Honors Theory of Computation, Spring'00.

HOMEWORK 5:

Out: April 5, 2000
Due: April 17, 2000

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	IMPORTANT:
	1) Answer each question within the page 
	limitation specified (use reasonable size handwriting,
	and leaving a 1-inch margin for grading).  
	If necessary, summarize your method or arguments.
	2) Be precise in your use of mathematical notations.
	Every sentence must be an English sentence, even if
	formulas are involved.
	3) This homework is available in postscript.
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(Q1) Problem 10.4.6 (p.237).
	This concerns $NP\cap co-NP$ and a notion
	called ``strong nondeterministic TM''.
	LIMIT: 1 page

(Q2) Let $n=65521$.  Give a certificate $C(n)$ (as in the text)
	for the fact that $n$ is prime.  Summarize what you did to
	find $C(n)$ (computer programs or hand calculators are allowed).
	LIMIT: 0.5 page

(Q3) Chernoff bounds are really a family of bounds based on a method.  
     LIMIT: 1 page
	(a) Show for any r.v. $X$ and real number $c$, and any $t>0$,

		$$Pr(X\ge c) \le E[e^{t(X-c)}]$$

	HINT: Use exactly the same argument as in beginning of
	of the proof of Lemma 11.9.

	(b) The trick in Chernoff bounds if to choose the
	$t$ which minimizes the expression $E[e^{t(X-c}]$.  
	Let us define

		$$m_X(c) = \min_{t>0} E[e^{t(X-c}].$$

	Determine $m_X(c)$ in case $X$ is a sum of $n$
	Bernoulli r.v.s $X_1, X_2,..., X_n$
	where $Pr(X_i=1)=p$ ($0 < p < 1$).
	HINT: Use calculus to determine $m_X(c)$.
	Differentiate the expression $f(t)=E[e^{t(X-c}]$ and
	equate to zero.

	(c) Conclude that 
		$$Pr(X \ge (1-\epsilon)np) \le
			\left( \frac{1}{1-\epsilon} \right)
			^ {(1-\epsilon)np}
			\left( \frac{1-p}{1-(1-\epsilon)p} \right)
			^ {n-(1+\epsilon)np}
		$$	

	SORRY, it is probably impossible to parse
	this formula, so we provide a postscript file for
	this homework: h5.ps

(Q4) Problems 11.5.14 and 11.5.15 (page 274):
	(a) BPP closed under union and intersection (1 page)
	(b) RP closed under union and intersection (1 page)
	(c) PP closed under complement and symmetric difference (1page)

     HINT: Use the (relaxed) definition of these computations in which
	a path of a computation tree of a BPP-, RP- or PP-machine
	is AT MOST (not EXACTLY EQUAL to) a polynomial of the input length.
	Also do Problem 11.5.13, but do not hand these in for grading.