• Lecture 1:(1/22) Motivation and overview of the class - definitions of alphabet, strings, and languages. Readings: Chapter 0, Homework 1 posted
  
  
• Lecture 2:(1/27) DFA definition - L is regular if it is recognized by a DFA - examples. Readings: Section 1.1.
  
  
• Lecture 3:(1/29) More examples of regular languages, Definition of NFA and examples. Readings: Section 1.2, Homework 1 due, Homework 2 posted
  
  
• Lecture 4:(2/3) NFA examples - Equivalence of NFAs and DFAs - Definition of regular operations on languages. Readings: Section 1.2
  
  
• Lecture 5:(2/5) Regular languages are closed under the regular operations. (union, concatenation, star), as well as under the unary operations of complement and +. Readings: Section 1.2. Homework 2 due. Homewok 3 posted.
  
  
• Lecture 6:(2/10) Definition of a Regular Expressions. Characterization that a language is regular if and only if it can be generated by a regular expression. Reading: Section 1.3
  
  
• Lecture 7:(2/12) Various examples of DFA to Regular expression conversion, the Pumping Lemma. Readings: Section 1.3, 1.4. Homework 3 due, Homework 4 posted.
  
  
• Lecture 8:(2/17) Non regular languages: Example applications of using pumping lemma and closure to prove non regularity. Readings: Section 1.4
  
  
• Lecture 9:(2/19) Context free grammars, ambiguity, examples, and a proof of correctness for a CFG. Readings: Section 2.1
  
  
• Lecture 10:(2/24) Complete proof of correctness, Pushdown automata definition and example. Readings: Section 2.2 Homework 4 due, Homework 5 posted
  
  
• Lecture 11:(2/26) PDA examples, equivalence of PDA and CFG. Readings Section 2.2
  
  
• Lecture 12:(3/3) The Tandem Pumping Lemma, and its use to show that languages are not context free. We did not show the proof, but did cover several examples of how to use the lemma. Readings: Section 2.3; Homework 5 due, Homework 6 posted
  
  
• Mid Term day (3/5) All material upto and including lecture 12
  
  
• Lecture 13:(3/10) Definition of a Turing machines. Some examples of Turing machines. Readings: Section 3.1
  
  
• Lecture 14:(3/12) More TM definitions (configurations, recognized language, decider, recognizable and decidable languages), mutli-tape TM and their equivalence to (standard) TM. Readings: Section 3.1, 3.2; Homework 6 due, Homework 7 posted
  
  
• Lecture 15:(3/23) Non-deterministic Turing Machines, and their equivalence to (standard, deterministic) TM. Church-Turing thesis Readings: Section 3.2, 3.3
  
  
• Lecture 16:(3/26) Universal TM, Definition of enumerators and proof that enumerators recognize the class of TM-recognizable languages, proof that L is decidable if and only if both L and complement of L are recognizable. Homework 7 due. Homework 8 posted. Readings Section 3.3
  
  
• Lecture 17:(3/31) Examples of decidable languages: acceptance, emptiness, and equivalence problems for DFAs; Mentioned status of same problems for CFGs; Acceptance problem for TM is recognizable (by universal TM) - we will show later that it's not decidable; Definition of countable sets. Readings: Section 4.1
  
  
• Lecture 18:(4/2) Countable Sets: rationals, Sigma* for any finite Sigma, any language (a set of strings) over any (finite) Sigma, the set of all TM, the set of all TM-recognizable languages. Uncountable Sets: the set of real numbers(Cantor) -- proof is by diagonalization. Another diagonalization proof to show that not all languages are TM-recognizable. Readings: Section 4.2 Homework 8 due, Homework 9 posted
  
  
• Lecture 19:(4/7) X_TM is not decidable, not even recognizable.(proof by contradiction or diagonalization). A_TM is recognizable but not decidable. (proof by reduction to X_TM). complement(A_TM) is not recognizable. (although it is "co-recognizable" - it's complement is recognizable). H_TM (halting problem) is not decidable (proof by reduction to X_TM). The notion of reduction ("subroutine") was discussed informally (and proofs by contradiction using a reduction were given as indicated above). Readings: Section 4.2, 5.1 (first Half)
  
  
• Lecture 20:(4/9) E_TM, REG_TM are undecidable. Formal definition of Turing reduction (A turing-reduces to B). if A turing-reduces to B and B is decidable then A is decidable. Corollary: if A turing-reduces to B and A is undecidable, then B is undecidable. HW 9 due, HW 10 posted. Readings: Section 5.1, 6.3.
  
  
• Lecture 21:(4/14) Definition of mapping reductions - Mapping reduction implies a turing reduction, where the subroutine is used just once and its output is used as the output of the program. if A is mapping-reducible to B and B is recognizable, then A is recognizable. Corollary: if A is mapping-reducible to B and A is not recognizable, then B is unrecognizable. Readings: Section 5.3
  
  
• Lecture 22:(4/16) A is mapping-reducible to B if and only if complement(A) is mapping-reducible to complement(B). EQ_TM is neither recognizable nor co-recognizable (each of the two directions is proved by showing that complement( A_TM ) is mapping-reducible to EQ (for first direction) or to complement of EQ (for second direction. Started to discuss complexity theory, defined running time of a machine. HW 10 due, HW 11 posted. Readings: Section 5.3, 7.1
  
  
• Lecture 23:(4/21) Define P (languages solvable in polynomial time -- identified with efficiently solvable).Define NP -- class of languages that have an efficient verifier. Show COMPOSITES is in NP, and mention (without proof) that it is actually in P too. Readings: Section 7.2, 7.3
  
  
• Lecture 24:(4/23) Discuss efficiently computable (P) vs efficiently verifiable (NP) problem. P, NP, EXP are subsets of each other. Example: HamCycle in NP. Alternative characterization of NP by non-deterministic Turing Machines. Define polynomial reduction. Prove that if A<pB then (1) if B is in P , so is A, (2) if B is in NP, so is A. Define NP-Complete languages. Readings: Section 7.4, 7.5. HW11 Due