

Factorials

))1)1(! nnn

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

),()(

yxisSquareyxsuccsquareequaloryxsuccisSquareyequalyisSquarexxisSquarexsquare

xxmultxsquare

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

yymultxequalysuccxhxyymultlessifyxhxhxsquare

Is a number a square?

Forward recursion (-recursion)

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number. naturalany of square anot is 50))5,0(

))5,0(.......................................................................))5,1(..........................................................................))5,2(..........................................................................

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

)5,5()5?(

isSquare

isSquareetcisSquareetcisSquareetc

isSquaresquareequalorsquareequalorisSquaresquareequalorisSquare

isSquaresquare

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etc

multequalhmultlessifhhsquare

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

)0,5()5?(

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Midterm Exam 3Restriction-Free Languages

Place: the LAMBDA examination hall Time: on Tuesday 2004-05-30, 10:15-12:00

It is OPEN BOOK. (This means you are allowed to bring in one book

of your choice.)It will cover Turing Machines/Restriction-free

Languages).

You will have the two hours to do the test.