1. Decidability and Enumerability
Lecture 27
Section 3.2
Wed, Oct 24, 2007

2. Turing Machines as Language Enumerators
To enumerate a language is to list all of its words in some order.
The order does not matter.
An enumerator is a Turing machine that will do this.
If the language is infinite, then the enumerator will loop.

3. Examples Let  = {0, 1}*. Design the following enumerators.
G enumerates all strings in *.
E1 enumerates all strings in * that contain an even number of 1s.
E2 enumerates all string in * that contain the substring 111.

4. Examples The first enumerator G begins by “writing” , followed by #.
Then it “copies”  to the right of # and replaces it with its successor 0, followed by #.
Then it copies 0, replaces it with 1, followed by #
This process is repeated indefinitely.

5. Examples
The second enumerator E1 uses the first enumerator G as a module.
G writes a string on Tape 2.
E1 writes 0 on Tape 3.
Then E1 scans the string on Tape 2.

6. Examples
Every time it reads 1, it changes the 0 on Tape 3 to a 1, or the 1 to a 0.
When it reaches the #, if the number on Tape 3 is 0, then copy the string from Tape 2 to Tape 1
Repeat the procedure on the next string.

7. Examples
The third enumerator E2 is similar to E1 except that it searches the string on Tape 2 for the substring 111.
If it finds it, it copies the string to Tape 1.
In either case, it then proceeds on to the next string.

8. Decidable  Canonically Enumerable
Theorem: A language is Turing decidable if and only if there is an enumerator that enumerates its member in canonical order.

9. Decidable  Canonically Enumerable
Proof ():
The previous examples contain the main idea.
If a language L is Turing decidable, then by definition there is a Turing machine D that decides L.
We will build enumerator E of L with G and D as modules.

10. Decidable  Canonically Enumerable
Start up the enumerator G.
G writes its strings on Tape 2.
Each time G writes a string, E copies it to Tape 3.
The decider D processes the string on Tape 3.
If D accepts the string, E writes it on Tape 1.

11. Decidable  Canonically Enumerable
Whether D accepts or rejects the string, E continues on with the next string.
Every string in L will eventually appear on Tape 1, and it will appear in its proper canonical place.

12. Decidable  Canonically Enumerablew E no
Write
yes G

13. Canonically Enumerable  Decidable
Proof ():
Suppose that L is enumerable in canonical order.
Let E be an enumerator of L that enumerates L in canonical order.
Let C be a Turing Machine that compares two strings and returns <, =, or >.

14. Canonically Enumerable  Decidable
We will build a decider D for L using E and C as modules.
D reads a string w (on Tape 1).
Then D starts up E, which writes strings on Tape 2.
Each time E produces a string x, D copies x and w to Tape 3.

15. Canonically Enumerable  Decidable
C processes the strings on Tape 3.
If C reports =, then D accepts w.
If C reports <, then D gets the next string from E and continues on.
If C reports >, then D rejects w.

16. Canonically Enumerable  Decidablex D
> = E w
<
yes no

17. Recognizable  Enumerable
Theorem: A language is Turing recognizable if and only if there is an enumerator that enumerates it.

18. Recognizable  Enumerable
Proof ():
The previous examples again contain the main idea.
If a language L is Turing recognizable, then by definition there is a Turing machine R that recognizes L.

19. Recognizable  Enumerable
We will build enumerator E of L using the generator G and the recognizer R as modules.
Start up G.
As G writes a string onto Tape 2, E copies it to Tape 3 and lets R process it.
If R accepts it, then E copies it to Tape 1.

20. Recognizable  Enumerable
In this way, E writes every string in L onto Tape 1.
There is a catch: What if R loops on one of the strings?
Then E will also loop.
Uh-oh.

21. Recognizable  Enumerable
To guard against this, E will
Run string 1 for 1 step.
Run string 1 for 2 steps and string 2 for 1 step.
Run string 1 for 3 steps, string 2 for 2 steps, and string 3 for 1 step.
Etc.
This way, for any i and j, we will eventually run string i for j steps.

22. Recognizable  Enumerable
If a string is accepted by R, it is accepted after a finite number of steps.
Eventually we will try that string for that number of steps, and accept it.
But if a string is not accepted by R, then E will not get hung up by looping on it.

23. Recognizable  Enumerablewi E
SUCC
M does not
halt on j
steps
M halts
on j steps j i G

24. Enumerable  Recognizable
Proof ():
Now suppose that a language L is enumerated by some enumerator E.
We will build a Turing machine M that recognizes L.

25. Enumerable  Recognizable
Write the input string w onto Tape 1.
Have the enumerator E begin writing strings in L onto Tape 2.
As each string is written, compare it to w.
If they match, then go to the accept state.
Otherwise, continue.

26. Enumerable  Recognizablex w
w = x
w  x
yes M