Learning Goals
Church-Turing Thesis
Definition of a Turing Machine
Understanding state diagram and execution of a TM
Understanding pseudocode description of a TM
Recognizing vs Deciding:
Def: deciders, recognizable languages, decidable languages
Copyrigh
Lecture 8
September-06-12
Copyright 2016 by Nicholas J.A. Harvey.
You may not distribute this document without permission.
8:38 PM
CFGs:
- Backus & Naur developed models for programming languages.
- Developed by Chomsky (interested in natural languages).
Learning Goals
LearningGoals
Definitions:
Definitions:DFA,NFA,
DFA NFA Language
LanguageAccepted
Accepted
By,RegularLanguage
What is non determinism?
Whatisnondeterminism?
Copyright2016byNicholasJ.A.Harvey.Youmaynotdistributethisdocumentwithoutpermissio
Copyright 2016 by Nicholas J.A. Harvey.
You may not distribute this document without permission.
Lecture 12
October-01-13
10:11 AM
Last Time:
- Formal definition of Turing machines
- Recognized / deciding
Restate again:
- Language recognized by M is L(M)
Learning Goals
LearningGoals
Whatistheoryofcomputationabout?
What is theory of computation about?
Whatisacomputationalproblem?
h i
i ?
Whatiscomputation?
Definitions:alphabet,string,computational
p
problem,decisionproblem,language
,
p
, g g
Copyright2
Learning Goals
LearningGoals
Theorem:AnNFAcanbeconvertedtoaDFA.
An NFA can be converted to a DFA.
Theorem:
Theorem:Regularlanguagesclosedunder
union,concatenation,star,complement
,
,
,
p
Definition: RegularExpressions
B ildi
Buildingregexps
f
forcertain
Lecture 9
September-24-13
8:37 PM
Copyright 2016 by Nicholas J.A. Harvey.
You may not distribute this document without permission.
Last time we discussed parse trees: a useful way to capture the structure when deriving a string from a grammar.
Ambiguous G
Copyright 2016 by Nicholas J.A. Harvey.
You may not distribute this document without permission.
Lecture 11
October-04-12
6:41 PM
Church-Turing Thesis:
"Intuitive notion of algorithm" equals "Turing machine"
This is not a theorem because it cannot be made
Learning Goals
The idea that an algorithm needs a formal definition
E.g., lambda-calculus or Turing Machines. (No details yet.)
The notion of Turing complete
The idea that some problems have no algorithmic
solution.
An easy proof using countable vs u
Learning Goals
Recognizable vs Decidable Languages
Hierarchy of languages
Can we make TMs more powerful by adding features?
Multitape Turing Machines
Non-deterministic Turing Machines
Copyright 2016 by Nicholas J.A. Harvey. You may not distribute thi
Learning Goals
LearningGoals
DefinitionofNFAs
of NFAs
Definition
Theorem:Regularlanguagesclosedunder
union concatenation star complement
union,concatenation,star,complement
(Tobeseenagainnexttime)
Copyright2016byNicholasJ.A.Harvey.Youmaynotdistributethi
Learning Goals
LearningGoals
Theconceptofnon
concept of nonregular
regularlanguages.
languages.
The
Alongwalkinagraphmustcontainacycle.
The pumping
pumpingcondition
condition:Anecessaryconditionfor
A necessary condition for
The
alanguagetoberegular.
Th
Learning Goals
Definition of leftmost derivation
Definition of ambiguous grammars
Theorem: Equivalence between PDAs and CFGs
We only discuss building a PDA to recognize the language
described by a CFG.
Copyright 2016 by Nicholas J.A. Harvey. You may n
Lecture 10
September-25-13
9:37 AM
Wrap-up of CFLs
- Every PDA can be converted to a CFG. The details are rather lengthy. There are many details, and
it's not clear that you will gain much from discussing it in class. See Textbook Lemma 2.27.
- Is every l
Learning Goals
Definition of CFGs and CFLs
Meaning of u v (Yields)
Meaning of u * v (Derives)
Understanding the language specified by a CFG
Parse Trees
Copyright 2016 by Nicholas J.A. Harvey. You may not distribute this document without permission.
Gramma
Learning Goals
LearningGoals
ProofofPumpingLemma.
of Pumping Lemma.
Proof
HowtousePumpingLemmatoprovethata
languageisnotregular.
g g
g
Copyright2016byNicholasJ.A.Harvey.Youmaynotdistributethisdocumentwithoutpermission.
ThePumpingLemma:Everyregularlangu
The P vs. NP problem
Madhu Sudan
May 17, 2010
Abstract
The resounding success of computers has often led to some common misconceptions about
computer science namely that it is simply a technological endeavor driven by a search
for better physical material
The Status of the P versus NP Problem
Lance Fortnow
Northwestern University
1.
INTRODUCTION
When Moshe Vardi asked me to write this piece for CACM,
my first reaction was the article could be written in two
words
Still open.
When I started graduate school
THE P VERSUS NP PROBLEM
STEPHEN COOK
1. Statement of the Problem
The P versus NP problem is to determine whether every language accepted
by some nondeterministic algorithm in polynomial time is also accepted by some
(deterministic) algorithm in polynomial
CPSC 421: Introduction to Theory of Computing
Assignment #0, due Monday September 12th, in class
[5] 1. Let L1 and L2 be finite sets. Suppose that C is a finite set with L1 C and L2 C.
Suppose that x L1 L2 . Is x C? If so, explain why; if not, give a coun
CpSc 421
Midterm 1
October 11, 2006
Do problems 0 and 1 and any two of 2, 3, or 4. Graded on a scale of 100 points.
0. (5 points) Your name: Mark Greenstreet
Your student #: 00000000
1. (35 points) (Sipser exercise 1.47)
Let = cfw_1, # and let
A
= cfw_w |
CpSc 421
Midterm 1
October 8, 2008
Do problem 0 and any three of problems 1-5.
If you attempt more than three of problems 1-5, please indicate which ones you want graded otherwise, Ill make an
arbitrary choice.
Graded on a scale of 100 points.
You can att