CSCI 5444: Theory of Computation


Logistics


Relevant Textbooks


Assignments


Course Objectives

The objective of this course is provide an introduction to the theory of computation covering the following three branches of theoretical computer science:
  1. Automata Theory
    • Formalization of the notion of problems via formal languages
    • Formalization of the notion of computation using "abstract computing devices" called automata
    • Understanding a hierarchy of classes of problems or formal languages (regular, context-free, context-sensitive, decidable, and undecidable)
    • Understanding a hierarchy of classes of automata (finite automata, pushdown automata, and Turing machines)
  2. Computability Theory
    • Understanding Church-Turing thesis (Turing machines as a notion of "general-purpose computers")
    • Understanding the concept of undecidability , i.e., when a problem can not be solved using computers
    • How to show undecidability using the concept of problem reduction
  3. Complexity Theory
    • Complexity classes : how to classify decidable problems based on their time and space requirements
    • Complexity classes P and NP
    • Intractability (NP-completeness)
    • How to prove NP-completeness?
    • Complexity Classes PSPACE and PSPACE-completeness

Topics Covered

  1. Regular Languages (3 weeks)
    • Deterministic finite-state machines
    • Nondeterministic finite-state machines
    • Regular expressions
    • Properties of regular languages
    • Languages that aren't regular: pumping lemma
  2. Context-Free Languages (2 weeks)
    • Context-free grammars
    • Pushdown automata
    • Properties of Context-free languages
    • Languages that aren't context-free: pumping lemma for CFLs
  3. Computability Theory (4 weeks)
    • Turing machines and their variants
    • Church-Turing thesis
    • Decidable languages
    • Undecidability
    • Proving Undecidability of a given problem using problem reductions
    • Rice's theorem
    • Famous undecidable problems such as Post Correspondence Problem (PCP), Tiling problem, halting problems for multistack and two-counter machines.
  4. Complexity Theory (4 weeks)
    • Time and space complexity
    • Complexity classes P and NP, and NP-Completeness
    • Famous NP-complete problems
    • Complexity class PSPACE and Pspace-Completeness
    • Complexity classes L and NL, and NL-completeness
  5. Advanced Topics for class projects (presentations in Week 16)
    • Polynomial, exponential, and arithmetical hierarchies
    • Approximation algorithms
    • Probabilistic complexity
    • Interactive proofs and complexity class IP
    • Probabilistically checkable proofs
    • Quantum algorithms (Dasgupta, Papadimitriou, and Vazirani)
    • Alternation
    • Automata on infinite words and S1S
    • Timed and hybrid Automata
    • Learning Finite Automata
    • Markov Decision Processes
    • Program termination analysis

Grading

The overall grade will be based on a cumulative score computed by adding together the grades from:

Schedule and Lecture Notes

# Date Description Chapter
1 January 17 Introduction to the theory of computation [Slides ] 0

Part One: Automata Theory

2 Week 1 — January 19 Regular languages and finite automata [ Slides ] 1.1, 1.2
3 Week 2 — January 24 Deterministic Finite Automata (Guest lecture by Dr. Sergio Mover) 1.2, 1.3
4 Week 2 — January 26 Nondeterministic Finite Automata (Guest lecture by Dr. Sergio Mover) 1.3
5 Week 3 — January 31 Subset Construction: Nondeterminism and Alternation [Slides ] 1.4
6 Week 3 — February 2 Regular expressions
7 Week 4 — February 6 Non-Regular languages: Pumping Lemma [Slides ] 1.4
8 Week 4 — February 9 Context-Free Languages: Grammars and Derivations 2.1
9 Week 5 — February 14 Pushdown Automata 2.2
10 Week 5 — February 16 Non-Context-Free Languages 2.3
11 Week 6 — February 21 Closure properties of CFLs
12 Week 6 — February 23 Wrap-up of Regular Languages and CFLs 2.1 — 2.3
13 Week 7 — February 28 In-Class Quiz I 1 and 2

Part Two: Computability Theory

14 Week 7 — March 2 Turing machines 3.1
15 Week 8 — March 7 Variants of Turing machines 3.2 and 3.3
16 Week 8 — March 9 Decidability: Decidable Languages 4.1
17 Week 9 — March 14 Halting Problem: Diagonalization and Reductions 4.2
18 Week 9 — March 16 Reductions: More undecidable problems 5.1, 5.2
19 Week 10 — March 21 Logics and Decidability 6.2
20 Week 10 — March 23 Wrap-up: Turing machines and decidability 3-4-5-6
21 Week 11 — March 27-31 No Class — Spring Break
22 Week 12 — April 4 In-class Quiz II 3-4-5-6

Part Three: Complexity Theory

23 Week 12 — April 6 Complexity 7.1 and 7.2
24 Week 13 — April 11 NP, co-NP, polynomial-time reductions and NP-completeness 7.3
25 Week 13 — April 13 NP-complete problems and reductions 7.4
26 Week 14 — April 18 Space Complexity Classes: Savitch's theorem
27 Week 14 — April 20 PSPACE and PSPACE-complete problems 7
28 Week 15 — April 25 L, NL, and NL-completeness 8.4-8.6
29 Week 15 — April 27 In-class Quiz III
30 Week 16 — May 2 Class Project Presentations
31 Week 16 — May 4 Class Project Presenttions

Notes

  1. Accommodation Statement. If you qualify for accommodations because of a disability, please submit to me a letter from Disability Services in a timely manner (for exam accommodations provide your letter at least one week prior to the exam) so that your needs can be addressed. Disability Services determines accommodations based on documented disabilities. Contact Disability Services at 303-492-8671 or by e-mail at dsinfo [AT] colorado.edu. If you have a temporary medical condition or injury, see Temporary Injuries under Quick Links at the Disability Services website and discuss your needs with me.
  2. Religious Observances. Campus policy regarding religious observances requires that faculty make every effort to deal reasonably and fairly with all students who, because of religious obligations, have conflicts with scheduled exams, assignments or required attendance. In this class, you should notify your instructor of any conflict at least two weeks in advance. See full details here .
  3. Classroom Behavior. Students and faculty each have responsibility for maintaining an appropriate learning environment. Those who fail to adhere to such behavioral standards may be subject to discipline. Professional courtesy and sensitivity are especially important with respect to individuals and topics dealing with differences of race, color, culture, religion, creed, politics, veteran's status, sexual orientation, gender, gender identity and gender expression, age, disability, and nationalities. Class rosters are provided to the instructor with the student's legal name. I will gladly honor your request to address you by an alternate name or gender pronoun. Please advise me of this preference early in the semester so that I may make appropriate changes to my records. For more information, see the policies on classroom behavior and the student code.
  4. Discrimination and Harassment. The University of Colorado Boulder (CU Boulder) is committed to maintaining a positive learning, working, and living environment. CU Boulder will not tolerate acts of sexual misconduct, discrimination, harassment or related retaliation against or by any employee or student. CU's Sexual Misconduct Policy prohibits sexual assault, sexual exploitation, sexual harassment, intimate partner abuse (dating or domestic violence), stalking or related retaliation. CU Boulder's Discrimination and Harassment Policy prohibits discrimination, harassment or related retaliation based on race, color, national origin, sex, pregnancy, age, disability, creed, religion, sexual orientation, gender identity, gender expression, veteran status, political affiliation or political philosophy. Individuals who believe they have been subject to misconduct under either policy should contact the Office of Institutional Equity and Compliance (OIEC) at 303-492-2127. Information about the OIEC, the above referenced policies, and the campus resources available to assist individuals regarding sexual misconduct, discrimination, harassment or related retaliation can be found at the OIEC website.
  5. Honor Code. All students enrolled in a University of Colorado Boulder course are responsible for knowing and adhering to the academic integrity policy of the institution. Violations of the policy may include: plagiarism, cheating, fabrication, lying, bribery, threat, unauthorized access, clicker fraud, resubmission, and aiding academic dishonesty. All incidents of academic misconduct will be reported to the Honor Code Council (honor@colorado.edu; 303-735-2273). Students who are found responsible for violating the academic integrity policy will be subject to nonacademic sanctions from the Honor Code Council as well as academic sanctions from the faculty member. Additional information regarding the academic integrity policy can be found at honorcode.colorado.edu.
  6. The web-page of a previous offering of the course is available here .