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

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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.

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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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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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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…

…

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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…

…

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Turing MachineTuring Machine

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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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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.

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

Languages

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regular languages

context free languages

all languagesdecidable

RE

decidable RE all languages

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A language is recursively enumerableif some Turing machine accepts it.

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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.

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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

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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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String Encoding:

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Coding Turing Machines

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Transition Rules are: Coding

Code for M

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Diagonalization Language:

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

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Fig. 10

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Decidable Languages about DFA

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Halting and Acceptance Problems:Acceptance Problem:

Does a Turing machine accept an input string?

ATM is recursively enumerable.

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END

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