Week 1

• Meeting 1: Introduction to the course. Finite automata: examples and definition. Pages 31--40 from Chapter 1 of text.
• Meeting 2: The process of designing finite automata. Operations on Languages. Closure under the union operation. Pages 39--47 from Chapter 1 of text.

Week 2

• Meeting 1: Nondeterministic finite automata, examples, and formal definition. Pages 47--54 of text.
• Meeting 2: Two views of what it means for an NFA to accept a string. Equivalence of NFA's and DFA's (Theorem 1.39). Pages 54--58 of text.

Week 3

• Meeting 1: Proof of the statement: if two languages are regular, their concatenation is regular. A similar result for the star operator. Introduction to regular expressions, from Section 1.3. Pages 58--64 of text.
• Meeting 2: Examples of regular expressions and the language corresponding to them. Proof of the statement: if a language is described by a regular expression, it is regular. Intro to GNFA's and what it means for a GNFA to accept a string. Pages 65--70.

Week 4

• Meeting 1: Given a GNFA with more than 3 states, how we can construct an equivalent GNFA with one less state. Proof of the statement: If a language is regular, then it is described by some regular expression. Pages 71--76.
• Meeting 2: The pumping lemma. Using it to show that certain langauages are not regular. Section 1.4, pages 77--81.

Week 5

• Meeting 1: Proof of the pumping lemma, and more examples of nonregular languages. pages 77--82. Introduction to context free grammars. Section 2.1, pages 102--103.
• Meeting 2: More examples of context free grammars. Formal definition of CFG and what it means to derive a string. Designing CFG's. Pages 103--107 from Section 2.1.

Week 6

• Meeting 1: Ambiguity in context free grammars. Pushdown automata (PDA's) and examples. Pages 107--116. Note that we skipped the part on Chomsky Normal Form.
• Meeting 2: One more PDA example. Equivalence of PDAs and context free grammars (Theorem 2.20). Proof of one direction of this, namely, that if a language is context free then there is a PDA that recognizes it. Pages 115--120 from Section 2.2.

Week 7

• Meeting 1: Equivalence of PDAs and context free grammars (Theorem 2.20). Proof of the other direction, namely, that if a PDA recognizes some language, then it is context free.
• Meeting 2: Midterm.

Week 8

• Meeting 1: Example illustrating the proof that if a PDA recognizes some langauage, then it is context free. The pumping lemma for CFL's, and an application to show that a particular language is context-free. (Section 2.3)
• Meeting 2: More examples of languages that are not context-free. Overview of pumping lemma proof. (Section 2.3)

Week 9

• Meeting 1: Turing Machines: Introduction, formal definition, and an example. Pages 165--173 from Section 3.1.
• Meeting 2: Another example, and definitions leading to Turing-recognizable and decidable languages. Pages 165--173 from Section 3.1.

Week 10

• Meeting 1: Example 3.11 from page 174: an implementation-level description of a Turing machine. Multi-tape and non-deterministic Turing Machines, and equivalence with deterministic Turing Machines. Pages 176--180 of Section 3.2.
• Meeting 2: Church-Turing thesis: TM as model for algorithm. The story of Hilbert's tenth problem. Representing graphs, automata, grammars, etc. using strings that can be input to a Turing machine. Pages 181--187 of Chapter 3.

Week 11

• Meeting 1: Decidable problems concerning automata. Pages 193--197 of Chapter 4.
• Meeting 2: Decidable problems concerning context-free grammars. Pages 198-200 from Chapter 4.

Week 12

• Meeting 1: Defining the notion of two sets having the same size. Countable sets and examples. The set of real numbers is uncountable. Diagonalization as a technique for showing this. Pages 201--2016 of Section 4.2.
• Meeting 2: Midterm.

Week 13

• Meeting 1: Proof that some languages are not Turing-recognizable. This is corollary 4.18 on page 206.
• Meeting 2: Proof of the undecidability of the language corresponding to the problem -- given a Turing machine and an input string, does the TM accept the string? The complement of this language is not Turing recognizable. Pages 207--210 of text.

Week 14

• Meeting 1: Undecidability of Halt_{TM}, E_{TM}, EQ_{TM}, and Regular_{TM}, shown via reductions. Pages 215--220 of Chapter 5. We skipped the portion of Section 5.1 devoted to Reduction via Computation histories, and Section 5.2.
• Meeting 2: Mapping Reducibility from Section 5.3. Examples, and consequences. Regular_{TM} is undecidable, and its complement is undecidable as well. Pages 234--238 of Section 5.3.

Week 15

• Meeting 1: EQ_{TM} is undecidable, and its complement is undecidable as well -- Theorem 5.30 in Chapter 5. A_{TM} is not mapping reducible to E_{TM} -- exercise 5.5. Review for Final.
• Meeting 2: Sample exam distributed, and parts of it reviewed for final.