CSCI 5444: Theory of Computation


Relevant Textbooks


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


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 Presentations


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