1. CSCI 2670 Introduction to Theory of Computing
November 2, 2005

2. Agenda Yesterday Today Tomorrow Two undecidability proofs
REGULARTM
Rice’s Theorem
Today
Reductions (Section 5.3)
Tomorrow
Section 6.3 & part of 6.4
November 2, 2005

3. Announcements Announcements
Quiz tomorrow
Undecidability proofs
Pizza party!
You are all invited to my house for pizza and snacks
Wednesday, November 9 at 6:30 PM
Please send me an and let me know if you want to come!
Let me know of any special dietary requirements so I can accommodate everyone
November 2, 2005

4. Reductions and decidability
To prove a language is decidable, we have converted it to another language and used the decidability of that language
Example – use decidability of EDFA to determine decidability of EQDFA
Thus, we reduce the problem of determining if EQDFA is decidable to the problem of determining if EDFA is decidable
November 2, 2005

5. Reductions and undecidability
To prove a language is undecidable, we have assumed it’s decidable and found a contradiction
Example – assume decidability of HALTTM and show ATM is decidable which is a contradiction
In each case, we have to do a computation to convert one problem to another problem
What kind of computations can we do?
November 2, 2005

6. TM’s and computation
TM’s can do more than just accept and reject strings
They can perform functions
Definition: A function f:* → * is a computable function if there is some TM M that, on every input w, halts with f(w) on the tape
November 2, 2005

7. Examples The copying TM discussed several weeks ago
Start with w on the tape, halt with ww on the tape
Finding intersection of two DFA’s
Start with <A,B> on the tape, where A and B are DFA’s, halt with <C> on the tape, where L(C) = L(A)  L(B)
November 2, 2005

8. Mapping reducibility
Definition: Language A is mapping reducible to language B, written AmB, if there is a computable function f:* → *, where for every w,
w  A iff f(w)  B f B A
November 2, 2005

9. Using mapping reductions
Test whether w  A by finding a mapping reduction f from A to B and determining whether f(w)  B
Example
ALLDFA = {<A> | A is a DFA with L(A)=*}
Let D = {<B> | B is a DFA} and let f:D→D where f(A) = Ā
Ā is the DFA that accepts L(A)
Then A  ALLDFA iff Ā  EDFA
Use membership of Ā in EDFA to determine membership of A in ALLDFA
November 2, 2005

10. Mapping reductions & decidability
Theorem: If A m B and B is decidable, then A is decidable.
Proof: Let M be a decider for B and let f be a reduction from A to B.
Consider the following TM, N:
N = “On input w:
Compute f(w)
Run M on f(w) and report M’s output”
Then N decides A
November 2, 2005

11. Example
Let EV = {<A>| A is a DFA s.t. every wL(A) has an even number of 1’s}
How can we prove EV is decidable using a mapping reduction?
Consider the following DFA B 1
\{1}
L(B) = {w * | w has an even number of 1’s}
November 2, 2005

12. Mapping reduction of L
Use EQDFA = {<A,B> | A and B are DFA’s with L(A) = L(B)}
Mapping from L to EQDFA
f(<A>) = <A,AB>
A has an even number of 1’s if and only if L(A) = L(AB)
I.e., A  EV iff f(A)  EQDFA
November 2, 2005