# Theory of Computation (Fall 2023)

**Time and Location:** TTh 1:30-2:45pm, Hodson 311

**Instructor:** Xin Li. **Office hours:** Wednesday 2:00pm-3:00pm in person, or by appointment

**TA:** Songtao Mao. **Email:** smao13@jhu.edu ** Office hours:** Friday 2:00pm-4:00pm, by Zoom.

**Syllabus**

**Course description:** This is a combined undergraduate upper-level and graduate-level course studying the theoretical foundations of computer science. Topics covered will be models of computation from automata to Turing machines, computability, complexity theory, randomized algorithms, inapproximability, interactive proof systems and probabilistically checkable proofs. Students may not take both 600.431 and 600.631, unless one is for an undergrad degree and the other for grad degree.

**Pre Requisite:** Discrete math or permission. Probability theory strongly recommended.

**Required Textbook:** Computational Complexity: A Modern Approach, by Sanjeev Arora and Boaz Barak.

**Recommended Textbook:** Introduction to the Theory of Computation, by Michael Sipser.

### Topics Covered

- Aug. 29 & Aug. 31: Inroduction and beginning Turing Machines.

- Sep. 5: Variants of Turing Machines.

- Sep. 7: Universal Turing Machines and Computability.

- Sep 12: Decidable/Recognizable/Undecidable/Unrecognizalbe languages.

- Sep. 14: Time complexity, Time Hierarchy theorem, starting P and NP.

- Sep. 19: Equivalence of two NP definitions, Polynomial time reduction and NP-completness, start Cook-Levin Theorem.

- Sep. 21: Proof of the Cook-Levin Theorem.

- Sep. 26: More NP-complete languages, dealing with NP-completeness.

- Sep. 28: CoNP, NEXP, Search to Decision reduction, starting space complexity.

- Oct. 3: Space hierarchy theorem and PSPACE-completeness.

- Oct. 5: Savitch's theorem and NL-completeness.

- Oct. 10: Certificate definition of NL, and NL=coNL.

- Oct. 12: The Polynomial Hierarchy.

- Oct. 24: Alternating Turing Machines and time-space trade-off for NP, starting Boolean Circuits.

- Oct. 26: P/poly, size hierarchy theorem, Karp-Lipton Theorem.

- Oct. 31 & Nov. 2: NC and AC circuits P/poly, starting interactive proofs. DIP=NP, starting IP

- Nov. 9: The class IP and some properties.

- Nov. 14: Public coin proof and AM, GNI and GI.

- Nov. 16: Sumcheck protocol.

- Nov. 28 & Nov. 30: IP=PSPACE, Variants of IP, PCP, BPP

- Dec. 5 & Dec. 7: Introduction to quantum computation, conclusion of class.

### Further readings

**Interactive proof with Perfect Completeness or Soundness**

**Introduction to mathematical arguments**

**Some Help on Reading Mathematics and Creating Proofs**

**A picture of some complexity classes**

**Wiki page of Hilbert's 10th problem**

**An article about Alan Turing**

**A website about Turing and the history of modern computing**

**Godel's letter to von Neumann**

**Clay Institute's official page about the P vs. NP problem**

Lance Fortnow's exposition of the P vs. NP problem