1. Turing-Enumerable (Part II)
Héctor Muñoz-Avila

2. Enumerability and Turing Machines
Definition: A language L is Turing-enumerable if there is a Turing machine that enumerates all words in L in its tape (and will run forever if L is infinite):
w1 w2 w3…
Theorem 1. If a language L is decidable then L is Turing-enumerable
Theorem 2. If a language L is Turing-enumerable then L is semi-decidable

3. * is Enumerable Lemma. If  is finite then * is enumerable 2
2 1 a1 a2 1 2 2
Increase word in second tape
copy word into first tape
Homework for Monday,
Nov Construct the two Turing machines
Such that * gets
enumerated:
abaaabbabb…

4. Decidable Implies Enumerable
Let L be a language over the alphabet 
If L is decidable there is a Turing Machine that decides M
Let * be the Turing machine that enumerates *
Construct a 3-tape Turing machine that enumerates L as follows:
M* will output words on the 2nd tape
Every time M* outputs a word w, it is copied in the 3rd tape, where M checks if w is in L
If w is in L, then w is appended to the end of the 1st tape

5. Enumerable Implies Semi-decidable
Let L be a Turing-enumerable language
Thus, there is a Turing machine M that enumerates words in L
We can construct a 3-tape Turing machine ML that semi-decides if a word w is in L as follows:
Put w in the first tape
Run M enumerating all elements in the 2nd tape:
Every time a new word w’ is added to the second tape, we copy it to the 3rd tape
If w = w’ then halt otherwise continue with Step 1