CMPSCI 401: Theory of Computation

David Mix Barrington

Spring, 2010

Questions and Answers on Homework Assignment #3

HW#3 is due on paper in class, Wednesday 10 March 2010.

Question text is in black, my answers in blue.

• Question 3.1, posted 8 March: On Question 4.4, does the language of G have to be {ε} or can it be any language that contains ε?

  L(G) can be any language that contains the empty string. The statement "G generates ε" doesn't say anything about what else G might or might not generate. If Sipser had meant "G generates ε, and does not generate any other string", he would have said that.

• Question 3.2, posted 8 March: In Question C-1, I want to accept if v is the u'th string among the w's. What do I do if u is 0, or if u is bigger than the number of w's in the input?

  You reject in both those cases. It says that an input is in the language if u is a number in the range from 1 through m, and the string w_u is equal to v. If u is not in the range from 1 through n, this compound statement is not true, and you should reject.

  Note that, as mentioned in lecture this morning, u might have leading zeros -- for example, if u = 000100, then you want to check whether v is equal to w₄. If you want to reject in the case where u has leading zeros, I don't mind, but I don't see that this makes the TM any simpler. I think you probably want to run a binary counter going downward in the place where u is, and as you count down you will develop leading zeros even if you didn't have them before...

  How about some examples?

  Sure. You should accept "1#01#01" because u = 1 and v = w₁. But you should reject "1#01#1" because "01" and "1" are not the same string even though they represent the same number. You should reject "10#01#01" because here u = 2 and there is no string w₂ -- u is out of range.

Last modified 9 March 2010