1. CSCI 2670 Introduction to Theory of Computing
October 26, 2004

2. Agenda Last week Today Tomorrow
Decidability of languages involving DFA’s & CFG’s
Non-decidability of the Turing machine acceptance problem ATM
Today
Revisit the ATM proof
Finish Chapter 4
Problems
Tomorrow
Begin Chapter 5 (pp. 171 – 176)
October 26, 2004

3. Announcements Announcements
Homework due next Tuesday (11/2)
4.4, 4.8, 5.1, 5.2, 5.13
No quiz this week
Quiz next Tuesday
No tutorials this week -- extra office hours instead
Tuesday & Wednesday 3:00 – 5:00
Or by appointment
October 26, 2004

4. Undecidability of ATM Theorem: ATM is undecidable
Proof: (By contradiction) Assume ATM is decidable and let H be a decider for ATM
H(<M,w>) = {
accept if M accepts w
reject if M does not accept w
October 26, 2004

5. Proof (cont.)
Consider the TM D that submits the string <M> as input to the TM M
D = “On input <M>, where M is a TM:
Run H on input <M,<M>>
If H accepts <M,<M>>, reject
If H rejects <M,<M>>, accept”
Since H is a decider, it must accept or reject
Therefore, D is a decider as well
D is a diagonalizing TM
D rejects if M accepts <M> and accepts if M does not accept <M>
October 26, 2004

6. Proof (cont.) What happens if D’s input is <D>? D(<D>) = {
D cannot exist!
Therefore, H cannot exist – i.e., ATM is undecidable
reject if D accepts <D>
accept if D does not accept <D>
October 26, 2004

7. Recap Assume H decides ATM Define D using H Consider D(<D>)
H(<M,w>) = accept if TM M accepts w, reject otherwise
Define D using H
D(<M>) returns opposite of H(<M,<M>>)
Consider D(<D>)
D accepts <D> if and only if D rejects <D>
October 26, 2004

8. Apply this method to ADFA
Assume H decides ADFA
H(<A,w>) = accept if DFA A accepts w, reject otherwise
Define D using H
D(<A>) returns opposite of H(<A,<A>>)
Consider D(<D>)
D only accepts DFA’s as input … D is a TM
Method does not apply
October 26, 2004

9. Apply this method to ATM-D
Assume H decides ATM-D
H(<M,w>) = accept if decider TM M accepts w, reject otherwise
Problem
What if M is not a decider and just a regular TM?
How can we test our input?
ATM-D is not decidable
October 26, 2004

10. Co-Turing recognizable
Definition: A language A is co-Turing-recognizable if Ā is Turing-recognizable.
Example:
EDFA = {<A> | A is a DFA and L(A) = }
EDFA = {S | S does not describe a DFA or S describes a DFA with a non-empty language}
How would we decide this language?
October 26, 2004

11. Decidability and recognizability
Theorem: The language A is decidable if and only if it is both Turing-recognizable and co-Turing-recognizable
Proof: In two parts
If A is decidable then A and Ā are Turing-recognizable
Follows from definitions
If A and Ā are Turing-recognizable then A is decidable
October 26, 2004

12. Decider for A Assume M1 recognizes A and M2 recognizes Ā
Consider the following TM
M = “On input w:
Run M1 and M2 on input w in parallel
If M1 accepts, accept; if M2 accepts, reject”
M must halt on w because at least on of the recognizers must halt on w
If w  A, M1 must halt; otherwise, M2 must halt
October 26, 2004

13. Corollary ATM is not Turing-recognizable
Proof: ATM is Turing-recognizable, but not decidable. By previous theorem, ATM cannot be co-Turing-recognizable.
October 26, 2004

14. Proving dedicability of L
Create a TM that decides L
Your TM may invoke another TM that decides a language you know is decidable
October 26, 2004

15. Languages we know are decidable
Input
ADFA
<D,w>, D is a DFA, w is a string
ANFA
<N,w>, N is an NFA, w is a string
AREX
<R,w>, R is an RE, w is a string
EDFA
<D>, D is a DFA
EQDFA
<C,D>, C and D are both DFA’s
ACFG
<G,w>, G is a CFG, w is a string
ECFG
<G>, G is a CFG
L(G)
G is a CFG
October 26, 2004

16. Some decidable languages
FDFA = {<A> | A is a DFA and L(A) is finite}
PRIME = { n | n is a prime number}
CONN = {<G> | G is a connected graph}
L10DFA = {D | D is a DFA that accepts every string w with |w| = 10}
INTCFG = {<G1, G2, w> | G1 and G2 are CFG’s and w is accepted by both}
INTLCFG = L(G1  G2), where G1 and G2 are CFG’s
October 26, 2004

17. Group project 1 FDFA = {<A> | A is a DFA and L(A) is finite}
October 26, 2004

18. Group project 2
PRIME = { n | n is a prime number}
October 26, 2004

19. Group project 3 CONN = {<G> | G is a connected graph}
October 26, 2004

20. Group project 4
L10DFA = {D | D is a DFA that accepts every string w with |w| = 10}
October 26, 2004

21. Group project 5
INTCFG = {<G1, G2, w> | G1 and G2 are CFG’s and w is accepted by both}
October 26, 2004

22. Group project 6 INTLCFG = L(G1  G2), where G1 and G2 are CFG’s
October 26, 2004