1. More undecidable languages
The Chinese University of Hong Kong
Fall 2009
CSC 3130: Automata theory and formal languages
More undecidable languages
Andrej Bogdanov

2. Summary of last lecture
input x
M on input x
program 〈M〉
The universal TM U takes as inputs a program M and a string x and simulates M on x
The program M itself is specified as a TM!

3. Summary of last lecture
recognizable:
undecidable:
We can simulate M on input w
by Turing’s Theorem
ATM = {〈M, w〉: M is a TM that accepts input w}
by complementation
not recognizable:
ATM = {〈M, w〉: M is a TM that does not accept w}
recognizable:
undecidable
We can simulate M on input x
by reduction from ATM
HALTTM = {〈M, w〉: M is a TM that halts on input w}

4. More undecidable problems
AEPSTM = {〈M〉: M is a TM that accepts input e}
SOMETM = {〈M〉: M is a TM that accepts some input}
EQTM = {〈M, M’〉: M and M’ accept the same inputs}
decidable
recognizable but
undecidable
unrecognizable

5. Example 1 AEPSTM = {〈M〉: M is a TM that accepts input e}
Step 1: You gotta believe it
To know if M accepts e, it looks like we have to simulate it
But then we might end up in a loop
Step 2: Use what you know
ATM is undecidable

6. Proof by “reduction” ? AEPSTM = {〈M〉: M is a TM that accepts input e}
Show that if AEPSTM can be decided, so can ATM
〈M〉 A
accept if M accepts e
reject if not ?
accept if M accepts w
〈M, w〉
reject if not

7. M’ on input e = M on input w
Proof by “reduction”
〈M〉 A
accept if M accepts e
reject if not
accept
if M accepts w A
〈M’〉
〈M, w〉
reject
if not
M’ is a Turing Machine such that:
M’ on input e = M on input w
If M accepts w, then M’ accepts e
If M does not accept w, then M’ does not accept e

8. Proof by “reduction” A M’ M’: On input z,
accept
if M accepts w
construct M’ A
〈M’〉
〈M, w〉
reject
if not M’ q1
M’: On input z,
☐/☐R
☐/☐L
If z = e, then simulate M on w
and return its answer
Otherwise, reject
write w q0
others
run M
qrej
qacc

9. The argument We assume AEPSTM is decidable. Let A be a decider.
We describe (in high level) a TM that decides ATM:
S: On input 〈M, w〉:
Construct the following TM M’:
Run A on input 〈M’〉 and return its answer.
M’: On input z,
If z = e, then simulate M on w and return answer
Otherwise, reject
S accepts 〈M, w〉
R accepts 〈M’〉
M’ accepts e
M accepts w

10. Recognizable or not? AEPSTM = {〈M〉: M is a TM that accepts input e}
decidable
recognizable but
undecidable
unrecognizable
Turing Machine that recognizes AEPSTM:
On input 〈M〉:
Simulate M on input e and return answer.

11. Example 2 SOMETM = {〈M〉: M is a TM that accepts some input}
Step 1: You gotta believe it
To know if M accepts, it looks like we have to simulate it
But then we might end up in a loop
Step 2: Use what you know
ATM is undecidable
AEPSTM is undecidable

12. Example 2 A A M’ is a Turing Machine such that:
accept if M accepts some input
reject if not
accept
if M accepts e A
〈M’〉
〈M, w〉
reject
if not
M’ is a Turing Machine such that:
If M accepts w, then M’ accepts some input
If M does not accept w, then M’ does not accept anything

13. Example 2 A M’: On any input, Simulate M on w Return its answer
accept
if M accepts w
construct M’ A
〈M’〉
〈M, w〉
reject
if not
M’: On any input,
Simulate M on w
Return its answer
M accepts w
M’ accepts some input
decidable
recognizable but
undecidable
unrecognizable

14. Is it recognizable?
SOMETM = {〈M〉: M is a TM that accepts some input}
Attempt to recognize SOMETM:
Simulate M on input e
Simulate M on input 0
Simulate M on input 1
Accept if one of them accepts
Simulate M on input 00
...
... but there are infinitely many!

15. Is it recognizable? SOMETM = {〈M〉: M is a TM that accepts some input}
Attempt to recognize SOMETM:
For all possible strings x (in lexicographic order):
Simulate M on input x
what if M loops on e but M accepts, say, 11?
If it accepts, accept.
If it rejects, reject.
lexicographic order: e, 0, 1, 00, 01, 10, 11, 000, 001, ...

16. Is it recognizable? SOMETM = {〈M〉: M is a TM that accepts some input}
Description of recognizer for SOMETM:
k := 1
For all possible strings x (in lexicographic order):
For all strings y that come before x
Simulate M on y for k steps
If it accepts, accept.
If it rejects or doesn’t finish, continue.
k := k + 1

17. Execution of Turing Machine
...
inputs e 1 00 01
Execution:
Simulate M on e for 1 step
If M accepts some w,
execution will see this
in some stage of the
simulation
Simulate M on e for 2 steps
Simulate M on 0 for 2 steps
Simulate M on e for 3 steps
Simulate M on 0 for 3 steps
Simulate M on 1 for 3 steps
...
decidable
recognizable but
undecidable
unrecognizable

18. Example 3 EQTM = {〈M1, M2〉: M1 and M2 accept the same inputs}
Step 1: You gotta believe it
Step 2: Use what you know
decidable
recognizable but
undecidable
unrecognizable
ATM is recognizable but undecidable
AEPSTM is recognizable but undecidable
SOMETM is recognizable but undecidable
ATM is unrecognizable

19. Example 3 ? A EQTM = {〈M1, M2〉: M1 and M2 accept the same inputs}
accept if M1, M2 accept same inputs
rej/loop if not
ATM = {〈M, w〉: M is a TM that does not accept w} ?
accept if M rej/loops on w
〈M, w〉
rej/loop if M accepts w

20. Example 3 A A M1, M2 accept same inputs M rej/loops on w 〈M1, M2〉
accept if M1, M2 accept same inputs
rej/loop if not
accept
if M rej/loops on w
〈M, w〉 A
〈M1〉
〈M2〉
M1: “On any input:
Run M on w”
M2: “Reject”
rej/loop
if M accepts w
M1, M2 accept same inputs
M rej/loops on w