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
Solution:
Define S(n) to be the predicate "f(n) <= 3*n*2^cf
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 only one state which is also an accetping state.
if R = a
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" (or "final") states, as well as
"transitions" from one
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: any finite sequence of symbols; the empty sequence is den
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 the solution:
The error is in the inudction step. The
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 and where E^S(x,y)
holds iff x is an element of y. Tran
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(s) on the transition arrow(s) from i to j.
If i = j, y
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).
 Formulas F and E are "logically equivalent" if every i
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.
 Examples:
. "forall x P(x) \/ exists y Q(x,y)": x is bound (i
General divideandconquer recurrences:
 Many algorithms written using "divideandconquer" technique: split up
problem, solve subproblems recursively, combine solutions. Worstcase
running times of such algorithms satisfy recurrences of the form:
cfw_K
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 the book)
 Using the truth table, give a DNF and CNF for