What is computation? What can be computed in principle with unbounded computational resources? What can be computed efficiently? What can we gain by formally modeling computation and how do different models relate to one another? How can models highlight different resources of computations, some obvious (such as time and memory) and others less so (such as communication and randomness). What is gained by considering natural and social phenomenon as computations and looking at central notions such as proofs, knowledge, learning, games, randomness, entropy and more through the computational lens?

We will consider these questions and others using a rigorous mathematical approach. We will discuss what we know as well as some of the central open problems in pure and applied mathematics, and specifically the P vs. NP problem.

Some specific topics: Finite Automata – Very Simple Models (constant memory), Non-determinism (power of guessing), Learning, communication complexity, Streaming algorithms, Powerful models – Turing Machines, Decidability, Kolmogorov Complexity, Time complexity, P vs. NP, NP-completeness, Other Resources: space, randomness, communication, power, … Crypto, Game Theory, … The Computational Lens.

Prerequisites: CS 103 or 103B.

**Current Offering: Fall 2020 **

**Instructor**: Omer Reingold, Gates 462, reingold (at stanford dot edu)

**CAs**:

Brian Axelrod, baxelrod (at stanford dot edu)

Celia Chen, xinuo (at stanford dot edu)

Tom Knowles. tknowles (at stanford dot edu)

Deon Jordan Richmond, deonrich (at stanford dot edu)

**Location and Times**:

Tue, 10:30 AM – 11:50 on zoom

**Book**: Michael Sipser, introduction to the theory of computation (2nd or 3rd edition)

– Extra reading: Boaz Barak, Introduction to Theoretical Computer Science (the approach is different from Sipser, but some parts could augment your understanding). Additional reading below.

**Office Hours**: See on Piazza

**HW assignments (see on Piazza)**:

Homework will be assigned almost every Tuesday and will be due one week later at the beginning of class. No late submission. We will drop your lowest homework grade.

You may (even encouraged to) collaborate with others, but you must:

1. Try to solve all the problems by yourself first

2. List your collaborators on each problem

3. If you receive a significant idea from somewhere, you must acknowledge that source in your solution.

4. Write your own solutions (important!)

Assignments and submissions through gradescope.com

Best to write in LaTex

The class is flipped. Pre-recorded lectures on Canvas and YouTube. The Tuesday meetings will be posted on Canvas only. A related and multi-instotutes project is here (could be a useful resource).

**Lectures**:

**Tuesday 9/15**: **Introduction**

Watch videos 0-4.

What are computations? The Computational Lens. Course information and topics. Why Theory? Class Preview. The most fundamental open question of CS: graph coloring ?!? Proof techniques (and an example).

**Reading**: Chapter 0, Introduction, Sipser

– extra reading: Barak’s chapters 0 and 1

**Additional**:

What between Ogres, Onions, Parfait, and good proofs?

- Mathematics and Computation, very interesting book by Avi Wigderson, I highly recommend Chapter 20
- An accessible article I wrote of this book (focusing on the computational lens)
- The Computational Lens plus a talk
- Four talks: lens of computation in the sciences
- Panel Discussion

And if my megalomaniac view of the computational lens is not megalomaniac enough, I suggest reading The Last Question by Isaac Asimov

**Thursday 9/17**: Quiz (survey)

**Tuesday 9/22**: Deterministic Finite Automata, Closure Properties, Nondeterminism, equivalence of DFSa and NFAs, regular expression and the languages they correspond to.

Watch videos 5-11

PPTX: 5p-DFA-overview 6p-DFAs 7p-DFAclosure1 8p-NFAs 9p-NFA2DFA 10p-Closure2 11p-RegExp

PDFs: 5p-DFA-overview 6p-DFAs 7p-DFAclosure1 8p-NFAs 9p-NFA2DFA 10p-Closure2 11p-RegExp

Reading (for next week as well): Chapter 1, Sipser

**Thursday 9/24 **Quiz

**Tuesday 9/29**: Non-Regular Languages, The Pumping Lemma, An Algorithm for Minimizing a DFA. The Myhill-Nerode Theorem, Learning DFAs

Watch videos 12-15

PPTX: 12p-pumping 13p-minimizingDFA 14p-Myhill-Nerode 15p-Learning-DFA

PDF: 12p-pumping 13p-minimizingDFA 14p-Myhill-Nerode 15p-Learning-DFA

Reading: Chapter 1.4, Sipser; A note on DFA minimization and Myhill-Nerode

Additional:

**Thursday 10/1**: Quiz

**Tuesday 10/6** Streaming Algorithms, Communication Complexity + Begin Turing Machines (deciding vs. recognizing)

Watch videos 16-19

PPTX: 16p-streaming 17p-Communication-Complexity 18p-TM-overview 19p-TMs

PDF: 16p-streaming 17p-Communication-Complexity 18p-TM-overview 19p-TMs

- A note on streaming Algorithms;
- Article on finding frequent elements
- Survey on Communication Complexity
- Turing’s paper.
- Sipser 3
- videos on asymptotic notation (among other preliminaries and relevant videos):

**Thursday 10/8**: Quiz

**Tuesday 10/13**: Continue Turing Machines: , Multitape TM, Universal Turing Machines, Nondeterministic Turing Machines, Undecidable and Unrecognizable, A_TM is unrecognizable, Mapping Reductions

Watch videos 20-24

PPTX: 20p-TM-variants 21p-Universal-TM 22p-counting-argument 23p-concrete-undecidable 24p-mapping-reductions

PDF: 20p-TM-variants 21p-Universal-TM 22p-counting-argument 23p-concrete-undecidable 24p-mapping-reductions

- Sipser 4

**Thursday 10/15**: Quiz

**Tuesday 10/20**: Rice’s Theorem, Oracle Machines, Hierarchy of Undecidable Problems, Self Reference, Self Reference, Foundation of Mathematics, Kolmogorov Complexity

Watch videos 25-29

PPTX: 25p-rices-theorem 26p-oracle-reductions 27p-self-reference 28p-logic 29p-Kolmogorov-Complexity

PDF: 25p-rices-theorem 26p-oracle-reductions 27p-self-reference 28p-logic 29p-Kolmogorov-Complexity

- Sipser 5, 6
- Note on Rice’s Theorem
- Note on Computability and Logic
- Rosencrantz & Guildenstern Are Dead (1990) – Heads Scene
- Note on Kolmogorov Complexity

**Thursday 10/22**: Quiz

**Tuesday 10/27**: Time Complexity, P, Time Hierarchy Theorems, NP and Polynomial-Time (Mapping) Reductions

Watch videos 30-33

PPTX: 30p-Complexity-overview 31p-time-complexity 32p-NP 33p-poly-time-reductions

PDF: 30p-Complexity-overview 31p-time-complexity 32p-NP 33p-poly-time-reductions

- Sipser 7-7.3, 9.1
- Note on NP-Completeness

**Thursday 10/29 **: Quiz

**Tuesday 11/3**: NP-Completeness, Cook-Levin Theorem, More NP-Completeness through Poly-Time Reductions, coNP, Oracles, Polynomial Hierarchy

Watch videos 34-36

**Thursday 11/5**: Quiz

**Tuesday 11/10**: Space Complexity, advanced topics on proofs (PCPs, Hardness of Approximation, IPs, Zero-Knowledge)

Watch videos 37-39

- Finish Sipser 7, Sipser 8.2, 9.1, 9.2

**Thursday 11/12**: Quiz

Advanced (not required) reading:

- Arora-Barak Chapters 8 and 18
- Where is Waldo (Applied Kid Cryptography)

**Tuesday 11/17**: Course Wrap-Up, Computational Lens, Randomness and Pseudorandomness, Algorithmic Fairness

Watch videos 40-42

Advanced (not required) reading:

- Arora-Barak Chapters 7, 16

**Thursday 11/19**: Quiz