1. Decidability
continued…

2. Theorem:
For a recursively enumerable language
it is undecidable to determine whether
is finite
Proof:
We will reduce the halting problem
to this problem

3. Let be the machine that accepts
Assume we have the finite language algorithm:
YES
finite
Algorithm for
finite language
problem NO
not finite

4. We will design the halting problem algorithm:
YES
halts on
Algorithm for
Halting problem
doesn’t
halt on NO

5. First construct machine :
Initially, simulates on input
When enters a halt state,
accept any input (inifinite language)
Otherwise accept nothing (finite language)

6. halts on
if and only if
is not finite

7. Algorithm for halting problem:
Inputs: machine and string
1. Construct
2. Determine if is finite
YES: then doesn’t halt on
NO: then halts on

8. Machine for halting problemNO
YES
Check if
construct
YES
is finite NO

9. Theorem:
For a recursively enumerable language
it is undecidable to determine whether
contains two different strings of same length
Proof:
We will reduce the halting problem
to this problem

10. Let be the machine that accepts
Assume we have the two-strings algorithm:
YES
contains
Algorithm for
two-strings
problem
Doesn’t
contain NO
two equal length strings

11. We will design the halting problem algorithm:
YES
halts on
Algorithm for
Halting problem
doesn’t
halt on NO

12. First construct machine :
Initially, simulates on input
When enters a halt state,
accept symbols or
(two equal length strings)

13. halts on
if and only if
accepts and
(two equal length strings)

14. Algorithm for halting problem:
Inputs: machine and string
1. Construct
2. Determine if accepts
two strings of equal length
YES: then halts on
NO: then doesn’t halt on

15. Machine for halting problem
YES
YES
Check if
construct NO NO
has two
equal length
strings

16. The Post Correspondence Problem

17. Some undecidable problems for
context-free languages:
Is context-free grammar ambiguous?
Is ?

18. We need a tool to prove that the previous
problems for context-free languages
are undecidable:
The Post Correspondence Problem

19. The Post Correspondence Problem
Input:
Two sequences of strings

20. There is a Post Correspondence Solution
if there is a sequence such that:
PC-solution

21. Example:
PC-solution:

22. Example:
There is no solution
Because total length of strings from
is smaller than total length of strings from

23. The Modified Post Correspondence Problem
Inputs:
MPC-solution:

24. Example:
MPC-solution:

25. 1. We will prove that the
MPC problem is undecidable
2. We will prove that the
PC problem is undecidable

26. 1. We will prove that the
MPC problem is undecidable
We will reduce the membership problem
to the MPC problem

27. Membership problem
Input: recursive language
string
Question:
Undecidable

28. Membership problem
Input: unrestricted grammar
string
Question:
Undecidable

29. The reduction of the membership problem
to the MPC problem:
For unrestricted grammar and string
we construct a pair such that
has an MPC-solution
if and only if

30. Grammar
: start variable
: special symbol
For every symbol
For every variable

31. Grammar
string
: special symbol
For every production

32. Example:
Grammar :
String

35. Grammar :

39. Theorem:
has an MPC-solution
if and only if

40. Algorithm for membership problem:
Input: unrestricted grammar
string
Construct the pair
If has an MPC-solution then
else

41. Membership machine
solution
construct
MPC
algorithm
No-solution

42. 2. We will prove that the
PC problem is undecidable
We will reduce the MPC problem
to the PC problem

43. : input to the MPC problem

44. We construct new sequences

45. We insert a special symbol between
any two symbols

47. Special Cases

48. Observation:
There is a PC-solution for
if and only if
there is a MPC-solution for

49. PC-solution
MPC-solution

50. MPC-algorithm
Input: sequences
Construct sequences
Solve the PC problem for

51. MPC algorithm
solution
construct PC
algorithm
No-solution