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Lecture 8
CSCI
6050
Problem Reduction Simple Examples
In this lecture we consider Reducibility, the
primary method for proving that problems are
computationally unsolvable, i.e., in our terms, that
a language is undecidable.
CSCI
6050
We see how reducibil
Lecture 7
CSCI
6050
Decidable Problems & Regular Languages
In this lecture we begin to investigate the power of
algorithms to solve problems. We'll demonstrate
certain problems that can be solved algorithmically and
others that cannot. Our objective is to
Lecture 10
CSCI
6050
Polynomial Time Computable/Reducible
In this lecture we
.introduce the notion of polynomial time reducibility;
Definition: A function f : * * is a polynomial
time computable function if some polynomial time
Turing machine M exists tha
Lecture 11
CSCI
6050
Space Complexity - Definitions
In this lecture we begin our study of Space
Complexity. Here we are concerned with the
space requirements of a computation.
CSCI
6050
It turns out that space behaves much nicer than time!
Definition: Let
Homework Set #5
Pass in Problems 1and 3
Due: March 30, 2004
Instructor: Bob LaBarre
Computability & Complexity
CSCI-6050
SOLUTIONS
(Graded) Problem 1: Let,
MAX_CLIQUE = cfw_ <G, k> | the largest clique in an undirected graph G has k vertices.
Whether MAX_
Computability and Complexity
CSCI-6050
Midterm Exam
SOLUTIONS!
1. No, the language L is not regular. (By contradiction.) Suppose that L is regular. Clearly
L is infinite since there are countably infinite prime numbers, which implies L has a
pumping lengt
Homework #2 Solutions
CSCI-6050: Spring 2004
(Graded) Problem 1: Draw an FA that recognizes the language of all strings of 0s and 1s of length
at least 1 that, if the strings were interpreted as binary representations of integers, would represent
integers
Homework #3
CSCI-6050: Spring 2004
(Graded) Problem 1: Using mathematical induction, prove that every string produced by the CFG with
productions
S 0 | S0 | 0S | 1SS | SS1 | S1S
contains more 0s than 1s.
Proof: We will use strong induction. The only strin
Computability & Complexity
CSCI
6050
Prerequisites
Rensselaer at Hartford - PDE
Spring 2004
CSCI
6050
This is a THEORY Course. As such, we
will spend time working with
DEFINITIONS, THEOREMS and
PROOFS. We will learn about
ABSTRACT computer structures and
Lecture 4
CSCI
6050
Kleenes Theorem
This lecture is devoted to the study of Kleenes
Theorem, which has to do with the equivalence of
regular languages and those languages that can be
accepted by finite automatons. We do this in two
parts, first showing th
Lecture 5
CSCI
6050
Palindromes Revisited
In this lecture we study context-free grammars,
context-free languages, and pushdown automata.
We see how these structures expand on the
capabilities of finite automata methods weve
already studied.
CSCI
6050
Cons
Lecture 9
CSCI
6050
Complexity Class - O
CSCI
6050
In this lecture we consider Time Complexity.
We start by defining a collection of complexity
classes and analyzing some algorithms. While
decidability is nice, we can ask the question, "Can
we compute the
Lecture 3
CSCI
6050
Lecture 3
We will investigate distinguishable and
indistinguishable strings, and use this concept to
generate bounds on the number of states in a finite
automaton.
CSCI
6050
We will also begin our study of Non-Determinism
Non-determini
Lecture 6
CSCI
6050
Turing Machines General Ideas
In this lecture we continue our studies of
computability theory. We introduce the Turing
Machine and some of its variants. Additionally,
we will develop a good understanding of
algorithms. We finish this l
Lecture 12
CSCI
6050
Quantifiers
In this lecture we consider three PSPACEComplete languages, TQBF, the formula game,
and the generalize geography game.
CSCI
6050
The first problem we will investigate is TQBF
True totally Quantified Boolean Formula.
Gener
Homework Set #1
Due: Jan. 20, 2004
Instructor: Bob LaBarre
Computability & Complexity
CSCI-6050
SOLUTIONS
1. (Graded) Let A, B be two languages with A. Suppose X is a language satisfying the relation X =
AX B. Prove that X = A*B.
Proof: To show set equali
Lecture 2
CSCI
6050
Regular Expressions
CSCI
6050
In this lecture we investigate regular expressions
and introduce the formal definition of finite state
automata.
Regular expressions are a finite way to describe
infinite languages. ("most typical string")
There have been a number of questions concerning
HW #3, Problem #2. Here is some information to
help you along.
Given the NFA- (this corresponds to the table given)
0,1
1
q0
q2
0
q3
0
1
q1
0
q5
q4
1
Prove that the equivalent NFA (no -transitions) is given
Homework Set #4
Pass in Problems 1and 2
Due: March 16, 2004
Instructor: Bob LaBarre
Computability & Complexity
CSCI-6050
SOLUTIONS
(Graded) Problem 1: Show that the following problem is undecidable: Given a Turing
machine M, a tape symbol a and non-empty
Computing is recognizing/describing/generating/accepting a language.
1
Context-free grammars: introduction
Two functions associated with languages:
reading: for language recognizers; writing: for language generators
One way to generate a language is to us
Notes on Finite Automata
Department of Computer Science
Professor Goldberg
Textbooks: Introduction to the Theory of Computation
by Michael Sipser
Elements of the Theory of Computation
by H. Lewis and C. Papadimitriou
These notes contain the material from
1
Basic Denitions of Graph Theory.
Denition 1
An undirected graph G = (V, E ) consists of a set V of elements called vertices, and a
multiset E (repetition of elements is allowed) of pairs of vertices called edges.
The order of a graph G(V, E ) is |V |; t
Satisability Problem:
x1, x2, . . . xn are boolean variables
xi =
T RU E 1
F ALSE 0
The negation x of x is TRUE (FALSE) i x is FALSE
(TRUE).
A literal is either a variable x or its negation x.
A clause is a set of literals: (x3 x4 x6)
Given an assignment
1
Vertex Coloring
A k vertex coloring of a graph G(V, E ) is a function
f : V [1, k ],
such that for any edge xy E , f (x) = f (y ).
Vertex Color (decision version): Given a graph G(V, E )
and an integer k > 0, is there a vertex coloring of G
which uses k
1
Summary
The running time of a computation by a Turing
Machine is dened to be the number of applications
of the transition function done by the Turing Machine
while executing the computation.
A deterministic Turing machine which halts on all inputs is ca
1
Examples of polynomial reducibility
Denition 1 A graph G(V, E ) is called connected,
if for any two distinct vertices u and v , there is a path
connecting u with v . A connected component of a
graph G is a maximal connected subgraph of G.
For a given gr
1
Decision problems vs Optimization Problems
Example:
Clique (decision version): Given a graph G(V, E ) and
an integer k > 0, is there a clique of size k in G ?
L = cfw_ G(V, E ), k : G contains a clique of size k .
Clique (optimization version): Given a