CS547T SPRING 2007

ASSIGNMENT 1

   January 2007
21 22 23 24 25 26 27
             assigned
28 29 30 31  1  2  3
             due in class, hardcopy


COLLABORATE ALL YOU WANT:

(60 points)
1.  Find the following on the internet and summarize the idea in a single
	sentence (I am using wiki, but you do not have to):

	More credit for more interesting sentences.
	More credit for more accurate sentences.
	No credit for more than one sentence.

	a.  (5 points) Stephen Kleene
	b.  (5 points) Kleene Closure
	c.  (5 points) Bijection
	d.  (5 points) Cardinality
	e.  (5 points) Countable Set
	f.  (5 points) Naive Set Theory 
		(ooh, I think I see a wiki error:
		"Alternatively, an ordered pair can be formally
		thought of as a set {a,b} with a total order."  
		What about <a,a>? 
	g.  (5 points) Regular Expression
	h.  (5 points) Quasiquotation
	i.  (5 points) Mathematical Induction
	j.  (5 points) Axiom of Choice
	k.  (5 points) Constructivism
	l.  (5 points) mu operator (try to go to the page where the mu
		is displayed in greek -- I clicked on the link in
		mu recursive functions)... otherwisee you may not be
		able to read the page properly.



YOU CAN HELP EACH OTHER, BUT DO NOT GIVE ANSWERS:

(20 points)
2.  Write out the following sets of strings (you may use "..." for an infinite set):

	a.  (5 points)

		{ "aa", "aaa" }^^+^^^

	b.  (5 points)

		{ "", "a", "b" }^5

	c .  (5 points)

		{ "a" } { "" } { "aa", "b" }^2 { "b" } 

	d.  (5 points)

		( U__i=0,1,2,3,4___ { "a" }^i ) - {} - { "aa" }^^*^^^


NO TALKY TALKY

(20 points)
3.  Say whether true or not true.  

For a and c:
If true, try a proof (you choose your style -- more points for better style).  
If not true, give a counterexample.  

For b and d:
If not true, give a proof.  If true, give the required sets.

	a.  (5 points)  For all S, a set of strings, 
		(S^^+^^^)^^+^^^ = S^^+^^^.

	b.  (5 points)  Consider the set of strings T:
		T = { x  in.gif  {a,b}^^*^^^ | x does not have "bb" as a substring }.
		There is an S such that S^^*^^^ = T

	c.  (5 points) For all sets of strings S and T,
		if ( S^^*^^^ = T^^*^^^ ) and ( S^2 = T^3 )
		then  S = T.

	d.  (5 points) There exists a finite set R s.t.
		for all infinite sets of strings S and T,
		R S R = R T R.