CS 151: Complexity Theory (Spring 2023)

Instructor: Chris Umans

Office: Annenberg 311

Times: Tu/Th 1:00-2:25 in Annenberg 213

TAs: Emile Anand and Joe Slote

Office hours:

• Tuesdays 8-9pm in Annenberg 213 (Emile)
• Wednesdays 1:30-2:30 pm in Annenberg 311 (Chris)
• Wednesdays 7:30-8:30pm in Annenberg 213 (Joe)

Announcements:

• The solutions to the final exam have been posted.
• Have a wonderful summer, everyone!

Handouts:

• Syllabus (pdf)

Lecture slides:

• Lecture 1: intro; languages, complexity classes, Turing Machines (pptx, pdf)
• Lecture 2: reductions and completeness, time and space classes, hierarchy theorems, relationships between classes (pptx, pdf)
• Lecture 3: a P-complete problem, padding and succinctness, nondeterminism, NP- and NEXP- complete problems, NTIME hierarchy theorem (pptx, pdf)
• Lecture 4: NTIME hierarchy theorem, Ladner's Theorem, unary languages and NP, nondeterministic space classes, STCONN (pptx, pdf)
• Lecture 5: Savitch's Theorem, I-S Theorem, nonuniformity and advice, NC hierarchy (pptx, pdf)
• Lecture 6: NC hierarchy, formula lower bound on Andreev function, Razborov's lower bound on monotone circuits for clique (pptx, pdf)
• Lecture 7: Razborov's lower bound on monotone circuits for clique, Schwartz-Zippel (pptx, pdf)
• Lecture 8: Valiant-Vazirani Theorem, randomized complexity classes, error reduction, BPP in P/poly (pptx, pdf)
• Lecture 9: Goldreich-Levin hard bit, Yao's Lemma, BMY generator (pptx, pdf)
• Lecture 10: Nisan-Wigderson generator, error-correcting codes, transforming worst-case hardness into average-case hardness (pptx, pdf)
• Lecture 11: finishing up worst-case-to-average-case reduction, extractors (pptx, pdf)
• Lecture 12: RL, oracles, the PH and alternating quantifiers, complete problems for levels of the PH and PSPACE (pptx, pdf)
• Lecture 13: Karp-Lipton Theorem, BPP in PH, the class #P, complete problems for #P, #Matching is #P-completes, interactive proof systems (pptx, pdf)
• Lecture 14: graph non-isomorphism, IP = PSPACE, Arthur-Merlin games, the classes MA and AM (pptx, pdf)
• Lecture 15: derandomization of MA and AM, optimization, approximation, and PCPs (pptx, pdf)
• Lecture 16: elements of the proof of the PCP Theorem (pptx, pdf)
• Lecture 17: finishing up PCPs, relativization (pptx, pdf)
• Lecture 18: natural proofs; course summary (pptx, pdf)

Problem sets:

• Problem Set 1: (pdf, LaTeX source) Solutions (pdf)
• Problem Set 2: (pdf, LaTeX source) Solutions (pdf)
• Problem Set 3: (pdf, LaTeX source) Solutions (pdf)
• Problem Set 4: (pdf, LaTeX source) Solutions (pdf)
• Midterm: (pdf, LaTeX source) Solutions (pdf)
• Problem Set 5: (pdf, LaTeX source) Solutions (pdf)
• Problem Set 6: (pdf, LaTeX source) Solutions (pdf)
• Problem Set 7: (pdf, LaTeX source) Solutions (pdf)
• Final: (pdf, LaTeX source) Solutions (pdf)

Resources:

• Here is a LaTeX template (tex, pdf) that you can use for your writeups if you wish.
• Videos of 2019 lectures (Caltech only) here.