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Study Guide for Discrete Structures II | COT 4210, Assignments of Discrete Structures and Graph Theory

University of Central Florida (UCF)Discrete Structures and Graph Theory

Material Type: Assignment; Class: Discrete Structures II; Subject: Computer Theory; University: University of Central Florida; Term: Fall 2000;

Typology: Assignments

Pre 2010

Uploaded on 02/24/2010

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koofers-user-491 🇺🇸

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COT 4210 Homework #1
Assigned: 8/28/00
Due: In class, 9/13, 14
Book: Sudkamp
Book Exercises:
Chapter 2(ppg 49-51): 2, 9, 10, 12, 19, 32, 38ade
Chapter 6(ppg 188-194): 2, 8, 14, 17, 33, 39
Chapter 7(ppg 223-226): 5, 8, 11acf, 13, 27
Other problems:
1) A one-DFA is defined exactly as a normal DFA, except that a one-DFA must have
exactly one accept state. Prove that a one-DFA is strictly weaker than a standard DFA.
(This means that you must show that there is a regular language L that can not be
accepted by any one-DFA.)
2) Create a DFA that accepts all binary strings over the alphabet {0,1} that are divisible
by 7. (You may assume that leading 0’s add no value to the number. For example, 00111
should be accepted by your DFA.)
COT 4210 Homework #2
Book Exercises:
Chapter 3: 2, 8, 10, 12, 14
Chapter 5: 18, 23, 28, 30
Chapter 8: 1, 3, 12, 14, 17
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Assigned: 8/28/

Due: In class, 9/13, 14

Book: Sudkamp

Book Exercises:

Chapter 2(ppg 49-51): 2, 9, 10, 12, 19, 32, 38ade

Chapter 6(ppg 188-194): 2, 8, 14, 17, 33, 39

Chapter 7(ppg 223-226): 5, 8, 11acf, 13, 27

Other problems:

1) A one-DFA is defined exactly as a normal DFA, except that a one-DFA must have

exactly one accept state. Prove that a one-DFA is strictly weaker than a standard DFA.

(This means that you must show that there is a regular language L that can not be

accepted by any one-DFA.)

2) Create a DFA that accepts all binary strings over the alphabet {0,1} that are divisible

by 7. (You may assume that leading 0’s add no value to the number. For example, 00111

should be accepted by your DFA.)

COT 4210 Homework

Book Exercises:

Chapter 3: 2, 8, 10, 12, 14

Chapter 5: 18, 23, 28, 30

Chapter 8: 1, 3, 12, 14, 17

Assigned: 10/24/

Due: 11/9/

1) (5 pts) Let a k-PDA be a pushdown automaton that has k stacks. Thus a 0-PDA is an NFA and a 1-PDA is a convensional PDA. You already know that 1-PDAs are more powerful (recognize a larger class of languages) than 0-PDAs. Show that a 2-PDA is more powerful than a 1-PDA. (Hint: Simulate a Turing machine tape with two stacks.) 2) A Turing machine with left reset is similar to an ordinary Turing machine except that the transition function has the form : Q x   Q x  x {R, RESET}. If (q,a) = (r,b,RESET), when the machine is in state q reading a, the machine’s head jumps to the left-hand end of the tape after it writes b in the tape and enters state r. Note that these machines do not have the usual ability to move the head one symbol left. Show that Turing machines with left reset recognize the class of Turing-recognizable languages(ie. they have the same power as a standard Turing machine.) 3) (6 pts) A Turing machine with stay put instead of left is similar to an ordinary Turing machine except that the transition function has the form : Q x   Q x  x {R, S}. At each point the machine can move its head right or let it stay in the same position. Show that this Turing machine variant is NOT equivalent to the usual version. (Bonus: What class of languages do these machines recognize?) 4) (3 pts each) Show that the collection of decidable languages is closed under the operations of a) union b) concatenation c) star d) complementation e) intersection 5) (2 pts) Let A be the language containing only the single string s, where s = 0, if God does not exists 1, if God exists Is A decidable? Why or why not? 6) (6 pts) Show that you can decide if two DFAs are equivalent(accept the exact same set of strings) or not. Give an algorithm that is guaranteed to terminate that correctly decides this problem. 7) (5 pts) Build a Turing machine that enumerates the set of even length strings over {a}. 8) (6 pts) Prove that there is no algorithm with input consisting of a Turing machine M = {Q, , , , q 0 , F), a state qi Q, and a string w  *^ that determines whether the computation of M with input w enters state

qi.

9) (5 pts) Show that the language {aibj^ | i,j 0  i  j} is not regular by using the the pumping lemma. 10) (5 pts) Show that the language {ak!^ | k > 0} is not context free by using the pumping lemma. (Note: Problems 1-6 come from Introduction to the Theory of Computation by Michael Sipser. The other two problems come from the text.)