Plan for Lectures
Topics to be Covered
These are tentative, and somewhat in order
- Turing Machines refresher, time hierarchy (1 lecture)
- diagonalization and relativization, Ladner’s theorem, Baker-Gill-Solovay (1 lecture)
- space complexity (2 lectures)
- polynomial hierarchy (1 lecture)
- Boolean Circuits (2 lectures)
- complexity of randomized computation (3 lectures)
- Derandomization (2 lectures)
- Natural Proofs (1 lecture)
- Complexity of counting (4 lectures)
- Proof complexity & Interactive Proofs (4 lectures)
- Probabilistic Checkable Proofs (2 lectures)
| Date | Topics | Slides | |
|---|---|---|---|
| Lecture 0 | September 7 | Introduction & Overview of Course | |
| Lecture 1 | September 7 | (Universal) Turing Machines, Time Hierarchy, Ladner’s theorem | |
| Lecture 2 | September 9 | Oracles, Relativization, Barriers to diagonalization (BGS) | |
| Lecture 3 | September 19 | Space Complexity I: configuration graphs, PSPACE | |
| Lecture 4 | September 21 | Space Complexity II: TQBF, TQBF is PSPACE complete | |
| Lecture 5 | September 23 | Polynomial Hierarchy (PH), Alternating TMs, alternating vs classical space-time | |
| Lecture 6 | September 26 | Non-uniform computation: boolean circuits, Karp-Lipton and Meyer | |
| Lecture 7 | September 28 | Algebraic complexity (uniform and non-uniform) | |
| Lecture 8 | September 30 | Randomized algorithms, probabilistic TMs, RP, coRP, ZPP, BPL, RL | |
| Lecture 9 | October 3 | BPP in $\Sigma_2$, Adleman’s theorem (BPP in P/poly) | |
| Lecture 10 | October 5 | Derandomization & Pseudorandom generators (PRGs) | |
| Lecture 11 | October 7 | Hardness vs randomness | |
| Lecture 12 | October 17 | Cryptography: computational security, one-way functions and PRGs | |
| Lecture 13 | October 19 | Barriers to lower bounds: natural proofs | |
| Lecture 14 | October 21 | Promise problems, Unique-SAT (Valiant-Vazirani) | |
| Lecture 15 | October 24 | #P, #P-completeness, Permanent is #P-complete (Valiant) | |
| Lecture 16 | October 26 | Toda’s theorem | |
| Lecture 17 | October 28 | approximate counting in $BPP^{NP}$, PP closed under intersection | |
| Lecture 18 | November 4 | Proof complexity and Interactive proofs (IPs) | |
| Lecture 19 | November 7 | $P^{\#P} \subset IP \subset PSPACE$ | |
| Lecture 20 | November 21 | IP = PSPACE | |
| Lecture 21 | November 23 | Arthur-Merlin (AM) | |
| Lecture 22 | November 25 | Probabilistic Checkable Proofs (PCP) and hardness of approximation | |
| Lecture 23 | November 28 | PCP theorem | |
| Lecture 24 | November 30 | Conclusion |
Suggested Reading
| Topics | Suggested Reading | |
|---|---|---|
| Lecture 0 | Introduction & Overview of Course | |
| Lecture 1 | (Universal) Turing Machines, Time Hierarchy, Ladner’s theorem | [AB09 - chapter 1] |
| Lecture 2 | Oracles, Relativization, Barriers to diagonalization (BGS) | [AB09 - chapter 3] |
| Lecture 3 | Space Complexity I: configuration graphs, PSPACE, TQBF | [AB09 - chapter 4, G06 - chapter 5] |
| Lecture 4 | Space Complexity II: TQBF is PSPACE complete | [AB09 - chapter 4, G06 - chapter 5] |
| Lecture 5 | Polynomial Hierarchy (PH), Alternating TMs, alternating vs classical space-time | [AB09 - chapter 5, G06 - chapter 3 ] |
| Lecture 6 | Non-uniform computation: boolean circuits, Karp-Lipton and Meyer | [AB09 - chapter 6, G06 - chapter 3.1] |
| Lecture 7 | Algebraic complexity (uniform and non-uniform) | [BCSS98 - chapters 1-5] |
| Lecture 8 | Randomized algorithms, probabilistic TMs, RP, coRP, ZPP, BPL, RL | [AB09 - chapter 7, G06 - chapter 6] |
| Lecture 9 | BPP in $\Sigma_2$, Adleman’s theorem (BPP in P/poly) | [AB09 - chapter 7] |
| Lecture 10 | Pseudorandom generators (PRGs) | [T02 - lecture 23] |
| Lecture 11 | Hardness vs randomness | [T02 - lecture 24] |
| Lecture 12 | Cryptography: computational security, one-way functions and PRGs | [T02 - lecture 11, AB09 Chapter 9] |
| Lecture 13 | Barriers to lower bounds: natural proofs | |
| Lecture 14 | Counting I: promise problems, Unique-SAT (Valiant-Vazirani), #P | |
| Lecture 15 | Counting II: #P-completeness, Permanent is #P-complete (Valiant) | |
| Lecture 16 | Counting III: Toda’s theorem | |
| Lecture 17 | Counting IV: approximate counting in $BPP^{NP}$, PP closed under intersection | |
| Lecture 18 | Proof complexity and Interactive proofs (IPs) | |
| Lecture 19 | $P^{\#P} \subset IP \subset PSPACE$ | |
| Lecture 20 | IP = PSPACE | |
| Lecture 21 | Arthur-Merlin (AM) | |
| Lecture 22 | Probabilistic Checkable Proofs (PCP) and hardness of approximation | |
| Lecture 23 | PCP theorem | |
| Lecture 24 | Conclusion |