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 2018**

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

**CAs**:

Andreas Garcia, andreas4 (at stanford dot edu)

Duligur Ibeling, duligur (at stanford dot edu)

Natalie Ng, nng1 (at stanford dot edu)

**Location and Times**:

Tue, Thu 10:30 AM – 11:50 AM at Gates B3

**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)

**Office Hours**: See on Piazza

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

Homework will be assigned every Tuesday (except for the week before the midterm) 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

**Grades: **Will be at least a combination of 45% HW assignments, 25% midterm, 30% finals

**Lectures (content for future dates is tentative)**:

**Tuesday 9/26**: **Introduction**

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 Lecture 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
- 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 9/27**: Deterministic Finite Automata, Closure Properties, Nondeterminism

Reading (for next few classes): Chapter 1, Sipser

**Tuesday 10/2**: Finish closure properties of regular languages, Show equivalence of DFSa and NFAs, define regular expression and characterize the languages they correspond to.

PPTX, PDF (to be finished next class)

**Thursday 10/4**: Non-Regular Languages, The Pumping Lemma, An Algorithm for Minimizing a DFA

PPTX, PDF (to be finished next class)

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

**Tuesday 10/9**: Finish Minimizing DFA, The Myhill-Nerode Theorem

PPTX, PDF (to be finished next class)

**Thursday 10/11**: Learning DFAs, Streaming Algorithms

No new slides

- Angluin’s classic paper on learning DFAs
- A note on streaming Algorithms;
- Article on finding frequent elements

**Tuesday 10/16**: Communication Complexity

**Thursday 10/18**: ; Finish CC. Turing Machines: deciding vs. recognizing,

PPTX, PDF (to be completed next week)

- Turing’s paper, and here
- Sipser 3

**Tuesday 10/23**: Multitape TM, Universal Turing Machines, Nondeterministic Turing Machines, Undecidable and Unrecognizable, A_TM is unrecognizable, Mapping Reductions

PPTX, PDF

**Thursday 10/25**:

A_TM is unrecognizable, Mapping Reductions, Rice’s Theorem, Oracle Machines, Hierarchy of Undecidable Problems, Self Reference

- Sipser 4, 5, 5.3

**Tuesday 10/30**: in-class midterm

**Thursday 11/1**: Rice’s Theorem, Oracle Machines, Hierarchy of Undecidable Problems, Self Reference

PPTX, PDF (continue next class)

- Note on Rice’s Theorem
- Sipser 6.1, 6.3

**Tuesday 11/6**: Foundation of Mathematics, Kolmogorov Complexity

**Thursday 11/8**: Finish Foundation of Math. Time Complexity, P

- Sipser 7 for the next few weeks, 9.1

**Tuesday 11/13**: NP and Polynomial-Time (Mapping) Reductions

- Note on NP-Completeness

**Thursday11/15**: Cook-Levin Theorem

**Tuesday 11/20**: Thanksgiving – no class

**Thursday 11/22**: Thanksgiving – no class

**Tuesday 11/27**: More NP-Completeness through Poly-Time Reductions

**Thursday 11/29**: coNP, Oracles, Polynomial Hierarchy, Space Complexity

- Finish Sipser 7, Sipser 8.2 and 9.2

**Tuesday 12/4**: Approximation and Hardness, PCPs, IPs, Zero-Knowledge + Wrap Up

Additional (not required) reading:

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

**Thursday 12/6**: “Ask Me Anything”

**Wednesday, December 12**, 12:15-3:15 p.m: **Finals**