PS #1, Due 09/18 (updated to reflect 9/11 class cancellation)

• Send me an email as per the syllabus.


• From Linz text: Ch 1.1 (p. 12) : 1, 7, 11, 20, 22, 23, 26

• What's wrong with the following "proof" by induction that all frogs are the same color?
  
  Base Case: Clearly for any set containing just 1 frog all frogs in the set have the same color.
  
  Inductive Assumption: Assume true for all sets containing i = 1, ..., N frogs.
  
  Inductive Step: To prove that all frogs in a set of N+1 frogs have the same color,
  consider removing the first frog. By the induction assumption, then, the last N
  frogs must all have the same color. Now put the first frog back in and consider
  removing the last frog. Again, by the inductive assumption the first N frogs
  must all have the same color. Thus each frog in this set of N+1 frogs has the same color, and consequently so will
  any set of N+2, N+3, ... frogs. In other words, all frogs must have the same color.

• From Linz text, Ch 1.2: 1, 3, 8, 10b,c, 12, 14a, 15, 19
• Note that for all problem sets in this course it is usually reasonable to use the results
  of other exercises in the current or previous section(s), even if they were not assigned, as
  an aid in solving assigned problems. It is bad form, however, to use material from future
  chapters.

• Ch 2.1: 2a, 2b, 7b,16
  For 2a, 2b, and 7b please draw the transition graph AND write the transition function explicitly as
  a state table in the manner given on p. 40-41.
• Ch 2.2: 11, 13
• Extra1: Suppose we have an alphabet and is as defined on p.16 in the text. Is it appropriate to
  consider ? Why or why not?
• Extra2: Given the NFA M = (Q,,,q0,F) where
  
  Q = {q0,q1,q2,q3,q4}
  = {a,b}
  F = {q4}
  = {(q0,a,q0),(q0,b,q0),(q0,b,q1),(q1,b,q2), (q1,a,q3),(q2,,q4),(q3,b,q4),(q4,a,q4), (q4,b,q4)}
  
  This is just another way of explicitly listing the value of the transition function for all
  relevant inputs, and is equivalent to the transition table form (again, as shown in pages
  40-41). Here each (q,,q') triplet denotes (q,) = q', for {a,b,}.
  
  a) Draw the transition graph for this NFA.
  b) Given a string w = bababab, show that both *(q0,w) = q0 AND *(q0,w) = q4 are possible.
  c) Is this w L(M)?

• E1) Consider the union of the two sets {0} and {1^n : n > 0}. Construct an NFA that
  accepts this union, with a single final state and no -transitions.
• E2) Convert the following NFA into a DFA:
  
  
• Ch 2.3: 5,6,13
• Ch 3.1: 1,3,5b,6,
• E3) Construct a DFA that accepts the language
  
  L = { w : (number_of_zeros(w) mod 2) = 0 }
  
  over the binary alphabet (ie, {0,1}). Write an equivalent regular expression.
• Extra Credit (optional): Suppose a regular expression r does not contain either or .
  What condition or conditions must r satisfy to ensure that L(r) is an infinite language?

• Ch 3.2: 1, 4c (but don't build a DFA, just construct an accepting NFA for the language, using the style of Theorem 3.1), 4d, 10a
• Ch 3.3: 1,15
• Ch 4.1: 2b, 6, 13

• Ch 4.2: 3, 6, 9
• Ch 4.3: 3, 4f, 5d (use Pumping Lemma for these 3 problems), 18

• Ch 5.1: 7c, 8c, 18
• Ch 5.2: 1, 13
• Ch 6.1: 5, 7
• Ch 6.2: 3,10
• Ch 7.1: 1,3a,4c,6,10

• Ch 7.2: 1, 10
• Ch 7.3: 1, 5, 6, 10
• Ch 8.1: 2, 7c, 8a, 8f

• Ch 8.2: 2, 7, 11, 12, 23
• Ch 9.1: 3, 5, 7e, 11a

• Ch 9.2: #3d
• Ch 10.4: 1, 2
• Ch 11.1: 3, 5 (see Linz Theorem 11.4)