MATH/COM 323 - Theory of Computation - Fall '15
Instructor: Perry Susskind
Office Hours: Tues, Thurs 9:15 – 10:00 AM, and by appointment.
Office: 409 Fanning
Phone: 439-2025 (office), 444-1086 (home)
Instructor: Christine Chung
Office Hours: Mon, Wed 9:30-11 AM (tentative). Sign up for a slot here. Or email for an appointment.
Office: 220 New London Hall
Phone: 439-2074 (office)
Texts: "Introduction to the Theory of Computation," by Michael Sipser, Second Edition, Thomson 2006 (required).
Prerequisites: MAT210 or permission of the instructor.
We shall study topics in chapters 1 – 8 of the text. These are: Finite Automata, Context Free Languages, Turing machines, Church's Thesis, Uncomputability, Computational Complexity.
[Stolen shamelessly from a similar class at Univ of Pitt]
There will be weekly problem sets for homework. These will be graded and returned to you. Attendance in class is required. You will also be expected to present problems and their solutions in class from time to time. There will be a midterm and a final exam. Your grade will be based on a weighted average of homework grades, test grades, and classroom performance. Numerical averages of your grades will not supersede the judgment of your instructor. If there is time we will also discuss recent applications of the theory.
What is a computation? Are there fundamental problems for which no computer algorithm will ever produce a solution? Are there computational problems which in principle can be solved but which are impracticable by virtue of the amount of time they take to solve?
In the theory of computation, these important questions in contemporary philosophy and mathematics are attacked by investigating the abstract theory of computing machines including the theory of deterministic and nondeterministic finite automata, formal languages, computability by Turing machines, uncomputability and computational complexity. The approach is mathematical and the subject matter is related to the incompleteness (Gödel) theorems of mathematical logic.
However, the viewpoint and motivation come from concerns in computer science; for instance, the theory of parsing, graph algorithms and many problems of a practical nature are studied. The course is a bridge between ideas in linguistics, logic, computer science, abstract mathematics, philosophy and more recently, physics; it culminates in a discussion of a most pressing open problem in current mathematical research: Does P = NP?
[Images from actual episodes of the Simpsons and Futurama]
[Note: All readings and exercises refer to the Sipser text.]
HW1 due Thurs, Sept 10
Reading: all of chapter 0 and 1.1
Exercises: 0.1, 0.2, 0.4, 0.5, 0.7, 0.11, 1.1, 1.3,
HW2 due Thursday Sept 17
Reading: HW1 grading notes and solutions, Sipser 1.2 and 1.3
Exercises: 1.4 g, 1.5 g, 1.6 a, c, f, j, 1.8 b, 1.10 b,
1.12, 1.14 a, b, 1.16, 1.18 a, c, f, j
1.1 Finite automata
1.3 Regular expressions
HW3 due Thursday Sept 24
Reading: HW 2 grading notes and solutions, and the rest of chapter 1
Exercises: 1.19, 1.21, 1.29 b, 1.31, 1.30, 1.32, 1.46 a, c, 1.48 (Hint: write D as the union of two regular expressions), 1.57.
1.4 The pumping lemma
Reading: HW2 and HW3 solutions, Sipser 2.1, 2.2
Exercises: 2.1, 2.4b,e,f, 2.5b,e,f, 2.11, 2.14, 2.16, 2.26, 2.27
2.1 Context free grammars
Thurs Sept 24
2.2 Pushdown automata
Reading: the rest of chapter 2 and HW 4 solutions
Exercises: 2.2, 2.25, 2.30a, 2.35, *Show that the complement of the language (from Example 2.38) is a context free language. Note, this example provides another proof that CFLs are not closed under complementation.
2.3 Equivalence of CFGs and PDAs
2.3 The pumping lemma for context-free languages
The Church Turing Thesis
Reading: HW 5 solutions, 3.1 and 3.2
Exercises: 3.1a,c, 3.2b, 3.9, 3.13, 3.15d,e, 3.16d
3.1 Turing machines
3.2 Turing machine variants
Reading: the rest of chapter 3
Exercises: 3.6, 3.7, 3.18, 3.21
Due on Thurs Oct 29
Take-home, open-book, open-note Midterm Exam on chapters 1-3.
4.1 Decidable languages
Reading: all of chapter 4
Exercises: 4.3, 4.6, 4.12, 4.16, 4.19, 4.22
5.1 The halting problem and other undecidable problems
More 5.1 and 5.2
Reading: all of chapter 5
Exercises: 5.1, 5.3, 5.9, 5.17
5.3 Mapping reducibility
7.1 Measuring complexity
Reading: chapters 7.1 and 7.2
Exercises: 7.1a,b,e, (show how each True answer satisfies def of big-O) 7.2a,b,e,f, (show how each True answer satisfies def of little-o) 7.3a,b, 7.4, 7.6, 7.10
7.2 The class P
7.3 The class NP
HW11 (due Dec 3)
Reading: chapters 7.3, 7.4 and 7.5
Exercises: 7.5, 7.7, 7.11, 7.17, 7.19, 7.26
7.4 NP-completeness, SAT is NP-complete
7.5 SAT is NP-complete continued, additional NP-complete problems
7.5 Additional NP-complete problems cont’d (e.g., VC, Ham Path, Subset Sum)
HW12 (due Dec 10)
Exercises: 7.27, 7.29, 7.34
See hints posted to moodle!
Please complete a course evaluation here:
8.1 Savitch’s Theorem
The Poly-time Hierarchy
Take-home Final Exam due by noon on Dec 22.
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