CMPSCI 601: Theory of Computation

David Mix Barrington

Spring, 2010

Questions and Answers on Homework Assignment #5

HW#4 is due on paper in class, Monday 29 March 2010.

Question text is in black, my answers in blue.

• Question 4.1, posted 25 March:

  Could you clarify the "simple padding argument" referred to in the proof of Theorem 5.11 in [AB]?

  Sure. The argument is used several times to say that an assumption such as NTIME(n) ⊆ DTIME(n²) implies that, say, NTIME(n⁷) ⊆ DTIME(n¹⁴), or in general that NTIME(m) ⊆ DTIME(m²) for any easily computable function m with m ≥ n.

  Let's prove the last implication. Let A be an arbitrary language in NTIME(m), so that A = L(M) for some NDTM M, running in O(m) time. To apply the assumption, we need to have a language in NTIME(n), not NTIME(m). So we define a padding function on strings, so that if w is any string of length n, pad(w) is the string w0^m (remembering that m is a function of n). Define A' to be the language {pad(w): w ∈ A}. I claim that A' is in the class NTIME(n), because it is L(M') for the following NDTM M'. On any input string x, M' first tests whether x is of the form w0^m for some string w -- if not it rejects. If it is, it puts w on a worktape and then simulates M on it. The running time of M' is O(m) to check the format and O(m) to simulate M on w, so O(m) in all. Since the length of the input x is also O(m), L(M') is in the class NTIME(n). By the assumption, then, A' is also L(D) for some DTM D that runs in time O(n²) on inputs of length n (or, therefore, time O(m²) on inputs of length m). Now, given any input w of length n, I can deterministically test whether w is in A by creating the string w0^m and simulating D on that string, and this takes time O(m²).

• Question 4.2, posted 27 March:

  To solve Problem 5.7, I would like to have a problem complete for EXP. Wikipedia says that the problem of evaluating a position in a generalization of chess, with an n by n board, is complete for EXP. May I use this result for my proof?

  Normally, I would discourage this because the point of the problem is for you to find your own proof. In this case I'll allow it, because the proof using generalized chess is about equal in difficulty to the proof I have in mind (which would use a version of CIRCUIT-EVAL as the EXP-complete problem). Note also that another EXP-complete problem occurs in Chapter 3 of [AB].

  Of course, the fact that Wikipedia says something is true is not worth anything by itself, because anyone might have put a false statement there. The reference to a journal paper is worth much more, but if you use the fact remember that you are responsible for relating what the journal paper actually says to what you want to use. You would be on firmer ground using definitions with which you are more familiar.

• Question 4.3, posted 27 March:

  In my proof of 6.14, it seems to be very important that the circuit has constant fan-in, that is, that every gate has at most a constant number of inputs. May I assume that this is true?

  Yes, it is a very important part of the definition of the NC classes that the circuits have a fan-in of two. Without this restriction, you get the AC classes instead, as we'll discuss on Monday. It is not known that L is contained in the class AC¹, and in fact this is believed to be false (among other things it would imply L = NL). So your proof should use the fact that the circuits have a fan-in of two.