General Information
- Instructor: Toniann Pitassi (toni@cs.columbia.edu)
- Time: Thursdays 1:10 – 3:30 PM
- Classroom: 337 Mudd Building
- TA: Tianqi Yang (tianqi@cs.columbia.edu)
- Office Hours:
- Tianqi: Thursdays 4 – 5 PM in open space 5th floor (501 CSB)
- Toni: Thursdays 12 – 1 PM in CSB 513 or by email/appointment
- Lecture Notes will be posted below
Description
Proof complexity studies the following question: Given a true mathematical statement, what is the length of the shortest proof ththe statement is true?
Proof complexity is fundamentally connected both to major open questions of computational complexity theory and to practical properties of automated theorem provers and reasoning systems. Moreover, new connections between proof complexity and circuit complexity have been uncovered, and the interplay between these two areas has become quite rich. In addition, concepts and techniques in proof complexity have borrowed from and contributed to several other areas of computer science, including: crytography, automated reasoning and AI, complexity of search classes within NP, the analysis of heuristics and algorithms for NP-hard problems, and bounded arithmetic.
Moreover in the last ten years, enormous progress has been made in proof complexity, with many open problems resolved, and many promising directions to attack other problems. For example, recent work (that we will cover) uses proof complexity to achieve breakthrough lower bounds on locally decodable codes, as well as the development of algorithms for a variety of distributional learning problems through the ``Sum-of-Squares'' paradigm.
This course is intended to serve as an introduction to modern proof complexity, emphasizing its connections with computational complexity and algorithms in optimization. We will cover the fundamentals of propositional proof complexity, starting with the study of the important proof systems that occur both in theory and practice. We will survey state-of-the-art lower bound techniques in proof complexity for a variety of proof systems, including algebraic and semi-algebraic systems. We will show how these proof systems are fundamentally tied to many areas, and will spend a significant portion of the course discussing exciting recent applications of both upper and lower bounds in proof complexity to other areas. A variety of open problems will be presented throughout the class.
Evaluation will be primarily based on a final project. I will provide a list of suggestions for the project, but students are encouraged to relate their project to their interests. There will also be occasional (at most 3) homework assignments, and students will be asked to scribe a lecture.
Prerequisites
This is an advanced topics class in theoretical computer science, but it is open to everyone. The most important prerequisite is mathematical maturity (comfort with understanding and writing mathematical proofs). Students should also be very comfortable with algorithmic concepts, basic complexity theory, and proofs.
Tentative Schedule
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January 23 |
Introduction to the course. Propositional proof complexity, examples, motivation, connections to complexity theory, algorithms and logic. Polynomial-simulations, and optimal proof systems. Scribe notes. |
[notes] |
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February 6 |
Resolution Lower Bounds : Resolution, Lower bounds via Random Restrictions plus Width, Size-Width Tradeoff, Expansion, PHP Resolution lower and upper bounds. Scribe notes. |
[notes] |
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February 13 |
Feasible Interpolation: Interpolation and Automatizability, Cutting Planes, Lower Bounds for Cutting Planes via Feasible Interpolation |
[notes] |
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February 20 |
Frege and Extended Frege: Frege, tree-like versus dag-like equivalence, Extended Frege and Equivalent Formulations (Extended Resolution, Substitution Frege) |
[notes] |
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February 27 |
Frege Systems: Frege and Extended Frege, Bounded Arithmetic, TFNP. |
[notes] |
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March 6 |
Bounded-depth Frege: Lower Bounds, Switching Lemma |
[notes] |
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March 13 |
Algebraic Proof Systems: Nullstellensatz, Polynomial Calculus, IPS. |
[notes] |
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March 27 |
Semi-algebraic Proof Systems: Sherali-Adams, SoS (Sum-of-Squares), Learning algorithms via SoS |
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April 3 |
Applications I Lower Bounds for Monotone Circuit Models and Secret Sharing Schemes, Extended Formulations |
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April 10 |
Applications II Random and Semi-Random CSPs, Lower Bounds for Locally Decodable Codes |
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April 17 |
Student Presentations |
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April 24 |
Student Presentations |
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May 1 |
Student Presentations and Open Problems/Discussion |
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Other Resources
There is no textbook for this course, but we will write lecture notes over the course of the semester. The following survey and similar courses could also be helpful resources.
- Semialgebraic Proofs and Efficient Algorithm Design, by Fleming, Kothari, Pitassi (link)
- Past Course at U of Toronto by Pitassi (link)
- Course by Robert Robere (link)
- Course by A. Razborov (link)
- Course by Paul Beame (link)
- Template for Scribe Notes (link)
- Project Presentation (PROPOSAL DUE FEB 26) (link)
- Homework 1 (DUE Mar 24) (link)
- Homework 1 with hints (DUE Mar 24) (link)