• Lecture 1:(1/18) Motivation and overview of the class - definitions of alphabet, strings, and languages. Readings: Chapter 0, Homework 1 posted
  
  
• Lecture 2:(1/20) DFA definition, Language recognized by a DFA, Regular languages; Examples; All finite languages are regular. Readings: Section 1.1.
  
  
• Lecture 3:(1/25) More examples of regular languages, intuition on when a language is not regular; Examples; Informal definition and examples of an NFA. Readings: Section 1.2
  
  
• Lecture 4:(1/27) Definition of NFA, its computation tree, the language recognized by an NFA; Every regular language has an NFA recognizing it. HW1 due, HW2 out. Readings: Section 1.2
  
  
• Lecture 5:(2/1) Equivalence of NFA and DFA (proof and example); Definitions of operations on regular languages (regular operations- union, concatenation, star and other operations- intersection, complement and power);Regular languages are closed under union.Readings: Section 1.2.
  
  
• Lecture 6:(2/3) Regular languages are closed under regular operations union, concatenation, star, as well as other operations such as complement. Readings: Section 1.2. Homework 2 due.
  
  
• Lecture 7:(2/8) Definition of regular expressions. A language is regular if and only if it has a regular expression describing it. Readings: Section 1.3. Homework 3 posted.
  
  
• Lecture 13:(3/1) Equivalence of PDAs and CFG (proved only one direction, constructing a PDA out of a CFG); Tandem pumping lemma for context free languages (skipped the proof); examples of how to use this to prove a language is not context free.
  
  
• Lecture 15: (3/8) TM definitions (configurations, recognized languate, decider, reconglizable and decidable languages).
  
  
• Lecture 16: (3/10) Equivalence of PDAs and CFG (proved only one direction, constructing a PDA out of a CFG); Tandem pumping lemma for context free languages (skipped the proof); examples of how to use this to prove a language is not context free.
  
  
• Lecture 17: (3/22) Review of TM and previously defined equivalent models. Definition of Non deterministic TM and the equivalence to TM
  
  
• Lecture 18: (3/31) examples of decidable languages: acceptance problem for DFA,CFG, emptiness problem for DFA, equivalence problem for DFA; mentioned without proof that emptiness problem for CFG is decidable, but equivalence problem for CFG is not
  
  
• Lecture 19: (4/5) universal TM, acceptance problem for TM is recognizable (by universal TM). countable sets (e.g., naturals, even numbers, positive rational numbers,...). uncountable sets: real numbers in [0,1], using Cantor's diagonalization method. We also showed the following are countable: the set of all strings over some alphabet; the set of all Turing Machines. and that the following are not countable (using diagonalization): the set of all "infinite strings", the set of all languages over a certain alphabet. conclusion: there exists a language that is not TM-recognizable (since there are more than countably many languages but only countably many TM)
  
  
• Lecture 20: (4/7) There exists a language that is not TM-recognizable (a non-constructive proof), a specific language (X_TM) that is not decidable, nor even recognizable. A_TM is not decidable. A language L is decidable if and only if both L and the complement of L are TM-recognizable.
  
  
• Lecture 21:(4/12) Proofs of undecidability for HALT_TM, REG_TM, E_TM.
  
  
• Lecture 22:(4/14) Turing reducibility: if A ≤ B and B is decidable then so is A. If A ≤ B and A is undecidable then so is B. Note: examples we saw in previous lectures in fact showed Turing-reducibility. Further examples including EQ_TM is undecidable.
  
  
• Lecture 23 (4/19) Mapping reducibility definition. if A ≤=m B and B is recognizable then so is A. If A ≤=m B and A is not recognizable then B is not recognizable. Examples, and properties of reductions.
  
  
• Lecture 24 (4/21) Finish mapping reductions. Introduce time complexity, big-Oh notation, P (class of polynomial time solvable problems), strong Church-Turing thesis.
  
  
• Lecture 25 (4/26) More on P (plus examples). EXP. Define NP as the class of polynomial time verifiable problems. Examples.
  
  
• Lecture 26 (4/28) Alternative equivalent definition of NP as class of languages with a poly time non-deterministic TM. On P vs NP problem. Definition of polynomial reductions and NP Complete languages. Cook Levin Theorem (without proof).