# Problem Set 4

Problem Set 4 is here: [PDF] [LaTeX template]

PS4 is due Tuesday, 23 March (one week from today).

### 4 comments to Problem Set 4

• moc8f

I was working with a group on the homework, and we were a bit confused by #5, specifically, the definitions of recognizable and decidable. The book (pg. 142) seems to suggest that:
1. A language is Turing-recognizable if some Turing machine “recognizes” it (accepts all strings in the language)
2. A language is Turing-decidable if some Turing machine always accept or reject the strings in the language.
Unless I’m interpreting the book/lecture incorrectly, this definition doesn’t seem to match up with what we covered in lecture. By the book’s definition, the set of Turing-recognizable languages seems to always be a subset of Turing-decidable languages.

Is there something we’re missing from our understanding?

• The definitions are identical, but you are misinterpreting something to conclude that the set of Turing-recognizable languages is a subset of the Turing-decidable languages — the opposite is true.

If there is a TM M that decides language L, that TM also recognizes language L. This follows from the definition of deciding and recognizing a language — a decider always gets the right Accept/Reject answer, a recognizers always gets Accept correctly for all strings in the language, but for a string not in the language either Rejects or does not terminate. Thus, the set of languages that can be recognized by a TM is a superset of the set of languages that can be decided by a TM.

We haven’t yet seen in class that it is a proper superset (that is, we haven’t seen a language that is TM-recognizable but not TM-decidable) but we will see this in class tomorrow.

• mmk2d

When creating a Turing Machine to decide the language of SQUAREFREE for problem 2, are we allowed to use nondeterminism?

• We’ll see soon that the class of languages recognized by nondeterministic TMs is equivalent to those recognized by deterministic TMs since we can simulate a nondeterministic TM with a deterministic TM. This means if you had a nondeterministic TM that decides SQUAREFREE, it could be converted to a deterministic TM that decides the same language.

But, for problem 2, for full credit you should endeavour to describe a deterministic TM.