CMPSCI 601: Theory of Computation

Offered through the PEEAS distance learning program

Homework Assignment #1

David Mix Barrington

Assignment posted Mon 9 June 2003

Make sure you have read the statement on academic honesty on the homework assignment index page.

• Question 1 (15): Let A be the set of strings over alphabet {a,b} such that every a in the string has a b immediately before it and a b immediately after it. (Note that a string with no a's at all satisfies this condition and is thus in A.) Design a DFA that recognizes the language A. Determine the four Myhill-Nerode equivalence classes of A and give a regular expression for each.

• Question 2 (30): Let M be a one-tape Turing machine that never writes on its tape. (Note that its one tape is the input tape -- it has no work tapes.) Show that L(M) is a regular language.

  Hint: M is in essence a two-way DFA, like an ordinary DFA except that it can move in either direction based on its state and the letter it sees. That is, its transition function is from (Q times Σ) to (Q times {L,R}) where Q is its state set and Σ its alphabet.

  To show that L(M) is regular, we define a set of questions about a string w and show that two strings with the same answers to all these questions are Myhill-Nerode equivalent with respect to L(M). Here are the questions:

  • If M is started on the left end of w in the start state, does it ever leave w to the right, and if so in what state does it do so?
  • If M is started on the right end of w in state s, does it ever leave w to the right and if so in what state does it do so?

  If you choose to follow this Hint, your job is to show (a) that if strings v and w have the same answers for each of these questions, then they are Myhill-Nerode equivalent for L(M), (b) there are only finitely many different possible answers for these questions, and (c) it follows that L(M) is regular.

• Question 3 (15): Using the result of Question 2 even if you did not prove it yourself, argue that the complexity class DSPACE(1) is equal to the class of regular languages. Recall that a DSPACE(1) machine has a read-only input tape and a read-write worktape, and the size of the latter is bounded by a number that does not depend on the input size.

• Question 4 (25):Define a two-stack PDA to be like an ordinary PDA except that it has two stacks. Thus each transition includes a push or pop from each stack as well as reading of an input character and a change of state. Prove that a language A is recursively enumerable iff there exists some two-stack PDA M such that A = L(M). (Hint: Use the two stacks to simulate the tape of a Turing machine. You may omit details if your general argument is convincing.)

• Question 5 (15): Let A be a recursively enumerable language. Prove that there is a Turing machine M such that if M is started with input consisting of two strings u and v, and exactly one of u and v is in A, then M halts and its output determines which of the strings is in A. Don't worry about what M does if both input strings are in A or neither input string is in A.

Last modified 16 June 2003