CS 3511  Additional Final Exam Review Exercises       Fall Semester, 2006

Here are some exercises similar to ones that might be on the final exam
in addition to the lists of exercises for the previous two exams.
The items in parentheses are the corresponding items in the 6th edition 
of the text (since I think everyone is using the 5th edition).

Note that * exercises were in the assignments, but are computationally
too hard (not enough time) to put on an exam.

Ex. 5, 7, 12-14, 19-21, 28, 41-43, 47, 51, 63, 67 of Section 3.3 pages 253-256
 (= 5, 7, 12-14, 19-21, 28, 37-39, 43, 47, 57, 59 of Section 3.2)

Ex. 12-15, 18, 19, 31, 32, 35-42 of Section 3.4 pages 270-274
 (= 10-13, 16, 17, 25,  ?, 27-34 of Section 3.3, and Ex. 32 of the 5th Ed.
  asks (a) to give a recursive definition of the function ones(s) which
  counts the number of 1's in the bit string s, and (b) use structural
  induction to prove that ones(st) = ones(s) + ones(t) for bit strings s, t)

Ex.   4-6,  9, 16-20, 29, 30, 37 of Section 3.5 pages 283-284
 (= 10-12, 15, 24-28, 37, 38, 49 of Section 4.4)

Ex. 1-4, 7, 8, 12 of Section 3.6 pages 283-284
 (= 1-4, 7, 8, 12 of Section 4.5)

Ex. 28, 35, 43, 51, 57, 58, 59, 60, 67 of Chapter 3 Supplement pages 292-297
 (= 20, 35, 43, 49, 55, 56, 57, 58, 65 of Chapter 4 Supplement)

  From Chapter 3 of 5th Edition, it would be good to know math induction
 (regular, strong, and structural), Fibonacci numbers, reversal of a string
 (Ex. 34-38 p. 272 = 4.2.10, 4.2.11, 4.1.14, 4.1.61 of 6th Ed.), and of
 course be able to write a simple recursive procedure.

Ex. 28 of Section 9.1 page 643
 (= 28 of Section 10.1)
  Also know the result of Theorem 5 of Section 9.1: There are at most m^h
  leaves in an m-ary tree of height h.  Most importantly, know Corollary 1:
  If an m-ary tree of height h has l leaves, h >= ceiling(log_m(l))

Ex. 11 of Section 9.2 page 657
 (= 11 of Section 10.2)
  Also know the result of Theorem 1 of Section 9.2: A sorting algorithm
  based on binary comparisons requires at least ceiling(log n!) comparisons.
  Also know Corollary 1: The number of comparisons used by a sorting
  algorithm to sort n elements using binary comparisons is Omega(n log n).

Ex.  4-9, 11 (easy ones), 12-15, 18, 22, 25, 27 of Section 11.1 pages 748-751
 (= 6-11, 15 (easy ones), 18-21, 24, 28, 31, 33 of Section 12.1)

Skip Section 11.2 of 5th Edition

Ex. 1, 2, 4, 5,      8-13, 18, 19, 27 of Section 11.3 pages 764-765
 (= 1, 2, 4, 5, 8,9,12,17, 44, 45, 55 of Section 12.3)

Ex.   2-4,  6,  7*, 15 (use Ex. 16), 16-19 of Section 11.4 pages 774-775
 (= 4,7,8, 10, 13*, 21 (use Ex. 22), 22-25 of Section 12.4)

Ex. 1-3, 5-7, 8 (= Example 1), 9-11, read 15-17 of Section 11.5 page 782
 (= 1-3, 7-9, 10,             11-13, read 17-19 of Section 12.5)

Ex. 3, 13-15 (only FA's) of Chap. 11 Supplement pages 784-786
 (= 3, 13-15 (only FA's) of Chap. 12 Supplement)

Also, know the following:

 The grammar/language and machine hierarchy:

  Languages generated by grammar    Machines recognizing them
  -----------------------------------------------------------
  Phrase-structure grammars         Turing machines

  Context-sensitive grammars        Linear bounded automata

  Context-free grammars             Pushdown automata

  Regular grammars                  Finite-state automata


 The idea of the Church-Turing Thesis: any problem that can be solved by
an effective algorithm can be solved by a Turing machine.