1. Reducibility
Sipser 5.1 (pages )

2. Reducibility

3. Driving directions
Boston
Cambridge
Western Mass

4. If you can’t drive to London…

5. If something’s impossible…
Theorem 4.11:
ATM = {<M,w> | M is a TM and M accepts w}
is undecidable.
Define:
HALTTM =
{<M,w> | M is a TM and M halts on input w}
Is HALTTM decidable?

6. The Halting Problem (again!)
Theorem 5.1: HALTTM is undecidable.
Proof Idea:
We know ATM is undecidable.
We need to reduce one of HALTTM or ATMto the other.
Which way to go?

7. HALTTM is undecidable. Proof: Suppose R decides HALTTM. Define
S = "On input <M,w>, where M is a TM and w a string:
Run TM R on input <M,w>.
If R rejects, then reject.
If R accepts, simulate M on input w until it halts.
If M enters its accept state, accept;
if M enters its reject state, reject."

8. What about emptiness? ETM = {<M> | M is a TM and L(M) = Ø}
Theorem 5.2: ETM is undecidable.

9. A step along the way
Given an input <M,w>, define a machine Mw as follows.
Mw = "On input x:
If x ≠ w, reject.
If x = w, run M on input w and
accept if M does.”

10. ETM = {<M> | M is a TM and L(M) = Ø}
Proof:
Suppose TM R decides ETM. Define a TM to decide ATM
S = "On input <M,w>:
Use the description of M and w to construct Mw.
Run R on input <Mw>.
If R accepts, reject; if R rejects, accept."

11. With power comes uncertainty
M accepts w
L(M) = Ø
L(M1) = L(M2)
Turing machines ✗
PDA ✔
Finite automata

12. Is there anything that can be done?
Rice’s Theorem:
Testing any nontrivial property of the languages recognized by Turing machines is undecidable!

13. We can’t even tell when something’s regular!
REGULARTM =
{<M> | M is a TM and L(M) is regular}
Theorem 5.3: REGULARTM is undecidable.

14. REGULARTM is undecidable
Proof:
Assume R is a TM that decides REGULARTM.
Define S = "On input <M,w>:
Construct TM
M2 = "On input x:
If x has the form 0n1n, accept.
Otherwise, run M on input w and accept if M accepts w."
Run R on input <M2>.
If R accepts, accept; if R rejects, reject."