CS547T SPRING 2007

ASSIGNMENT 2

   February 2007
 S  M Tu  W Th  F  S
            15 16 17
             assigned
18 19 20 21 22 23 24
            NO CLASS
25 26 27 28
      due in class, hardcopy


COLLABORATE ALL YOU WANT:

(50 points)
1.  Give the definitions as precisely, formally, and tersely as possible.  Style points may be
	given or taken away.

	a.  (5 points) Define  Lambda.gif(S),the lambda closure of a set of states S in an NFA- Lambda.gif

	b.  (5 points) If <Q,Sigma.gif,q_0,A,delta.gif> is a NFA- Lambda.gif, define the NFA that reduces it;
		i.e., give Q' in terms of Q, give A' in terms of A, etc. for the NFA that accepts the same language

	c.  (5 points) Define the equivalence class of x, [x], 
		under an equivalence relation R, where x is in the domain of R

	d.  (5 points) Define distinguishability w.r.t. L

	e.  (5 points) Given FSM's M_1 and M_2, specify M s.t. L(M) = L(M_1)^^*^^^L(M_2);
		i.e., give Q in terms of Q_1 and Q_2 and any new states, etc.

	f.  (5 points) Define the minimal DFA of a language L, in terms of equivalence classes of strings under I_L

	g.  (5 points) Define the reachable states in a DFA, Q' subset.gif Q, 
		(a state is reachable iff there is at least one string that can get to the state)

	h.  (15 points) Suppose an NFA- Lambda.gif, M, defined (as in class) so that  delta.gif  is a relation on Q times.gif Sigma.gif  → Q;
		define a computation of M on input x to be a sequence of states that M could follow while processing x 
		(each sequence starts at q_0);

		AND

		define  Omega.gif__M,x___ the set of all such possible computations of M on x

		You will need some method for specifying sequences:
		i suggest either using a function or a concatenation operation;
		you can use <q_1, q_2, ..., q_n>, but you may lose style points
		for using "...". 

		You may refer to the i-th character in x as x_i.



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

(30 points)
2.  Give these as required

	a.  (5 points)

		Give the minimal DFA for the language (0+01)0^^*^^^1

	b.  (5 points)

		Give an NFA for the language (1+11)(110)*11

	c.  (5 points)

		Give the DFA that reduces the NFA above using the Q' = 2^Q construction

	d.  (5 points)

		Number your states in question 2a, starting with q_0 receiving number 1

	e.  (5 points)

		Give L(p,q,2) for all p,q, pairs for your machine in question 2a
		with numbering in question 2d

	f.  (5 points)

		Give an inductive definition of an NFA- Lambda.gif, where  delta.gif
		is a relation on Q times.gif Sigma.gif  → Q (as in class);
		starting with the simplest machine, <{q_0}, {}, q_0, {}, {}>


NO TALKY TALKY

(20 points)
3.  Give required proofs or disproofs.

	a.  (5 points)  

		I_L is an equivalence relation

	b.  (5 points)  

		{ x c y | x in.gif {a,b}^^*^^^, x neq.gif y^r } is regular

	c.  (5 points) 

		For a set of computations,  Omega.gif__M,x___, defined in question 2h above,
		there is always a finite subset  Omega.gif'__M,x___ subset.gif   Omega.gif__M,x___
		that contains an accepting computation iff
		the original set contains an accepting computation

	d.  (5 points) 

		Given two regular expressions, R_1 and R_2, there is a test
		for the equivalence of their denotations that always terminates in
		finite time