Lectures, Course Outline, and Learning Objectives

Key to ASCII notation

'{}' = ∅ = "empty set"
'(-' = ∈ = "element of"
'(_' = ⊆ = "subset of" (not strict)
'u' = ∪ = "union"
'n' = ∩ = "intersection"
'~L' = L = "complement of L"
'-]' = ∃ = "there exists"
'\-/' = ∀ = "for all"
'/\' = ∧ = "and"
'\/' = ∨ = "or"
'->' = → = "implies"
'<->' = ↔ = "if and only if (iff)"
'!' = ¬ = "not", e.g., 'a != b' = a ≠ b = "a is not equal to b", 'w !(- L' = w ∉ L = "w is not an element of L", etc.
'\sum' = ∑ = summation sign
'\prod' = ∏ = product sign
'\Sigma' = Σ = capital greek letter Sigma, '\delta' = δ = lowercase greek letter delta, etc.
'|_x_|' = ⌊x⌋ = floor(x)
'|^x^|' = ⌈x⌉ = ceiling(x)
'_' indicates a subscript, e.g., 'q_1' = q₁
'^' indicates a superscript, e.g., 'n^2' = n²
curly braces '{}' surround longer subscripts/superscripts, e.g., '\sum_{0 <= i <= n} 2^{i/2}' = ∑_{0 ≤ i ≤ n} 2^i/2

Lecture summaries

Every week, specific sections of the textbook will be posted as readings. You will be expected to read these sections to prepare for the following week's lectures and tutorials.
At the end of each week, a short summary of the material covered during lecture will be posted.

Week 1 (Jan 7–11); readings: section 3.1 (and review chapters 0, 1, 2 as needed).
Week 2 (Jan 14–18); readings: sections 3.2, 3.3.
Formal details of the equivalence between regular TMs and 2-way infinite TMs.
Week 3 (Jan 21–25); readings: section 4.1.
Week 4 (Jan 28–Feb 1); readings: section 4.2.
Week 5 (Feb 4–8); readings: section 5.1.
Week 6 (Feb 11–15); readings: section 5.3.
Week 7 (Feb 25–29); readings: problem 5.28, section 7.1.
Week 8 (Mar 3–7); readings: sections 7.2, 7.3.
Updated on Mon 31 Mar: Week 9 (Mar 10–14); readings: section 7.4.
Detailed write up of reduction CNF-SAT ≤_p 3SAT
Week 10 (Mar 17–21); readings: section 7.5.
Week 11 (Mar 24–28); readings: sections 8.1, 8.2.
Week 12 (Mar 31–Apr 4); readings: sections 8.3, 8.4.
Week 13 (Apr 7–11); readings: sections 9.1, 10.1.
See the Tests/Exam page for advice about studying for and writing the final exam.

Course outline

Lecture/tutorial topics

The following topics will be covered in this course, in the order listed. For each topic, we have indicated the approximate number of weeks required to cover that topic as well as a list of the relevant sections in the textbook.

Computability Theory [6 weeks]
(Chapters 3, 4, 5 in the textbook)
- Turing machines: definitions and examples (section 3.1)
- Other models of computation, the Church-Turing thesis (sections 3.2, 3.3)
- Diagonalization, the Halting problem (sections 4.1, 4.2)
- Decidability and recognizability, examples (sections 4.2, 5.1)
- Reducibility, examples (sections 5.1, 5.2)
- Mapping reducibility, examples (section 5.3)
Complexity Theory [7 weeks]
(Chapters 7, 8, 10 in the textbook)
- Models of efficient computation (sections 7.1, 7.2)
- P, NP, coNP, examples (section 7.2, 7.3)
- Polytime reducibility, NP-completeness (section 7.4)
- Cook's theorem, more NP-completeness (section 7.5)
- Self-reducibility (not in textbook)
- Space complexity and other complexity classes (sections 8.1, 8.2, 8.3, 9.1)

Learning objectives

By the end of this course, students should