**Question 1 (15):**We defined the classes NC[t(n)], AC[t(n)], and ThC[t(n)] to have negation only at the leaves. That is, gates above the leaves were all AND, OR, or positive threshold gates and the leaves were inputs or negated inputs.We might have defined these classes in terms of circuits that had NOT gates of fan-in one anywhere in the circuit. Prove that under this definition the classes NC[t(n)], AC[t(n)], and ThC[t(n)] would be the same for any function t(n). (Don't worry about uniformity.) (Hint: Look at the proof that MCVP is P-complete.)

**Question 2 (15):**Let a = Σ_{i}a_{i}2^{i}and b = Σ_{i}b_{i}2^{i}be two numbers in redundant binary notation, where each a_{i}and b_{i}is in the set {0,1,2,3}. We want to add these two numbers to get a redundant-notation result r = Σ_{i}r_{i}2^{i}with r = a + b. As shown in Lecture 24, we do this by choosing a carry c_{i}for each i such that:- a
_{i}+ b_{i}+ c_{i}= 2c_{i+1}+ r_{i}for each i, and - each c
_{i}and r_{i}is in the set {0,1,2,3}.

Here is the rule to choose c

_{i+1}given only the numbers a_{i}, b_{i}, a_{i-1}, and b_{i-1}. First compute the floor of (a_{i-1}+ b_{i-1})/2 and call it f. Then compute the floor of (a_{i}+ b_{i}+ f)/2 and call it g. Set c_{i+1}to be the minimum of g and 3.Prove that this rule meets the conditions above and thus allows us to add two redundant-notation numbers in NC

^{0}.- a
**Question 3 (15):**Prove that the class sAC^{1}is closed downward under log-space many-one reductions. That is, if B is in sAC^{1}and f is function in F(L) such that w is in A iff f(w) is in B, then A is in sAC^{1}.(Typo corrected 11:15 am Thu 6 May 2004.)

**Question 4 (20):**Recall that PARITY is the set {w: w is a binary string with an odd number of ones}. Define the language ITADD to be {(n_{1},...,n_{n},i): the i'th bit of the sum of the n_{i}'s is a one}. Define the language MULT to be {(a,b,i): the i'th bit of the product of the numbers a and b is a one}.Prove that PARITY is circuit-reducible to each of the languages ITADD and MULT, following the definition in HW#7. Explain why this fact and the theorem that PARITY is not in AC

^{0}imply that neither ITADD nor MULT is in AC^{0}.**Question 5 (10):**Let A be the language of a family of constant-depth poly-size threshold gates with the following property: the threshold of every gate is either less than 12 or greater than s - 12, where s is the fan-in of the gate. Prove that A is in AC^{0}.**Question 6 (10):**Consider the game of chess, modified so that Black wins in the case of a draw. (Recall that Black may claim a draw, and thus win, if the board position is the same on three different moves. So a game cannot proceed forever if Black is smart enough to notice this.)Define CHESS-WIN to be the set of board positions from which White can force a win. Prove that CHESS-WIN can be decided by a Turing machine in O(1) time. (I am not claiming that the constant is a practical one.)

**Question 7 (15):**Now consider a variant of chess where the board is n by n rather than 8 by 8, but there are still only the usual 16 pieces on each side. Prove that the corresponding set CHESS-WIN is in the class**P**.

Last modified 6 May 2004