meljun cortes automata theory 15

8/13/2019 MELJUN CORTES Automata Theory 15

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(MSCS) Theory of Automata & Formal

Turing Machines

Fall 2008

MELJUN P. CORTES MBA MPA BSCS ACS

MELJUN CORTES


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Are DFAs and PDAscomputers?

0, e / a

#, e / e e, Z0 / e

1, a / e 0

1

0, 1

A computer is a machine that manipulates dataaccording to a list of instructions .

yes

no!


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A short history of computingdevices

Leibniz calculator (1670s)

slide rule (1600s)

abacus(Babylon 500BC, China 1300)

Babbagesmechanicalengine

(1820s)


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A brief history of computingdevices

Z3 (Germany, 1941) ENIAC (Pennsylvania, 1945)

PC (1980) MacBook Air (2008)


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Which of these are realcomputers?

not programmable

programmable


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Computation is universal

In principle, all these computershave the same problem-solving ability

C++ java python LISP


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The Turing Machine

C++ java python LISP

Turing Machine


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The Turing Machine

The Turing Machine is an abstractautomaton that ismeant to model any physically realizablecomputer

Using any technology (mechanical, electrical,nano, bio)

Under any physical laws (gravity, quantum,E&M)

At any scale (human, quantum, cosmological)


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Computation and mathematicalproofs

Prove that...2 is irrational.

The equationx n + y n = z n , n 2

has no integer solutions.

G = ( L , R ) has a perfect matchingiff | G( S )| | S | for every L S .

L {1}* is regular iffL = L 0 L q 1,r ... L qk,r

The language 0n 1n is not regular.

Math is hard, lets go shopping!

(Archimedes Theorem)

(Fermat-Wiles Theorem)

(Halls Theorem)


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Hilberts list of 23 problems Leading mathematician in the1900s

At the 1900 InternationalCongressof Mathematicians, proposeda list of23 problems for the 20 th century

David Hilbert(1862-1943)

Find a method for telling if an equation likexyz 3xy + 6 xz + 2 = 0 has integer solutions


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Automated theorem-proving

computer

input data

instructions

output

The equationx n + y n = z n , n 2

has no integer solutions.

check ifthey are true! true / false


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Automated theorem proving

How could automated theorem provingwork?

Lets look at a typical proof: 2 is irrational. Assume not.Then 2 = m / n .Then m 2 = 2 n 2 But m 2 has an even number of prime factors

and 2n 2 has an odd number of prime factors.Contradiction.

Theorem:

Proof:


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First idea of automated theoremproving

Checking that a proof is correct is mucheasier thancoming up with the proof

Assume not. Then 2 = m / n .Then m 2 = 2 n 2 But m 2 has an even ...

and 2n 2 has an odd ...Contradiction.

Proof:

... if we write out the proof in sufficient detail

2 = m / n both sides n

( 2)n = m square ( 2)2n 2 = m 2 def of sqrt 2n 2 = m 2


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First idea of automated theoremproving

Checking that a proof is correct is mucheasier thancoming up with the proof

In principle this is uncontroversialIn practice, it is hard to do because writingproofs indetail sometimes requires a lot of work

A proof, if written out in sufficient detail,canbe checked mechanically


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Second idea of automatedtheorem proving

To find if statement is true, why dont wetry allpossible proofs

After all, a proof is just a sequence ofsymbols (over

alphabet {0, 1, 2, m , n , , 2, etc.})

If statement is true, it has a proof of

finite length, say k

2 = m / n ( 2)n = m ( 2)2n 2 = m 2 2n 2 = m 2

...


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Knigsberg, 1930


Kurt Gdel(1906-1978)

September 8: Hilbert gives a famous lectureLogic and the understanding of nature


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Knigsberg, 1930


Kurt Gdel(1906-1978)

John von Neumann(1903-1957)

I reached the conclusion that in any reasonable formalsystem in which provability in it can be expressed as aproperty of certain sentences, there must be propositions

which are undecidable in it.

It is all over!


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What did Gdel say?

To find if statement is true, why dont wetry allpossible proofs

After all, a proof is just a sequence ofsymbols (over

alphabet {0, 1, 2, m , n , , 2, etc.})

If statement is true, it has a proof of

finite length, say k

NO!


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Gdels incompletenesstheorem

Some mathematicalstatementsare true but not provable .

What are they?


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Example of incompleteness

Recall Hilberts 10th problem:

Matyashievich showed in 1970 that there

exists apolynomial p such that

Find a method for telling if an equation likexyz 3xy + 6 xz + 2 = 0 has integer solutions

p( x 1 , ..., x k ) = 0 has no integer solutions, butthis cannot be proved


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Another example ofincompleteness

Recall ambiguous context free grammars

We did exercises of the type

Show thatG is ambiguous

However there exists a CFG G such thatG is unambiguous, but this cannot beproved


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Gdels incompletenesstheorem

Some mathematicalstatementsare true but not provable .

What does this have to do with computer science?


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The father of modern computerscienceInvented the Turing Test totellhumans and computersapartBroke German encryptioncodesduring World War IIThe Turing Award is theNobel prize of Computer

Alan Turing(1912-1954)

On Computable Numbers, with an Application to the

Entscheidungsproblem

1936:


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Turings angle onincompleteness

computerinput data

instructionsoutput

check if true! true / false

mathematical statement

Gdels theorem says that a computer cannotdetermine the truth of mathematical statements

But there are many other things!


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Computer program analysispublic static void main(String args[]) {

System.out.println("Hello World!");}

What does this program do?

How about this one?

public static void main(String args[]) {int i = 0;for (j = 1; j < 10; j++) {

i += j;if (i == 28) {

System.out.println("Hello World!");}

}}


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Computers cannot analyzeprograms!

computerinput data

instructions

output

does it print Hello, world!

javaprogram

The essence of Gdels theorem is that computers cannot analyze computer programs


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How do you argue things likethat?

To argue what computers cannot do,we need to have a precise definitionof what a computer is.

On Computable Numbers, with an Application to the Entscheidungsproblem

1936:

Section 1. Computing Machines


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Turing Machines

state control

A Turing Machine is a finite automaton with atwo-way access to an infinite tape

Initially, the first few cells contain the input, andtheother cells are blank

infinite tape0 1 0

input


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Turing Machines

state control

At each point in the computation, the machineseesits current state and the symbol at the head

It can replace the symbol on the tape, changestate ,and move the head left or right

0 1 a 0

head


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Example

0 1 a 0q 1 q 2

a/bL

Replace a with b, and move head left

current state

0 1 b 0q 1 q 2

a/bL

current state


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Formal Definition

A Turing Machine is (Q, , G, , q 0, q acc, q rej ):Q is a finite set of states ;

is the input alphabet ;G is the tape alphabet ( G) including theblank symbol q 0 in Q is the initial state ;

q acc, q rej in Q are the accepting and rejectingstate ;

is the transition function

: (Q {q acc, q rej}) G Q G {L, R}

Turing Machines are deterministic


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Language of a Turing Machine

The language recognized by a TM is theset of allinputs that make it reach q acc

Something strange can happen in a TuringMachine:q 0

q acc

q rej

= {0, 1}input: e

/ R

This machinenever halts


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Looping behavior

Inputs can be divided into three types:1. Reach the state q acc and halt2. Reach the state q rej and halt3. Loop forever

The language recognized by a TM = type1 inputs


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The Church-Turing Thesis

Turing Machinequantum computing

DNA computing

cosmic computing


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The Church-Turing Thesis

On Computable Numbers, with an Application to the Entscheidungsproblem

1936:

Section 9. The extent of the computable numbers

All arguments [for the CT Thesis] which can be given arebound to be, fundamentally, appeals to intuition, and for thisreason rather unsatisfactory mathematically. The arguments which I shall use are of three kinds:

1. A direct appeal to intuition2. A proof of the equivalence of two definitions

(In case the new definition has greater intuitive appeal)3. Giving examples of large classes of numbers which are

computable.

meljun cortes automata theory 15

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