Theory of Computation

Source of Slides: Introduction to Automata Theory, Languages, and ComputationBy John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman

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

Dept. of Computer Science & IT, FUUAST Theory of Computation 2

DecidabilityDecidability

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.

N, Z and Q are countable.R and C are uncountable.



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



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, ...)



R is uncountable

Proof: o Suppose RR is countableo List RR according to the bijection f:

n f(n) _

1 3.14159…

2 5.55555…

3 0.12345…

4 0.50000…

…



R is uncountable

Proof: o Suppose RR is countableo List RR 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…

…


Turing MachineTuring 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.


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.




• Recursion and Recursive Functions• Enumerable Sets• Recursively Enumerable Languages• Resursive Languages• Non-Recursively Enumerable

Languages



regular languages

context free languages

all languagesdecidable

RE

decidable RE all languages



A language is recursively enumerableif some Turing machine accepts it.



A language is recursive if some Turing machine accepts it and halts on any input string.

ORA language is recursive if there is a membership algorithm for it.



A language is recursively enumerableif and only if there is an enumeration procedure for it.

If a language L is recursive then there is an enumeration procedure for it







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 } … }



String Encoding:









Coding Turing Machines



Transition Rules are: Coding

Code for M



Diagonalization Language:



Theorem: Ld is not a recursively enumerable language. That is there is no Turing Machine that accepts Ld





Fig. 10









Decidable Languages about DFA







Halting and Acceptance Problems:Acceptance Problem:

Does a Turing machine accept an input string?

ATM is recursively enumerable.








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Theory of Computation

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Theory of Computation