source of slides: introduction to automata theory, languages, and computation by john...

Source of Slides: Introduction to Automata Theory, Languages, and Computation

By John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman

And

Introduction to Languages and The Theory of Computation by J. C. Martin

Dept. of Computer Science & IT, FUUAST Theory of Computation

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Decidability

Basic Mathematical Definitions

Countable Set:

A set is countable if there is a one-to-one correspondence between the set and N, the natural numbers. It can be

enumerated.

Numbers: N = natural numbers = {1, 2, 3, …}

Z = integers = {…, -2, -1, 0, 1, 2, …}

Q = rational numbers ( expressed in ratios, 1/5, 3/7, 2/9 etc.)

R = real numbers ( floating point, 1.5, 2.5674 etc. )

C = complex numbers (2+5i, 27- 3i etc. )

{ a, b, c, d, e, ……… }

1 2 3 4 5

N, Z and Q are countable.

R and C are uncountable.


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Decidability

1/1 1/2 1/3 1/4 1/5 1/6 …

2/1 2/2 2/3 2/4 2/5 2/6 …

3/1 3/2 3/3 3/4 3/5 3/6 …

4/1 4/2 4/3 4/4 4/5 4/6 …

5/1 …

…….

Rational Numbers, Q are countable


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Decidability

s1 = (0, 0, 0, 0, 0, 0, 0, ...)

s2 = (1, 1, 1, 1, 1, 1, 1, ...)

s3 = (0, 1, 0, 1, 0, 1, 0, ...)

s4 = (1, 0, 1, 0, 1, 0, 1, ...)

s5 = (1, 1, 0, 1, 0, 1, 1, ...)

s6 = (0, 0, 1, 1, 0, 1, 1, ...)

s7 = (1, 0, 0, 0, 1, 0, 0, ...)

……….

Uncountable Set: Cantor’s Diagonalization

s0 = (1, 0, 1, 1, 1, 0, 1, ...)


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Decidability

R is uncountable

Proof: o Suppose R is countable

o List R according to the bijection f:

n f(n) _

1 3.14159…

2 5.55555…

3 0.12345…

4 0.50000…

…


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Decidability

R is uncountable

Proof: o Suppose R is countable

o List R according to the bijection f:

set x = 0.a1a2a3a4…

where digit ai ≠ ith digit after decimal

point of f(i) (not 0, 9)

e.g. x = 0.2312…

x cannot be in the list!

n f(n) _

1 3.14159…

2 5.55555…

3 0.12345…

4 0.50000…

…


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Turing Machine

Decidability

Turing decidability

L is Turing decidable (or just decidable) if there exists a

Turing machine M that accepts all strings in L and

rejects all strings not in L. Note that by rejection we

mean that the machine halts after a finite number of

steps and announces that the input string is not

acceptable. Acceptance, as usual, also requires a

decision after a finite number of steps.


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Decidability

Turing Recognizability

L is Turing recognizable if there is a Turing machine M

that recognizes L, that is, M should accept all strings in

L and M should not accept any strings not in L. This is

not the same as decidability because recognizability

does not require that M actually reject strings not in L.

M may reject some strings not in L but it is also

possible that M will simply "remain undecided" on

some strings not in L; for such strings, M's

computation never halts.

Decidability


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Decidability

• Recursion and Recursive Functions

• Enumerable Sets

• Recursively Enumerable Languages

• Resursive Languages

• Non-Recursively Enumerable

Languages


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Decidability

regular

languages

context free

languages

all languages decidable

RE

decidable RE all languages


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Decidability

A language is recursively enumerable

if some Turing machine accepts it.


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Decidability

A language is recursive if some Turing machine

accepts it and halts on any input string.

OR

A language is recursive if there is a membership

algorithm for it.


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Decidability

A language is recursively enumerable

if and only if there is an enumeration

procedure for it.

If a language L is recursive then there

is an enumeration procedure for it


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Decidability


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Decidability


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Decidability

Theorem:

S is an infinite countable set, the

powerset 2s of S is uncountable

Since S is countable,

we can write S = { s1 , s2 , s3 , s4 , …….. }

where S consists of s1 , s2 , s3 , s4 , ….. elements

and the powerset 2s is of the form:

{ {s1}, {s2 }, … {s1 , s2 }, ….. {s1 , s2 , s3 , s4 } … }


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Decidability

String Encoding:


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Decidability


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Decidability


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Decidability


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Decidability

Coding Turing Machines


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Decidability

Transition Rules are: Coding

Code for M


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Decidability

Diagonalization Language:


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Decidability

Theorem: Ld is not a recursively enumerable language.

That is there is no Turing Machine that accepts Ld


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Decidability


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Decidability

Fig. 10


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Decidability


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Decidability


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Decidability


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Decidability

Decidable Languages about DFA


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Decidability


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Decidability


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Decidability

Halting and Acceptance Problems:

Acceptance Problem:

Does a Turing machine accept an input string?

ATM is recursively enumerable.


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Decidability


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Decidability


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Decidability


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END


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Decidability


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Decidability


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Decidability


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Decidability

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