CS 898: Kolmogorov complexity and its applications Winter 2021 Home

   

Instructor:
Ming Li	Contact me by email	mli@uwaterloo.ca
Course time and location:	The official course time is Fridays 2:00-4:50pm. We will hold the first meeting at this slot on Jan. 15, 2pm EST, on zoom: zoom meeting id: 863 4846 7016. (There is no passcode) Regular meeting time for Q and A will be back to Fridays 2:00pm--whenever we are done, at the same zoom meeting id: 863 4846 7016, with no passcode. The students will view the recorded lectures online at their convenience (but please do this timely, taking the proper lectures for each week).
Office hours:	Any time by email or phone 519-500-3026 (I am back to Waterloo now)
Textbook:	Ming Li and Paul Vitanyi, An introduction to Kolmogorov complexity and its applications. 4th edition, Springer, 2019 (Other editions are ok too)

The course material will be mainly from the textbook, research papers, the ppt lecture notes and recorded zoom session presentations. The students will do three homeworks and a research project at the end of the term, and present it via zoom (recorded).

Kolmogorov complexity is a modern theory of information and randomness. The goal of this course is to teach the students the basic concepts of Kolmogorov complexity and how to use this tool in their own research.

Topics include: plain Kolmogorov complexity, randomness, prefix Kolmogorov complexity, incompressibility method, information distance, applications in various fields ranging from average-case analysis of algorithms, Lovasz Local Lemma, zero-shot learning. It is impossible to cover the complete textbook, rather we will focus more on recent developments but we will focus on the applications, especially I wish to explore potential applications of Kolmogorov complexity in deep learning and zero-shot learning.

Marking Scheme: Each student is expected to do three homeworks (20% each) and a final project and presentation in class (via zoom recording) which can be either theoretical or programming (40%).

Course announcements and lecture notes will appear on this page. You are responsible for reading it regularly.

Assignments:

• The assignment 1 (Due Jan. 31) contains the following problems from the textbook (4th edition): Exercises 2.1.1[a], 2.1.2, 2.1.3, 2.1.5 (page 113) and 2.2.5, 2.2.6 (page 122).
• Assignment 2 (Due Feb. 28)). Do Exercises 2.4.2 (page 141), 2.5.1, 2.5.2, 2.5.3 (page 160),from the textbook (4th edition) and prove the following statement: If a binary sequence x of length n has exactly n/2 0's, then x is not c-incompressible, for large enough n.
• Assignment 3 (Due March 25). Problem 1: Use Kolmorogov complexity to directly prove that a 1 head 2-way DFA cannot accept L={w#w | w in {0,1}*}. Note, please do not use other theorems, do a direct Kolmogorov complexity proof. Problem 2: Using Kolmogorov complexity, obtain the average case lower bound for binary search. Problem 3: Using Kolmogorov complexity (directly, without using theorems from Chapter 6), prove that {0^n 1^m | m>n} is not regular. Problem 4: Do Excercise 6.8.5 (page 504 in the textbook, 4th edition). Problem 5: ShakerSort sorts a list of numbers by comparing each adjacent pair of items in a list in turn, swapping them if necessary, and alternately passes through the list from the beginning to the end then from the end to the beginning. It stops when a pass does no swaps. Using incompressibility argument, prove a tight average case lower bound for ShakerSort. Problem 6. Do Exercise 6.3.1 (Page 465 of the textbook, 4th edition), using Kolmogorov complexity. Problem 7. Do Exercise 6.3.3 (Page 465, 4th edition), using Kolmogorov complexity.

Possible project topics:

• Review and extend the Bill Gasarch's recent work: Gasarch recent manuscript on context-free grammar size using Kolmogrov complexity.
• Kolmogorov complexity and zero shot learning
• Simplicity of Deep learning models. Take a look at this paper.
• Replace mutual information argument in this and next paper by Kolmogorov complexity? This paper extends the usage of mutual information to NLP
• Do average analysis of an algorithm, using Kolmogorov complexity.
• Interesting applications of Kolmogorov complexity to LLL type of problems.
• Simplify and improve average-case analysis of Quicksort using Kolmogorov complexity (I will provide the papers).
• Improve the Shellsort average-case lower bound (Open). This is very difficult.
• Close the gap of logn upper bound and (1/2)logn lower bound for the following problem: given a sequence of k stacks linearly connected from left to right. The input is a permutation of n integers to the leftmost stack. The integers move in order (not passing each other unless they go into stacks) from left to right and any of them can go into any stack when they arrive at that stack and popped out at some subsequent steps. The output is from the rightmost stack and should be sorted in increasing order. How many stacks do we need? Or demonstrate why Kolmogorov complexity cannot be used for further improvement. Note: Luke Schaeffer has improved (1/2)logn to 0.53logn.
• Please do not hesitate to discuss with me on both finding and solving your project problems. Please let me know what you wish to work on. I generally prefer people working on the different projects. A project can be 1) Original theoretical research; 2) An application of Kolmogorov complexity (proving a theorem, analyzing an algorithms, or implement a program); 3) In depth analysis of one potential application of Kolmogorov complexity, when you failed 1) or 2); 4) In depth survey / review of one area.

Lectures (The videos will be uploaded gradually):

• Lecture 1. History, definition of Kolmogorov complexity, basics (Chapters 1 and 2). 1 week. Lecture 1 ppt. || Lecture 1 video, Part 1 || Lecture 1 video, Part 2
• Lecture 2: Randomness. Definitions and Martin-Lof tests (Sections 2.4, 2.5). 1 week. Lecture 2, Part 1 ppt || Part 2 ppt. || Lecture 2 video, Part 1 || Lecture 2 video, Part 2.
• Lecture 3: Symmetry of Information (Section 2.8). 1 week. Lecture 3 ppt || Lecture 3 video CNCC video
• Lecture 4: Information Distance and zero shot learning. (Sections 8.2, 8.3, 8.4, 8.6 this paper, this paper, and this paper). 2 weeks. Lecture 4 ppt || Lecture 4 video
• Lecture 5: The incompressibility method and its applications in average case analysis and lower bounds (Chapter 6) Lecture 5 ppt || Lecture 5 video, part 1|| Lecture 5 video, part 2|| Lecture 5 video, part 3|| Lecture 5 video, part 4|| Some nice additional notes by Tao Jiang 2 weeks.
• Lecture 6: Prefix Kolmogorov complexity (Chapter 3) Lecture 6 ppt || Lecture 6 video
• Lecture 7: Inductive Inference (Chapter 5) Lecture 7 ppt || Lecture 7 video
• Lecture 8: Kolmogorov complexity and the nature (Chapter 8) Lecture 8 ppt || Lecture 8 video
• Lecture 9: Resource bounded Kolmogorov complexity (Chapter 7). Lecture 9 ppt || Lecture 9 video || For those who are interested in reading more, you can look at this Lance Fortnow notes).
• Student zoom presentations: (45 min. each student). Watching other student presentations is also part of this course. Please make sure you watch student presentations as well. Here are the presenations by the students Ben Baral: https://drive.google.com/file/d/1ka8NVk6gzWHtG3VrC4nBQJKWGQ5yHyRu/view?usp=sharing, Stan Ivashkevich: https://drive.google.com/file/d/1pujUc7iAIxiPPsgimfY-60jf_bYYd1U6/view?usp=sharing, Adam Jaffe: https://www.icloud.com/attachment/?u=https%3A%2F%2Fcvws.icloud-content.com%2FB%2FARWRLCcelqd05DLofbpcBUd3OVnqAc0bG5xzK5A7MldMT3muJFpn435o%2F%24%7Bf%7D%3Fo%3DArsXCUBOyO8IrshPxrW49_tY0raom7QM8PNLzGQkRjgH%26v%3D1%26x%3D3%26a%3DCAog0yAe8pylW9VEzK2sa9LLz9CqnVa2yqWMD7wQG_x1ZiYSeBDu5ayEkS8Y7vWn2JovIgEAKgkC6AMA_1CCRkVSBHc5WepaBGfjfmhqJhBC14toUn9kA7NJFXwCnfPCEMTBhl9byMWFuZaAC4TroX_JZwXqciYMTjnhQodwrLgibVmkbX3Cql8P6Xt-VaQ2NA78rjPds1kvBywj3A%26e%3D1622072228%26fl%3D%26r%3D59285E69-1902-4879-A3F5-0E1FBD699B9D-1%26k%3D%24%7Buk%7D%26ckc%3Dcom.apple.largeattachment%26ckz%3D3C01E263-3A02-4A65-87BD-5F5505BDBB89%26p%3D32%26s%3D3XrQK1dGEWih_orgAvj6H_hXERY&uk=99qgI8ndRMjBdTEEF6tESw&f=ajaffe_presentation.mp4&sz=83509966 Zhiying Jiang: https://drive.google.com/file/d/1jQKB0cBTTGKe7LZpyj8XGtt4ne38STor/view?usp=sharing, Josh Lemmon: https://drive.google.com/file/d/1_6Y4gb0JbiFJnPdMcEYEhEvhfkOVBFZV/view?usp=sharing, Joseph Meleshko: https://drive.google.com/drive/folders/13h6WHsb4j4WpwLmSLKqTQPpu8yd8zYri?usp=sharing, Haolin Yu https://drive.google.com/drive/folders/1TXsktk8a9bzr9FId_bHaOettjBB6Dq_8?usp=sharing

Announcements: