1. Decidability

2. Decidable Languages Recall: A language is decidable (recursive),
if there is a Turing machine (decider)
that accepts the language and
halts on every input string
Decision
On Halt:
Turing Machine
Accept
Input
string
Decider for
Reject

3. Problem:
Is number prime?
Corresponding language:

4. Decider for :
On input number :
Divide with all possible numbers
between and
If any of them divides
Then reject
Else accept

5. Thus, PRIMES is decidable
We also say that the corresponding
prime number problem is solvable:
we can give an answer (positive or negative)
for every input instance of the problem

6. the decider for the language solves the corresponding problem
Decider for PRIMES
Accept
YES
(Accept)
Input number
is number prime?
(Input string) NO
Reject
(Reject)

7. Corresponding Language:
Problem:
Does DFA accept
the empty language ?
Corresponding Language:
Description of DFA as a string
(For example, we can represent as a
binary string, as we did for Turing machines)

8. Decider for :
On input :
Determine whether there is a path from
the initial state to any accepting state
DFA
DFA
Decision:
Accept
Reject

9. Problem:
Does DFA accept
a finite language?
Corresponding Language:
Decidable language

10. Decider for :
On input :
Check if there is a walk with cycle
from the initial state to an accepting state
DFA
DFA
infinite
finite
Decision:
Accept
Reject

11. Problem:
Does DFA accept string ?
Corresponding Language:
Decidable language

12. Decider for :
On input string :
Run DFA on input string
If accepts
Then accept (and halt)
Else reject (and halt)

13. Problem:
Do DFAs and
accept the same language?
Corresponding Language:
Decidable language

14. Decider for :
On input :
Let be the language of DFA
Construct DFA such that:
(combination of DFAs)

15. and

16. or

17. Therefore, we only need
to determine whether
which is a solvable problem for DFAs:

18. Undecidable Languages
language is not decidable
There is no decider for the language:
there is no Turing Machine
that reaches a decision (halts)
for all input strings of the language
(decision may be reached for some strings)

19. For an undecidable language, the corresponding problem is unsolvable:
there is no Turing Machine (Algorithm)
that gives an answer (solves the problem)
for all input instances of the problem
(answer may be given for some input instances)

20. We have shown before that there are undecidable languages:
Turing recognizable
Decidable
is Turing recognizable but not decidable

21. We will prove that two particular problems
are unsolvable:
Membership problem
Halting problem

22. Membership Problem
Input:
Turing Machine
String
Question:
Does accept ?
Corresponding language:

23. We will assume that is decidable; We will then prove that
Theorem:
is undecidable
(The membership problem is unsolvable)
Proof:
Basic idea:
We will assume that is decidable;
We will then prove that
every decidable language
is also Turing recognizable
A contradiction!

24. Suppose that is decidable
Input
string
Decider
for
accepts
YES NO
rejects

25. Let be a Turing recognizable language
Let be the Turing Machine that accepts
We will prove that is also decidable:
we will build a decider for

26. Decider for YES accept accepts ? NO reject Input string Decider for
(and halt)
accepts ? NO
reject
Input
string
(and halt)

27. Since is chosen arbitrarily, every Turing recognizable language
Therefore,
is decidable
Since is chosen arbitrarily, every Turing recognizable language
is also decidable
But there are Turing recognizable
languages which are not decidable
Contradiction!!!!
END OF PROOF

28. We have shown:
Undecidable
Decidable

29. We can actually show:
Turing recognizable
Decidable

30. is Turing recognizable
Turing machine that accepts :
Run on input
If accepts
then accept

31. Halting Problem
Input:
Turing Machine
String
Question:
Does halt while
processing input string ?
Corresponding language:

32. Suppose that is decidable; we will prove that every decidable language
Theorem:
is undecidable
(The halting problem is unsolvable)
Proof:
Basic idea:
Suppose that is decidable;
we will prove that
every decidable language
is also Turing recognizable
A contradiction!

33. Suppose that is decidable
Input
string
halts on
input
YES
Decider for NO
doesn’t halt
on input

34. Let be a Turing recognizable language
Let be the Turing Machine that accepts
We will prove that is also decidable:
we will build a decider for

35. Decider for NO reject halts on ? YES accept Run with input reject
and halt
halts on ?
YES
Input
string
halts
and accepts
accept
and halt
Run
with input
halts
and rejects
reject
and halt

36. Since is chosen arbitrarily, every Turing recognizable language
Therefore
is decidable
Since is chosen arbitrarily, every
Turing recognizable language
is also decidable
But there are Turing recognizable
languages which are not decidable
Contradiction!!!!
END OF PROOF

37. An alternative proof Theorem: is undecidable Proof: Basic idea:
(The halting problem is unsolvable)
Proof:
Basic idea:
Assume for contradiction that
the halting problem is decidable;
we will obtain a contradiction
using a diagonilization technique

38. Suppose that is decidable
Input
string
Decider
for
YES
halts on NO
doesn’t
halt on

39. Looking inside
Decider for
YES
Input string:
halts on ? NO

40. Construct machine :
Loop forever
YES
halts on ? NO
If halts on input Then Loop Forever
Else Halt

41. Construct machine :
Copy
on tape If
halts on input
Then loop forever
Else halt

42. Then loops forever on input
Run with input itself
Copy
on tape If
halts on input
Then loops forever on input
Else halts on input
CONTRADICTION!!!
END OF PROOF

43. We have shown:
Undecidable
Decidable

44. We can actually show:
Turing recognizable
Decidable

45. is Turing recognizable
Turing machine that accepts :
Run on input
If halts on
then accept