Other Models

8/9/2019 Other Models

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Other Models of

Computation


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Models of computation:

Turing Machines

Recursive Functions

Post SystemsRewriting Systems


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Churchs Thesis:

All models of computation are equivalent

Turings Thesis:

A computation is mechanical if and only ifit can be performed by a Turing Machine


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Churchs and Turings Thesis are similar:

Church-Turing Thesis


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Recursive Functions

An example function:

1)( 2 ! nnfDomain Range

3 1010)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)( !xz

Successor function: 1)( ! xxs

Projection functions: 1211 ),( xxxp !

2212 ),( xxxp !

Basic Primitive Recursive Functions


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Building complicated functions:

Composition: )),(),,((),( 21 yxgyxghyxf !

Primitive Recursion:

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

)()0,( 1 xgxf !


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Any function built from

the basic primitive recursive functions

is called:

Primitive Recursive Function


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A Primitive Recursive Function: ),( yxadd

xxadd !)0,( (projection)

)),(()1,( yxaddsyxadd !

(successor function)


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5

)4())3((

)))0,3(((

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

!

!

!

!

!

sss

addss

addsadd


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),( yxmult

0)0,( !xmult

)),(,()1,( yxmultxaddyxmult !

Another Primitive Recursive Function:


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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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Theorem

there is a function that

is not primitive recursive

Proof:

Enumerate the primitive recursive functions

-,,, 321 fff


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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:

Ackermanns function:

)),(,1()1,(

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

yxAxAyxA

xAxAyyA

!

!

!

Grows very fast,

faster than any primitive recursive function


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Q Recursive Functions

0),(such thatsmallest)),(( !! yxgyyxgyQ

Accermans function is a

Q Recursive Function


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Primitive recursive functions

Q Recursive Functions


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Post Systems

Have Axioms

Have Productions

Very similar with unrestricted grammars


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Example: Unary Addition

Axiom: 1111 !

Productions:

11

11

321321

321321

VVVVVV

VVVVVV

!p!

!p!


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111111111111111111 !!!

11 321321 VVVVVV !p!

11 321321 VVVVVV !p!

A production:


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Post systems are good for

proving mathematical statements

from a set of Axioms


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Theorem:

A language is recursively enumerableif and only if

a Post system generates it


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Rewriting Systems

Matrix Grammars

Markov Algorithms

Lindenmayer-Systems

They convert one string to another

Very similar to unrestricted grammars


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Matrix Grammars

Example:

PP pp

ppp

213

22112211

,:

,::

SSP

cbSSaSSPSSSP

A set of productions is applied simultaneously

Derivation:

aabbccccbbSaaScbSaSSSS 212121


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PP pp

pp

p

21322112

211

,:,:

:

SScbSSaSS

SSS

}0:{ u! ncbaL nnn

Theorem:A language is recursively enumerable

if and only if

a Matrix grammar generates it


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Markov Algorithms

Grammars that produce

P

Example:

P.p

p

p

S

SaSb

Sab

Derivation:

P SaSbaaSbbaaabbb


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P.p

p

p

S

SaSb

Sab

}0:{u!

nbaL

nn


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In general: }:{*P! wwL

Theorem:

A language is recursively enumerable

if and only ifa Markov algorithm generates it


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Lindenmayer-Systems

They are parallel rewriting systems

Example: aaap

aaaaaaaaaaaaaaa Derivation:

}0:{ 2 u! naLn


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Lindenmayer-Systems are not general

As recursively enumerable languages

Extended Lindenmayer-Systems: uyax p),,(

Theorem:

A language is recursively enumerableif and only if an

Extended Lindenmayer-System generates it

context


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Computational Complexity


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Time Complexity:

Space Complexity:

The number of steps

during a computation

Space used

during a computation


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Time Complexity

We use a multitape Turing machine

We count the number of steps until

a string is accepted

We use the O(k) notation


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Example: }0:{ u! nbaL nn

Algorithm to accept a string :w

Use a two-tape Turing machine

Copy the on the second tape

Compare the and

a

a b


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|)(|wOCopy the on the second tape

Compare the and

a

a b

Time needed:

|)(| wO

Total time: |)(| wO

}0:{ u! nbaLnn


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For string of length

time needed for acceptance:

}0:{ u! nbaLnn

n

)(nO


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Language class: )(nDTIME

A Deterministic Turing Machineaccepts each string of length

in time

)(nDTIME

1L

2L 3L

)(nO

n


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)(nDTIME

}0:{ unba nn

}{ww


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In a similar way we define the class

))(( nTDTIME

)(nTfor any time function:

Examples: ),...(),( 32 nInI


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Example: The membership problem

for context free languages

}grammarbygeneratedis:{ GwwL !

)( 3nDTIMEL (CYK - algorithm)

Polynomial time


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Theorem:

)( knI

)( 1knI

)()( 1 kk nInI


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Polynomial time algorithms: )( knI

Represent tractable algorithms:

For small we can compute the

result fast

k


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)( knIP ! for all k

The class P

All tractable problems

Polynomial time


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P

CYK-algorithm}{ nnba

- }{ww


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Exponential time algorithms: )2( nDTIME

Represent intractable algorithms:

Some problem instances

may take centuries to solve


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Example: the Traveling Salesperson Problem

Question: what is the shortest route that

connects all cities?

5 3

21

8 10

24

3

6


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Question: what is the shortest route that

connects all cities?

5 3

21

8 10

24

3

6


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)2()!( nDTIMEnDTIMEL }

Exponential time

Intractable problem

A solution: search exhuastively all

hamiltonian paths

L = {shortest hamiltonian paths}


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Example: The Satisfiability Problem

Boolean expressions in

Conjunctive Normal Form:

ktttt .321

pi xxxxt ! .321

Variables

Question: is expression satisfiable?


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)()( 3121 xxxx

Satisfiable: 1,1,0 321 !!! xxx

1)()( 3121 ! xxxx

Example:


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2121 )( xxxx

Not satisfiable

Example:


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e}satisfiablisexpression:{ wwL !

)2( nILFor variables:n

Algorithm:search exhaustively all the possible

binary values of the variables

exponential


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Non-Determinism

Language class: )(nNTIME

A Non-Deterministic Turing Machineaccepts each string of length

in time

)(nNTIME

1L2L

3L

)(nO

n


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Example: }{wwL !

Non-Deterministic Algorithmto accept a string :ww


Guess the middle of the string

and copy on the second tape

Compare the two tapes

w


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|)(| wO

Time needed:

|)(|wO

Total time: |)(|wO

}{wwL !


Guess the middle of the string

and copy on the second tape

Compare the two tapes

w


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)(nI

}{wwL !


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In a similar way we define the class

))(( nTNTIME

)(nTfor any time function:

Examples: ),...(),( 32 nInI


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Non-Deterministic Polynomial time algorithms:

)( knIL


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)( knIP ! for all k

The class NP

Non-Deterministic Polynomial time


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Example: The satisfiability problem

Non-Deterministic algorithm:

Guess an assignment of the variables


Check if this is a satisfying assignment


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Time for variables:n

)(nO

e}satis iablisexpression:{ wwL !

Total time:

Guess an assignment of the variables

Check if this is a satisfying assignment )(nO

)(nO


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NPL

The satisfiability problem is an - ProblemNP


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Observation:

NPP

Deterministic

Polynomial

Non-Deterministic

Polynomial


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Open Problem: ?NPP!

WE DO NOT KNOW THE ANSWER


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Example: Does the Satisfiability problemhave a polynomial time

deterministic algorithm?

WE DO NOT KNOW THE ANSWER

Open Problem: ?NPP!

NP C l t


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NP-Completeness

A problem is NP-complete if:

It is in NP

Every NP problem is reduced to it

(in polynomial time)


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Observation:

If we can solve any NP-complete problemin Deterministic Polynomial Time (P time)

then we know:

NPP!


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Observation:

If we prove thatwe cannot solve an NP-complete problem

in Deterministic Polynomial Time (P time)

then we know:

NPP{


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Cooks Theorem:

The satisfiability problem is NP-complete

Proof:

Convert a Non-Deterministic Turing Machine

to a Boolean expression

in conjunctive normal form


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Other NP-Complete Problems:

The Traveling Salesperson Problem

Vertex cover

Hamiltonian Path

All the above are reduced

to the satisfiability problem

Observations:


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Observations:

It is unlikely that NP-completeproblems are in P

The NP-complete problems haveexponential time algorithms

Approximations of these problems

are in P

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Other Models