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CS 208: Computing Theory
Assoc. Prof. Dr. Brahim Hnich
Faculty of Computer Sciences
Izmir University of Economics
Computability Theory
Decidability
Motivation
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Turing machines:
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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
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Try to understand the limitation of computers
Is studying decidability useful?
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An example of a decidable problem:
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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
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Acceptance problem: Does DFA B accept input
string w?
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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
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ADFA = {<B,w>| B is a DFA that accepts input
string w}
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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
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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?
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On input <B,w>
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Simulate B on input w
If simulation ends in accepting state, accept, otherwise reject
Examples of Decidable Languages
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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?
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On input <B,w>
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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
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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?
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On input <R,w>
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Convert R into equivalent NFA C
Run previous TM on input <C,w>
If that TM accepts, accept, otherwise reject
Examples of Decidable Languages
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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?
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(Interested students can read the book p. 156)
Examples of Decidable Languages
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Theorem: Every CFG is a decidable
Relationships among classes of
languages
Context-free
regular
decidable
enumerable
The Haling Problem
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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
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Does a program sort an array of integers?
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Both program and specification are precise
But, automating the verification is undecidable
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No computer program can perform the task of checking the
program against the specification!
Halting Problem
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Halting problem: Does a Turing machine accept a
string?
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ATM = {<M,w>| M is a Turing machine that accepts
string w}
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Theorem: ATM is undecidable
Halting Problem
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Before proving that ATM is undecidable, note that
ATM is enumerable
How? Design a Turing Machine U that
recongnizes ATM
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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
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Diagonalization: a very crucial technique that is
useful to prove undecidability ATM
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Question: what does it mean to say that two
infinite sets are the same size?
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Answered by Georg cantor in 1873
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How? Pair them off!
Correspondence
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Recall a correspondence f: A B is a bijection:
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Injective
Surjective
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Question: what does it mean to say that two
infinite sets are the same size?
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Answer: A and B are the same size if there is a
correspondence from A to B
Correspondence
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Question: in a crowded room, how can we tell if
there are more people than chairs, or more chairs
than people?
Correspondence
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Question: in a crowded room, how can we tell if
there are more people than chairs, or more chairs
than people?
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Answer: Establish a correspondence: ask
everyone to sit down!
Correspondence
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Claim: The set of Natural numbers has the same
size as the set of even numbers!
Correspondence
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Claim: The set of Natural numbers has the same
size as the set of even numbers!
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Proof: Establish a correspondence
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Let f(i)=2i
Remark: a proper subset of A can be the same size
as A!!!!
Countable
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Definition: A set A is countable iff
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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:NZ by
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f(i)=i/2
f(i)=-(ceil(i/2) +1)
If i is even
if i is odd
Challenge
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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
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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
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Then a countable number of guests arrive, all
angrily demanding a room
Now What to do?
Challenge
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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
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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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.
.
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.
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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
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Theorem: R, the set of reals is uncountable
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Cantor introduced the diagonalization method to
prove this theorem!
The Real Numbers
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Theorem: R, the set of reals is uncountable
Proof: By contradiction
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Assume there is a correspondence between N and R
Write it down
n
f(n)
1 3.143……
2 55.435….
3 3456.75…
4 456.655…
We show now that there a number x not in the list!
Diagonalization
Proof: By contradiction
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n
1
2
3
4
f(n)
3.143……
55.435….
3456.75…
456.655…
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 ….
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X=0.2487….
Thus x is not the image of any natural which is a contradiction
So, R is uncountable!
Important implications
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Previous theorem has an important application to
the theory of computation
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It shows that some languages are not decidable!
Or even Turing machine recognizable
There are languages that are not enumerable
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The set of Turing machines is countable
The set of languages is uncountable
Important implications
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Theorem: The halting problem is undecidable
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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 Turingrecognizable (enumerable)
A non-enumerable language
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Corollory: if L is not decidable, then either L or its
compliment is not enumerable
????
Co-enumerable
enumerable
decidable
Conclusions
Decidable languages
Halting problem
diagonalization method
undecidable example