CS 208: Computing Theory

Assoc. Prof. Dr. Brahim HnichFaculty of Computer SciencesIzmir University of Economics

Computability Theory

Decidability

Motivation

Turing machines: a general purpose computer; Formalizes the notion of an algorithm (Church-Turing

thesis)

We now turn our attention into the power of algorithms (i.e. Turing machines) to solve problems Try to understand the limitation of computers

Is studying decidability useful?

An example of a decidable problem: Is a string a member of a context-free language?

This problem is at the heart of the problem of recognizing and compiling programs in a programming language

Preliminaries

Acceptance problem: Does DFA B accept input string w? For convenience we use languages to represent

various computational problems So, the acceptance problem can be expressed as a

language

ADFA = {<B,w>| B is a DFA that accepts input string w}

Preliminaries

ADFA = {<B,w>| B is a DFA that accepts input string w} The problem of testing whether a DFA B accepts a

string w is the same as testing whether <B,w> is a member of the language ADFA

Thus, showing that the language is decidable is the same as showing that the computational problem is decidable!

Examples of Decidable Languages

ADFA = {<B,w>| B is a DFA that accepts input string w}

Theorem: ADFA is a decidable language

Proof idea: Design a TM M that decides ADFA

How? On input <B,w>

Simulate B on input w If simulation ends in accepting state, accept, otherwise reject

Examples of Decidable Languages

ANFA = {<B,w>| B is a NFA that accepts input string w}

Theorem: ANFA is a decidable language

Proof idea: Design a TM M that decides ANFA

How? On input <B,w>

Convert NFA B into equivalent DFA C Run previous TM on input <C,w> If that TM accepts, accept, otherwise reject

Examples of Decidable Languages

AREX = {<R,w>| R is a regular expression that generates string w}

Theorem: AREX is a decidable language

Proof idea: Design a TM M that decides AREX

How? On input <R,w>

Convert R into equivalent NFA C Run previous TM on input <C,w> If that TM accepts, accept, otherwise reject

Examples of Decidable Languages

ACFG = {<G,w>| G is a CFG that generates string w}

Theorem: ACFG is a decidable language

Proof idea: Design a TM M that decides ACFG

How? (Interested students can read the book p. 156)

Examples of Decidable Languages

Theorem: Every CFG is a decidable

Relationships among classes of languages

regular

Context-free

decidable

enumerable

The Haling Problem

One of the most philosophically important theorems of the theory of computation

Computers (and computation) are limited ina very fundamental way

Common, every-day problem are unsolvable Does a program sort an array of integers?

Both program and specification are precise But, automating the verification is undecidable

No computer program can perform the task of checking the program against the specification!

Halting Problem

Halting problem: Does a Turing machine accept a string?

ATM = {<M,w>| M is a Turing machine that accepts string w}

Theorem: ATM is undecidable

Halting Problem

Before proving that ATM is undecidable, note that ATM is enumerable

How? Design a Turing Machine U that recongnizes ATM

On input <M,w> Simulate M on w If M ever enters its accept state, accept, and if M ever

enters its reject state reject

U is called a universal Turing machine

Diagonalization

Diagonalization: a very crucial technique that is useful to prove undecidability ATM

Question: what does it mean to say that two infinite sets are the same size?

Answered by Georg cantor in 1873 How? Pair them off!

Correspondence

Recall a correspondence f: A B is a bijection: Injective Surjective

Question: what does it mean to say that two infinite sets are the same size?

Answer: A and B are the same size if there is a correspondence from A to B

Correspondence

Question: in a crowded room, how can we tell if there are more people than chairs, or more chairs than people?

Correspondence

Question: in a crowded room, how can we tell if there are more people than chairs, or more chairs than people?

Answer: Establish a correspondence: ask everyone to sit down!

Correspondence

Claim: The set of Natural numbers has the same size as the set of even numbers!

Correspondence

Claim: The set of Natural numbers has the same size as the set of even numbers!

Proof: Establish a correspondence Let f(i)=2i

Remark: a proper subset of A can be the same size as A!!!!

Countable

Definition: A set A is countable iff Either it is finite, or It has the same size as N, the set of natural numbers

We have just seen that the set of even numbers is countable

Claim: The set Z of integers is countable Proof: Define f:NZ by

f(i)=i/2 If i is even f(i)=-(ceil(i/2) +1) if i is odd

Challenge

In Heaven, there is a hotel with a countable number of rooms

One day, the society of Prophets, Oracles, and AI researchers hold a convention that books every room in the hotel

Then one more guest arrives, angrily demanding a room

You are the manager. What to do?

Challenge

Then one more guest arrives, angrily demanding a room

You are the manager. What to do? Answer: Ask the guest in room i to move to room

i+1, and put the new comer in room 1!

Challenge

Then a countable number of guests arrive, all angrily demanding a room

Now What to do?

Challenge

Then a countable number of guests arrive, all angrily demanding a room

Now What to do? Answer: Ask the guest in room i to move to room

2i, and put the new comers in odd numbered rooms !!!!

(Infinity is such an amazing thing!!)

Rational Numbers

Let Q ={m/n | m and n are natural numbers} Theorem: Q is countable What is the correspondence between N and Q?

Rational Numbers

1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ……..

2/1 2/2 2/3 2/4 2/5 2/6 2/7 2/8 ……..

3/1 3/2 3/3 3/4 3/5 3/6 3/7 3/8 ……..

4/1 4/2 4/3 4/4 4/5 4/6 4/7 4/8 ……..

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Rational Numbers

1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ……..

2/1 2/2 2/3 2/4 2/5 2/6 2/7 2/8 ……..

3/1 3/2 3/3 3/4 3/5 3/6 3/7 3/8 ……..

4/1 4/2 4/3 4/4 4/5 4/6 4/7 4/8 ……..

Enumerate numbers along Northeast and Southwest diagonals, and skip duplicates

Rational Numbers

1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ……..

2/1 2/2 2/3 2/4 2/5 2/6 2/7 2/8 ……..

3/1 3/2 3/3 3/4 3/5 3/6 3/7 3/8 ……..

4/1 4/2 4/3 4/4 4/5 4/6 4/7 4/8 ……..

Does this mean that every infinite set is countable?

The Real Numbers

Theorem: R, the set of reals is uncountable

Cantor introduced the diagonalization method to prove this theorem!

The Real Numbers

Theorem: R, the set of reals is uncountable Proof: By contradiction

Assume there is a correspondence between N and R Write it down

We show now that there a number x not in the list!

n f(n)1 3.143……2 55.435….3 3456.75…4 456.655…

Diagonalization

Proof: By contradiction Pick x between 0 and 1, so non-zero digits follow decimal point First fractional digit of f(1) is 1 Pick first fractional digit of x to be different, say 2 Second fractional digit of f(2) is 4 Pick second fractional digit of x to be different, say 6 And so on ….

X=0.2487…. Thus x is not the image of any natural which is a contradiction So, R is uncountable!

n f(n)1 3.143……2 55.435….3 3456.75…4 456.655…

Important implications

Previous theorem has an important application to the theory of computation It shows that some languages are not decidable! Or even Turing machine recognizable

There are languages that are not enumerable The set of Turing machines is countable The set of languages is uncountable

Important implications

Theorem: The halting problem is undecidable Proof uses diagonalization technique (see bok those

interested)

Theorem: A language is decidable if and only if it is both Turing-recognizable and co-Turing recognizable

In other words, a language is decidable if and only if it and its compliment are Turing-recognizable (enumerable)

enumerable

A non-enumerable language

Corollory: if L is not decidable, then either L or its compliment is not enumerable

Co-enumerable

decidable

????

Conclusions

Decidable languages

Halting problem

diagonalization method

undecidable example