Homework 2 solutions
1. (10 + 5 = 15 points)
(i) Dene Rev(L) = cfw_xR | x L . Note xR is the reversal of the string x.
Thus, e.g., if L = cfw_ab, aba, babb then Rev(L) = cfw_ba, aba, bbab.
Let L1 be a regular language that is recognized by a DFA M1 = (Q1

Homework 10 CISC 303
Timo Ktzing (tkoe@udel.edu)
Monday, May 4.
Friday, May 8.
Handed out:
Due Date:
(2 or 8 points)
Let f : N2 N, (x, y ) x + y (f is addition on natural numbers). If you have showed that f is computable when all
inputs are unary, show th

Homework 11 CISC 303
Timo Ktzing (tkoe@udel.edu)
Friday, May 8.
Friday, May 15.
Handed out:
Due Date:
Problem 1.
(12 points)
(i) Prove informally that L0 = cfw_x|y | x2 + 15 y 300 x3 is decidable.
(ii) Prove informally that L1 = cfw_code(M ) | M is TM an

Lecture Notes
CISC 303 Automata Theory
Spring 2009
Timo Ktzing
May 6, 2009
These are the ocial lecture notes as used for the introductory course on Automata Theory at the University of
Delaware (CISC 303) in spring 2009.
The main topics of the course are:

Midterm Study Guide CISC 303
Timo Ktzing (tkoe@udel.edu)
The midterm will basically consist of questions similar to homework problems of the homework sets 1-4. Hence,
you should make sure to understand all homework problems. In particular, the following w

Multi State Machines
a.k.a.: The Powerset Construction; The Subset Construction
Timo Ktzing
o
March 3, 2009
General Idea
Given an NFA M = (A, Q , , F , q0 ), we dene the multi state
machine M for M as described below. This multi state machine
M will be a

Homework 5 CISC 303
Timo Ktzing (tkoe@udel.edu)
Handed out: Monday, April 6.
Deadline for Version 2 language approval: Monday, April 20.
Due Date: Wednesday, May 20.
This document describes the (optional) project. The project exists in two versions: Versi

Concept Map for Regular Languages CISC 303
Timo Ktzing (tkoe@udel.edu)
are
Automata
are
are
DFAs
compile to
NFAs
compile to
-NFAs
1
Concept Map for Regular Languages CISC 303
Timo Ktzing (tkoe@udel.edu)
are
Automata
are
are
DFAs
dene
compile to
NFAs
compi

CISC 303 Automata Theory
Spring 2009
Instructor: Timo Ktzing (tkoe@udel.edu)
o
Time/Classroom: MWF, 12:20PM-1:10PM, 235 Purnell Hall.
Instructors Oce Hours (Room 102 Smith Hall): Thursday, 2:30-4:30.
Additional times may be arranged with the instructor if

Homework 5 solution
3. Show L5 = cfw_ Mx #My | L(Mx ) L(M y ) is co-re but not re. (actually, it turns out that its not co-re.)
To show that L5 is not recursive:
Assume that L5 is recursive, we can generate a machine K that takes Mx #My as input
and retu

Homework 4 solutions
1. L = cfw_an bm ck | n + k = m
Accept on empty stack
Comment: This pda pops a for each b. Once all the as are popped o (i.e.
is exposed as top of stack), the bs are pushed in. Then, a b is popped for each c in the input,
with accept

Homework 3 solutions
For each of the languages below, give a context-free grammar that will generate it.
1. L1 = cfw_an bm ck | n + m = k
Must add a c for each a and b.
Production Rules
S aSc
S S1
S
S1 bS1 c
S1
2. L2 = cfw_an bm ck | n + k = m
Must add

Homework 1 solutions
1. (10 + 5 = 15 points)
a. Let M1 = (Q1 , , 1 , q0 1 , F1 ) and M2 = (Q2 , , 2 , q0 2 , F2 ) be two DFA.
Construct a DFA that accepts L(M1 ) - L(M2 ). (If A and B are two sets then
A - B = cfw_ x | x A and x B .
/
Give a precise desc

Homework 9 CISC 303
Timo Ktzing (tkoe@udel.edu)
Monday, April 27.
Friday, May 1.
Handed out:
Due Date:
(8 points)
Let f : N2 N, (x, y ) x + y (f is addition on natural numbers). Show that f is computable by explicitly giving a
TM computing f .
Problem 1.

Homework 8 CISC 303
Timo Ktzing (tkoe@udel.edu)
Monday, April 20.
Friday, April 24.
Handed out:
Due Date:
Problem 1.
(8 points)
(i) Design a total acceptor-TM M such that L(M) = cfw_an bn cn | n 0.
(ii) Design a total function-TM f such that, for any word

PBL on closure properties of (D)CFL CISC 303
Timo Ktzing (tkoe@udel.edu)
The following table gives closure properties for various collections of languages.
closure under
reg
DCFL
CFL
complement
set dierence
?
dot product
Kleene-
(HW5)
We dene the followin

Final Study Guide CISC 303
Timo Ktzing (tkoe@udel.edu)
The nal will basically consist of questions similar to homework problems of the homework sets 5-11. In addition
to this, there will be some general questions regarding regular languages, and regarding

Homework 1 CISC 303
Timo Ktzing (tkoe@udel.edu)
Wednesday, February 11.
Monday, February 16.
Handed out:
Due Date:
(8 points)
Give the DFA from Example 1.1.4 in the lecture notes in set notation.
Problem 1.
(8 points)
Give a DFA M in graphical notation on

Homework 2 CISC 303
Timo Ktzing (tkoe@udel.edu)
Monday, February 16.
Friday, February 27.
Handed out:
Due Date:
Note that the Problem 1 gives you a choice of two parts, you don't need to submit both (no extra credit for
submitting both parts). Further not

Homework 2 add-on CISC 303
Timo Ktzing (tkoe@udel.edu)
Friday, February 20.
Friday, February 27.
Handed out:
Due Date:
The following problem is an add-on to Homework 2.
This problem is an extra-credit problem.
(8 points)
Use the pumping lemma to show that

Homework 3 CISC 303
Timo Ktzing (tkoe@udel.edu)
Handed out: Friday, February 27.
Due Date: Friday, March 6.
Problem 1. (8 points)
Let
A = cfw_a
be our alphabet.
(i) Give an NFA
M
in graphical notation only such that
L(M) = cfw_w | |w|
is divisible by
3
or

Homework 4 CISC 303
Timo Ktzing (tkoe@udel.edu)
Friday, March 6.
Friday, March 13.
Handed out:
Due Date:
(8 points)
Give a regular expression for the following languages.
Problem 1.
(i) L0 = cfw_w cfw_a, b | the second to last symbol of w is a;
(ii) L1 =

Homework 4 add-on CISC 303
Timo Ktzing (tkoe@udel.edu)
Monday, March 16.
Friday, March 27.
Handed out:
Due Date:
Problem 1.
(8 points) Use the algorithm from class1 to minimize the following DFAs. Show all work.
(i)
a
0
1
a
(ii)
a
b
a
a
b
0
1
2
b
Problem

Homework 5 CISC 303
Timo Ktzing (tkoe@udel.edu)
Handed out: Monday, March 16.
Due Date: Friday, March 27.
Problem 1. (2 points)
(i) What is the singular of automata?
(ii) Describe, in your own words, rst the syntactic, then the semantic dierence of the wo

Homework 5 Mathy Version CISC 303
Timo Ktzing (tkoe@udel.edu)
Handed out: Monday, March 16.
Due Date: Friday, March 27.
Problem 1. (8 points)
in graphical notation only
(i) Give a PDAs
accepting
L0
as dened just below.
L0
is the set of all strings from th

Homework 6 CISC 303
Timo Ktzing (tkoe@udel.edu)
Monday, April 5.
Friday, April 10.
Handed out:
Due Date:
(8 points)
Let A = cfw_a, b. Give two CFGs accepting L0 and L1 as dened just below, respectively.
Problem 1.
(i) L0 = cfw_w A | w = wR is the set pal

Homework 7 CISC 303
Timo Ktzing (tkoe@udel.edu)
Monday, April 13.
Friday, April 17.
Handed out:
Due Date:
The total number of points for this assignment is 32. If you work on problems from more that 32 points, you may
get additional points as extra credit

Basic Math Extra Homework CISC 303
Timo Ktzing (tkoe@udel.edu)
Handed out: Wednesday, March 11.
Due Date: Friday, March 27.
Please consider working through this homework if you have trouble with math it is designed to make you a little
bit more comfortabl