Summer 1, 2014
CIS 511
Introduction to the Theory of Computation
Jean Gallier
Homework 3
June 13, 2014; Due June 23, 2014
A problems are for practice only, and should not be turned in.
Problem A1. Giv
Summer 1, 2014
CIS 511
Introduction to the Theory of Computation
Jean Gallier
Homework 1
May 30, 2014; Due June 6, 2014
Beginning of class
A problems are for practice only, and should not be turned in
CIS 511 Spring 2006: Homework 3 Solutions
Problem 1
Consider L1 rst. Observe that x = y i either
(a) |x| < |y|, which can be expressed as x = u1 , y = u2 w, where |u1 | = |u2 |
and w = ; or
(b) |x| >
Summer 1, 2014
CIS 511
Introduction to the Theory of Computation
Jean Gallier
Homework 4
June 24, 2014; Due July 2, 2014
A problems are for practice only, and should not be turned in.
Problem A1. (i)
Spring, 2010
CIS 511
Introduction to the Theory of Computation
Jean Gallier
Final Exam
May 11, 2010
Note that this is a closed-book exam
Read all the questions before starting solving any of them!
Pro
Summer 1, 2014
CIS 511
Introduction to the Theory of Computation
Jean Gallier
Homework 2
June 7, 2012; Due June 13, 2014
Beginning of class
A problems are for practice only, and should not be turned i
CIS 511: Spring 2012
Problem Set 2 Solutions
(1) Exercise 2.6 part (c) has its answer given in the book. That is, the book gives a
grammar generating the language, but does not prove that this grammar
CIS 511: Spring 2011
Problem Set 1 Solutions
(1) In class we discussed the concept of diagonalization, which can be used to show that some
function cannot be computed in some model of computation. Her
CIS 511: Spring 2012
Problem Set 3 Solutions
(1) Exercise 3.8 (b).
Answer:
1. Scan the tape till you read rst zero. If you reach end of input, scan again to
see if there is any remaining one, if there
Spring, 2010
CIS 511
Introduction to the Theory of Computation
Jean Gallier
Homework 3
February 18, 2010; Due March 4, 2010,
beginning of class
A problems are for practice only, and should not be turn
Spring, 2010
CIS 511
Introduction to the Theory of Computation
Jean Gallier
Homework 2
February 4, 2010; Due February 18, 2010
A problems are for practice only, and should not be turned in.
Problem A1
CIS 511: Spring 2012
Midterm Practice Problems
Some problems are harder than exam problems, but solving them will sharpen your thinking and
help you tackle the exam questions better. Solutions, perhap
Spring 2010
CIS 511
Introduction to the Theory of Computation
Jean Gallier
Homework 1
January 14, 2010; Due February 4, 2010
Beginning of class
A problems are for practice only, and should not be turn
CIS 511 Spring 2006: Homework 4 Solutions
Problem 1
Consider the binary operator on languages dened as follows: given two languages L1 and L2 over , L1 L2 consists of words of the form uv such that
u
CIS 511: Spring 2012
Problem Set 4 Solutions
(1) Exercise 5.4
Answer: No. Consider following counter-example.
A = cfw_0n 1n |n > 0
B = cfw_1
Dene f : as below
f (s) = 1 if s A
f (s) = 0 otherwise.
Cle
CIS 511: Spring 2012
Midterm Solutions
(1) Let B be any language over the alphabet . Prove that B = B i BB B and B .
Answer: One direction is easy. If B = B , then BB B = B and B = B .
Suppose now tha
CIS 511: Spring 2012
Midterm Practice Solutions
We have given sketches for the solutions to the practice problems. You should make sure you
know how to complete the proofs.
(1) Let L be a regular lang
Acceptable Indexings
Undecidable Problems
Chapter 6
Elementary Recursive
Function Theory
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6.1.
Acceptable Indexings
In a previous Sec
CIS511
Introduction to the Theory of Computation
Public Key Cryptography and RSA
Jean Gallier
April 30, 2010
2
Chapter 1
Public Key Cryptography; The RSA
System
1.1
The RSA System
Ever since written c
3.10. TREE DOMAINS AND GORN TREES
3.10
211
Tree Domains and Gorn Trees
Derivation trees play a very important role in parsing
theory and in the proof of a strong version of the pumping
lemma for the c
6.3.
Recursively Enumerable Sets
Recursively . . .
Consider the set
Reducibility and . . .
A = cfw_x N | x (a) is dened,
where a N is any xed natural number.
By Rices Theorem, A is not recursive (chec
3.17. LR(0)-CHARACTERISTIC AUTOMATA
3.17
259
LR(0)-Characteristic Automata
The purpose of LR-parsing, invented by D. Knuth in
the mid sixties, is the following: Given a context-free
grammar G, for any
The Class P
More Problems
Propositional Logic . . .
The Class N P, . . .
Chapter 7
Computational Complexity
The Cook-Levin . . .
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7.1
6.5.
The Recursion Theorem
The Recursion . . .
The recursion Theorem, due to Kleene, is a fundamental result in recursion
theory.
Extended Rice . . .
Creative and . . .
Theorem 6.5.1 (Recursion Theore
Spring, 2010
CIS 511
Introduction to the Theory of Computation
Jean Gallier
Homework 6
April 8, 2010; Due April 22, 2010
A problems are for practice only, and should not be turned in.
Problem A1. Prov
CIS511
Notes on the Closure Denition
Of the Regular Languages
Jean Gallier
Department of Computer and Information Science
University of Pennsylvania
Philadelphia, PA 19104, USA
[email protected]
Chapter 3
Context-Free Languages and PDAs
3.1
Context-Free Grammars
A context-free grammar basically consists of a nite set
of grammar rules. In order to dene grammar rules, we
assume that we have two
Pairing Functions
Coding of RAM . . .
Kleenes T -Predicate
Chapter 5
Universal RAM Programs and
Undecidability of the Halting
Problem
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182
3.6
CHAPTER 3. CONTEXT-FREE LANGUAGES AND PDAS
The Greibach Normal Form
Every CFG G can also be converted to an equivalent
grammar in Greibach Normal Form (for short, GNF).
A context-free grammar