North Carolina Central University
Department of Mathematics & Computer Science
CISG 5115 - Theory of Computation
Fall 2008
Topics Covered
Date
Aug 19
Aug 21
Aug 26
Aug 28
Sep 02
Sep 04
Sep 09
Sep 11
Sep 16
Sep 18
Sep 23
Sep 25
Sep 30
Oct 02
Oct 07
Oct 09
Oct 14
Topics Covered
Introduction to the Course; Methods of Proof – Overview
More on Methods of Proof; Proving Equivalence; Proof and Disproof; General Proof and (Counter)Example; Mathematical Induction; Structural and Mutual Induction
Central Concepts of Automata Theory: Alphabets, Strings, Languages; Problems as Languages; Informal Introduction to Finite
Automata; Marble Rolling Toy Example
Deterministic Finite Automata - Formal Definition; Transition
Function; Generalized Transition Function
Non-Deterministic Finite Automata: Formal Definition, Need of
Non-Determinism in Computation; Transition Function; Generalized Transition Function; Equivalence of NDFA and DFA; Subset
Construction
Greedy Version of the Subset Construction; Application: Automata Recognizing Patterns in Text; Finite Automata with
²-transitions: Formal Notation; Examples; ²-closures
Generalized Transition Function for ²-NDFA’s; Constructing a
DFA, Equivalent to Given ²-NDFA; Examples
Regular Expressions; Operations: Closures, Concatenation, Union;
Languages Corresponding to Regular Expressions; Building Regular Expressions; Examples
Converting Finite Automata to Regular Expressions: General Recursive Construction; Time and Space Analysis
Converting Finite Automata to Regular Expressions by Eliminating States; Converting Regular Expressions to ²-NDFA
Algebraic Laws for Regular Expressions; The Test for a RegularExpression Algebraic Law
The Pumping Lemma for Regular Languages; Proving that a
Language is Not Regular; Applications of the Pumping Lemma:
L01 = {0i 1i |i ≥ 0} is Not a Regular Language
Examples of Non-Regular Languages and Use of Pumping Lemma
to Prove them Not Being Regular; Closure Properties of Regular
Languages; Closure Under Union, Intersection, Complement, Difference, Reversal, Closure (Kleene Star) and Concatenation
Homomorphism; Inverse Homomorphism; Closure of the Regular Languages Under Homomorphism and Inverse Homomorphism;
Automata Accepting Languages Defined with Respect to a Regular
Language: half (L)
Time and Space Complexity of the Conversions Between Different Representations of Regular Languages; Decision Properties of
Regular Languages; Testing for Emptiness; Membership Testing
Midterm Exam
Minimization of DFA’s; State Equivalence; Table-Filling Algorithm; Testing Equivalence of Regular Languages
Exercises
1.16, pp. 20–21;
1.21, pp. 24–25
1.23 pp. 26–27
Examples from p. 32;
Ex. 2.2.1, pp. 52–53
Ex. 2.2.4(a), p. 53
Ex. 2.2.8, p. 54
Ex. 2.3.1, p. 65
Ex. 2.4.1, 2.4.2, p. 72;
Ex. 2.5.1(a, b), p. 79
Ex.
Ex.
Ex.
Ex.
2.5.1(c), 2.5.2, p. 79;
2.5.3(b), p. 80
3.1.1(b, c), p. 91;
3.1.4(b), p. 92
Ex. 3.2.1(a, b, c, d),
p. 107
3.2.1(e), p. 107;
3.2.4, p. 108
3.4.1(d, f, g, h),
3.4.2(a, c), 3.4.3, p. 122
4.1.2(f), 4.1.4, p. 132
Ex. 41.5, p. 143,
4.2.8, p. 148
4.3.1–4.3.5, p. 155
4.4.1, 4.4.2, p. 165
Oct 16
Oct 21
Oct 23
Oct 28
Oct 30
Nov 04
Nov 06
Nov 11
Nov 13
Nov 18
Nov 20
Nov 25
Nov 27
Dec 02
Dec 04
Context-Free Grammars: Definition, Notational Conventions, Examples; Grammars for 0n 1n and the Palindromes; Derivations;
Leftmost and Rightmost Derivations; Sentential Forms; Languages
of Context-Free Grammars
Parse Trees; Definition, Examples; Equivalence Between Parse
Trees and Derivations; Remarks on Parsing; General Task of Parsing, Requirements; Ambiguity; Ambiguous Grammars, Examples;
Inherently Ambiguous Grammars; Removing Ambiguity
Pushdown Automata(PDA): Informal Introduction, Discussion of
Their Computational Power, and Differences with the Finite Automata; The Language LwwR , and the PDA for it; Formal Description of a PDA; Graphical Representation of a PDA; Configurations
and Instantaneous Descriptions of a PDA; Computation on a PDA
Instantaneous Descriptions of Pushdown Automata (Configurations); Legal Computations; Languages of Pushdown Automata;
Acceptance by Final State
Class Canceled
Discussion of the Online Homeworks on Context-Free Grammars;
Proofs of Equivalence of Languages
Acceptance by Empty Stack; Equivalence of Acceptance by Empty
Stack and by Final State; Restriction on the Size of the Stack
Alphabet (Binary Coded Stack Alphabet); Restrictions on the Size
of the String Pushed; Examples
Equivalence of Pushdown Automata and Context-Free Grammars;
Constructing a PDA that Accepts the Language of a CFG; Constructing a CFG, whose Language is the Language of a Given PDA
Constructing a Context-Free Grammar from a Given PDA: Complete Discussion of Exercise 6.3.3, p. 251; Deterministic PDA and
Their Languages
Chomsky Normal Form of a Context-Free Grammar; Converting a CFG to CNF; Eliminating Useless Symbols; Eliminating ²Productions; Eliminating Unit Productions; Generating, Reachable and Nullable Symbols
Comments on Some Problems from Gradiance Homework;
Greibach Normal Form (GNF); Converting a Grammar to a
Greibach Normal Form
Pumping Lemma for the Context-Free Languages; Proving that Languages are not Context-Free,
Examples:
0n 1n 2n , 0i 1j 2i 3j , Lww ; Closure Properties of the ContextFree Languages; The Intersection of two CFL’s is Not Necessarily
a CFL; Intersection of a CFL and Regular Language is a CFL;
Closure Under Substitution; Closures Under Union, Concatenation, Homomorphism, and Closure; Closure Under Reversal;
Inverse Homomorphism
Thanksgiving - No classes
Algorithmic Issues in Context-Free Languages; Complexity of Conversions from PDA to CFG and vice versa; Complexity of Conversions Between PDA’s; Complexity of Converting a CFG to CNF;
Linear Time Test for Emptiness; Testing Membership in a CFL:
CYK Algorithm; Examples
Turing Machines: Definition, Notation, Instantaneous Description;
Classes of Problems Computable on Turing Machines; Halting
5.1.2, p. 182
5.1.8, p. 183
5.2.2, p. 193;
5.4.5, p. 216
Examples 6.1-6.4,
pp.227–231
Example 6.7, p. 235
6.2.1, p. 241;
6.2.4, 6.2.7, 6.2.8, p. 242
6.3.1, 6.3.3, p. 251
7.1.1, 7.1.2, p. 275
7.1.11, 7.1.12,
pp. 278–279
Examples 7.19–7.21,
pp. 284–285