CMPSCI 601: Theory of Computation

David Mix Barrington

Spring, 2003

Homework Assignment #1

Due Wednesday 12 Feb 2003 in class

(Mod 24 Feb for off-campus students -- deadline extended on 18 Feb)

• Question 1 (20): Find regular expressions for the following two languages. It may be helpful to design a DFA and then use either state-elimination or the construction from lecture to create a regular expression equivalent to the DFA. In both cases the alphabet is S = {a,b}. Note: You are responsible for convincing the grader that your regular expression is correct.
  • (a,10) The set of all strings that never have two b's in a row. (Note that this includes strings with no b's at all.)
  • (b,10) The set of all strings with an even number of a's and an even number of b's.

• Question 2 (10):If A is a language, define A^R to be the set {w: w^R is in A}. (If w is a string, w^R is that string written backward.) Prove that if A is regular, then A^R is also regular. (Hint: There are at least two good proofs, one using induction on all regular expressions and one using NFA's. For the latter, it may be useful to prove a lemma that any regular language is the language of an NFA that has exactly one final state, not equal to its start state.)

• Question 3 (20): In Lecture 2 we defined the language A₇ of decimal strings that evaluate to non-negative integers that are divisible by seven. This language has a DFA with seven states and no DFA with fewer than seven states. Let S be the alphabet {0,1} and let strings over S represent integers in the usual binary representation. Let A₆ be the set of binary strings that represent numbers divisible by six.
  • (a,5) Give a six-state DFA for A₆.
  • (b,15) Can A₆ be decided by a DFA with fewer than six states? Prove your answer. (If yes, exhibit such a DFA, if no, use the Myhill-Nerode Theorem.)

• Question 4 (20): In Lecture 1 there is an NFA N whose language A is given by the regular expression S^*1Sⁿ, where S represents the alphabet {0,1}. This NFA has n+2 states. Using the Myhill-Nerode Theorem, prove that any DFA for the language A has at least 2ⁿ⁺¹ states. (Hint: Find a set of 2ⁿ⁺¹ different strings such that no two strings in the set are A-equivalent.) Can you find a DFA with exactly 2ⁿ⁺¹ states?

• Question 5 (30): Let NONCFL be the language {aⁿbⁿcⁿ : n &ge 0}. (If you see "ampersand-g-e" here (&ge), it should be a "greater-than-or-equal sign" but your browser is too ignorant to implement that.) We show in Lecture 3 that NONCFL is not a context-free language.
  • (a,10) Prove that the complement of NONCFL is a context-free language. (Hint: Show it is the union of several context-free languages.) Explain why this shows that context-free languages are not closed under complement.
  • (b,10) Find two context-free languages whose intersection is NONCFL, and show that they are context-free. Explain why this shows that the context-free languages are not closed under intersection.
  • (c,10) Prove that the intersection of a context-free language and a regular language must be context-free. Let SAMECOUNT be the set of all strings over alphabet {a,b,c} that have equal numbers of a's, b's, and c's. Use the closure property you just proved (not the CFL Pumping Lemma directly) to prove that SAMECOUNT is not context-free.

  Last modified 18 February 2003