Problem Set 4 is here: [PDF] [LaTeX template]
PS4 is due Tuesday, 23 March (one week from today).


Problem Set 4Problem Set 4 is here: [PDF] [LaTeX template] PS4 is due Tuesday, 23 March (one week from today). 4 comments to Problem Set 4 

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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 Turingrecognizable if some Turing machine “recognizes” it (accepts all strings in the language)
2. A language is Turingdecidable 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 Turingrecognizable languages seems to always be a subset of Turingdecidable 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 Turingrecognizable languages is a subset of the Turingdecidable 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 TMrecognizable but not TMdecidable) but we will see this in class tomorrow.
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.