CS103
HO#37
Slides--Finite Automata II
May 3, 2010
1
CS103
Mathematical Foundations of Computing
5/3/10
Midterm Exam: Tuesday, May 4, 7 – 9 pm
Coverage is through induction (no automata)
Rooms by last name:
A – F
Educ 128
G – Ke
Gates B12
Ki – Z
420-041
Please go to the correct room and sit in
alternate seats.
Note:
It is important to do the reading in the Sipser text.
You should try to understand it line by line.
Come to office hours
, write us with questions,
or talk to your classmates!
Let
= {a, b, c, d}
Let L
Missing
= {w | there is a symbol from
not in w}
Start state: all letters missing
After one character, the state could be
a read, b, c, d still missing
b read, a, c, d still missing
c read, a, b, d still missing
d read, a, b, c still missing
After two characters, the state could be any of the previous, or
a, b read, c, d still missing
a, c read, b, d still missing
...
After three characters, ...
Some Important Definitions
Let M = (Q,
,
, q
0
, F) be a finite automaton and let w = w
1
, w
2
, ..., w
n
be
a string where each w
i
.
Then
m accepts w
if there exists a sequence of states r
0
, r
1
, ..., r
n
in Q
such that:
1.
r
0
= q
0
,
2.
(r
i
, w
i+1
) = r
i+1
for i = 0, 1, ..., n – 1, and
3.
r
n
F.
We say that
M recognizes language A
if A = { w | M accepts w }
A language is called a
regular language
if some finite automaton recognizes it.
More Important Definitions
Let A and B be languages.
We define the
regular operations
union,
concatenation, and star as follows:
Union
:
A
B = { x | x
A or x
B }
Concatenation
:
A
B = { xy | x
A and y
B }
Star
:
A
*
= { x
1
, x
2
, ..., x
k
| k
0 and each x
i
A }
A set is
closed
under an operation if applying that operation to members
of the set returns an object still in the set.
E.g:
N
= { 1, 2, 3, ... } is closed under addition but not under division.