1. By John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman
Theory
of Computation
Decidability
Recursive and
Recursively Enumerable Languages
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
Prof. Muhammad Saeed

2. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Basic Mathematical Definitions
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, etc. )
C = complex numbers (2+5i, 27- 3i etc. )
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.
{ a, b, c, d, e, ……… }
N, Z and Q are countable.
R and C are uncountable.
Dept. of Computer Science & IT, FUUAST Theory of Computation

3. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Rational Numbers, Q are countable
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 …
…….
Dept. of Computer Science & IT, FUUAST Theory of Computation

4. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Uncountable Set:
Cantor’s Diagonalization
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, ...)
……….
s0 = (1, 0, 1, 1, 1, 0, 1, ...)
Dept. of Computer Science & IT, FUUAST Theory of Computation

5. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
R is uncountable
Proof:
Suppose R is countable
List R according to the bijection f:
n f(n) _ … … … … …
Dept. of Computer Science & IT, FUUAST Theory of Computation

6. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
R is uncountable
Proof:
Suppose R is countable
List R according to the bijection f:
n f(n) _ … … … … …
set x = 0.a1a2a3a4…
where digit ai ≠ ith digit after decimal point of f(i) (not 0, 9)
e.g. x = …
x cannot be in the list!
Dept. of Computer Science & IT, FUUAST Theory of Computation

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

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

9. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Recursion and Recursive Functions
Enumerable Sets
Recursively Enumerable Languages
Resursive Languages
Non-Recursively Enumerable Languages
Dept. of Computer Science & IT, FUUAST Theory of Computation

10. decidable  RE  all languages
Decidability
decidable
all languages
regular languages
context free languages RE
decidable  RE  all languages
Dept. of Computer Science & IT, FUUAST Theory of Computation

11. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
A language is recursively enumerable
if some Turing machine accepts it.
Dept. of Computer Science & IT, FUUAST Theory of Computation

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

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

14. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
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15. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Dept. of Computer Science & IT, FUUAST Theory of Computation

16. Dept. of Computer Science & IT, FUUAST Theory of Computation
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 } … }
Dept. of Computer Science & IT, FUUAST Theory of Computation

17. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
String Encoding:
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18. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
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19. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Dept. of Computer Science & IT, FUUAST Theory of Computation

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

21. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Coding Turing Machines
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22. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Transition Rules are:
Coding
Code for M
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23. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Diagonalization Language:
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24. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Theorem: Ld is not a recursively enumerable language. That is there is no Turing Machine that accepts Ld
Dept. of Computer Science & IT, FUUAST Theory of Computation

25. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
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26. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Fig. 10
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27. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Dept. of Computer Science & IT, FUUAST Theory of Computation

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

29. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
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30. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Decidable Languages about DFA
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31. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
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32. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
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33. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Halting and Acceptance Problems:
Acceptance Problem:
Does a Turing machine accept an input string?
ATM is recursively enumerable.
Dept. of Computer Science & IT, FUUAST Theory of Computation

34. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
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35. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Dept. of Computer Science & IT, FUUAST Theory of Computation

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

37. Dept. of Computer Science & IT, FUUAST Theory of Computation
END
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38. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
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Decidability
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40. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
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41. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
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42. Dept. of Computer Science & IT, FUUAST Theory of Computation
Decidability
Dept. of Computer Science & IT, FUUAST Theory of Computation