Honors Theory of Computation, Spring'99

Homework 4

-- DATE DUE: March 29, 1999
-- PROBLEMS:
  1. Show that EXPT has complete language. ONLY SKETCH THIS PROOF (you must not use more than one page).
  2. Consider the many-one reducibility using transformations computed in deterministic log space and linear time (this is a subclass of of the transformations is denoted Llin in chapter 4). Show the following:
    (A) For each i, show that if L is in DTIME-SPACE(ni,log n) and L' is reducible to L, then L' is in DTIME-SPACE(ni+1,log n).
    (B) Show that NTIME(n2) has a complete language under this reducibility. HINT: The proof of Theorem 3 in Chapter 5 may be useful in thinking about this problem.
    (C) NTIME(n2) is different from DLOG.
  3. A alternating finite automata (afa) is an alternating Turing machine with no work tapes, and whose input tape is one-way.
    (A) Show that we can assume that the afa moves its input head on reading each input symbol, and that it hals upon reading the first blank symbol after the input.
    Please pick up your HINTfor this part!
    (B) Show that the language accepted by an afa is regular language.
    HINT: Assume that the afa has k states. View the afa as a function
    H: \Sigma -> [Bk -> Bk]
    where B={0,1} and \Sigma is the input alphabet.