Function Problem
Problem : Let f: N > N be the function defined by
/ 1,
if n = 0

f(n) = < 2,
if n = 1

\ 4*f(n  2) + 2^n
if n > 1
Prove that for every integer n >= 3, f(n) <= 3*n*2^cfw_n  2
Solu
Equivalence of Regular Expressions
R.E. to FSA.
 We use the defintion of a R.E. and induction:
. Basis: if R = empty then M contains only one state and no accepting state.
if R = e then M contains on
Finite State Automata
 Simple models of computing devices used to analyze strings. A F.S.A.
has a fixed, finite set of "states", one of which is the "initial state"
and some of which are "accepting"
Formal Language Theory
Basic definitions:
 Alphabet: any finite, nonempty set of atomic symbols (meaning
"compound" symbols like "ab" are not allowed)
(e.g., cfw_a,b,c, cfw_0,1, cfw_+).
 String: an
Examples of wrong inductive proofs:
 Example 1.8 on page 31 of the notes.
 Exercise 13 on page 58 of the notes. First state the problem clearly
and let them think about it for a few minutes.
Here is
First Order Language L Questions
Consider the firstorder language L that consists of only one
binary predicate symbol E, and consider the structure S for L
whose domain is the collection of all sets
FSA to R.E Example
Here is an example: for L^0_cfw_i,j, if i <> j the only string that
takes M from i to j without going through any other state (because k = 0) is the
string which has only the symbol
Logical equivalences:
 Formula F "logically implies" formula E if and only if every interpretation
that satisfies F also satisfies E.
Example: forall x(A(x) > B(x) and A(c) logically implies B(c).

Free vs bound variables:
 Variable x is "bound" in formula A if it appears in A within the scope
of a quantifier on x. Variable x is "free" if it appears outside the
scope of any quantifier.
 Exampl
General divideandconquer recurrences:
 Many algorithms written using "divideandconquer" technique: split up
problem, solve subproblems recursively, combine solutions. Worstcase
running times of
DNF and CNF Formulas Question
Consider the following formula:
(~x > (y /\ x) /\ (~y > (x /\ z)
 Give the truth table of this formula.
(to do this use the method described on pages 146 and 147 of th