Some Hardness Escalation Results in Computational Complexity Theory by Pritish Kamath B.Tech. Indian Institute of Technology Bombay (2012) S.M. Massachusetts Institute of Technology (2015) Submitted to Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering & Computer Science at Massachusetts Institute of Technology February 2020 ⃝c Massachusetts Institute of Technology 2019. All rights reserved. Author: ............................................................. Department of Electrical Engineering and Computer Science September 16, 2019 Certified by: ............................................................. Ronitt Rubinfeld Professor of Electrical Engineering and Computer Science, MIT Thesis Supervisor Certified by: ............................................................. Madhu Sudan Gordon McKay Professor of Computer Science, Harvard University Thesis Supervisor Accepted by: ............................................................. Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science, MIT Chair, Department Committee on Graduate Students Some Hardness Escalation Results in Computational Complexity Theory by Pritish Kamath Submitted to Department of Electrical Engineering and Computer Science on September 16, 2019, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science & Engineering Abstract In this thesis, we prove new hardness escalation results in computational complexity theory; a phenomenon where hardness results against seemingly weak models of computation for any problem can be lifted, in a black box manner, to much stronger models of computation by considering a simple gadget composed version of the original problem. For any unsatisfiable CNF formula F that is hard to refute in the Resolution proof system, we show that a gadget-composed version of F is hard to refute in any proof system whose lines are computed by efficient communication protocols. This allows us to prove new lower bounds for: − Monotone Circuit Size : we get an exponential lower bound for an explicit monotone function computable by linear sized monotone span programs and also in (non-monotone) NC2. − Real Monotone Circuit Size : Our proof technique extends to real communication protocols, which yields similar lower bounds against real monotone circuits. − Cutting Planes Length : we get exponential lower bound for an explicit CNF contradiction that is refutable with logarithmic Nullstellensatz degree. Finally, we describe an intimate connection between computational models and communication complexity analogs of the sub-classes of TFNP, the class of all total search problems in NP. We show that the communication analog of PPAp captures span programs over Fp for any prime p. This complements previously known results that communication FP captures formulas (Karchmer– Wigderson, 1988) and that communication PLS captures circuits (Razborov, 1995). Thesis Supervisor: Ronitt Rubinfeld Title: Professor of Electrical Engineering and Computer Science, MIT Thesis Supervisor: Madhu Sudan Title: Gordon McKay Professor of Computer Science, Harvard University 3 Acknowledgements I have been very fortunate to be at MIT with great advisors, collaborators, friends and family. I am extremely grateful to Madhu for taking me as his student, guiding me carefully and helping me navigate different kinds of situations. Madhu adapted his advising style for me over the years: very hands on initially, when I was figuring out research areas, but later, encouraging me to pursue directions on my own with other collaborators. I’m always amazed at Madhu’s profound insights on virtually any topic; something I’ll greatly miss after MIT. I have been truly blessed to also have Ronitt as an advisor. Ronitt’s guidance has been very valuable; she has provided me unconditional support and has been extremely caring and encouraging in whatever directions I wished to pursue. The work in this thesis has been a result of guidance I found rather serendipitously. My initiation to the topics studied in this thesis happened when I started a reading group on query & communication complexity with Shalev Ben-David and Robin Kothari among others. Later, I had the opportunity to learn from Raghu Meka, who hosted me at UCLA on two occasions. Raghu’s clear thinking and emphasis on foundational problems is something I aspire for. I’m also grateful to Raghu for serving on my committee. Finally, I had the fortune of working with Mika Göös, with whom all the work in the thesis was done (along with other collaborators). Despite working with Mika closely for so long, I haven’t been able to fully absorb his graphic design skills, crisp writing style and bouldering techniques (I can barely complete a V2). I often try to compensate for this by using Mika’s signature hyperlink style. I would like to thank all the collaborators I have had the good fortune of working with while at MIT: Jayadev Acharya, Mohammad Bavarian, Arnab Bhattacharyya, Ankit Garg, Badih Ghazi, Mika Göös, Bernhard Haeupler, Elad Haramaty, Shen Li, Toniann Pitassi, Prasad Raghavendra, Ronald Rivest, Robert Robere, Ankit Shah, Julie Shah, Dmitry Sokolov, Katerina Sotiraki, Madhu Sudan, Ameya Velingker, Tom Watson and Manolis Zampetakis. A special shout-out to Badih, with whom I have had numerous fruitful collaborations. I’m yet to assimilate Badih’s resourceful- ness, coolness and optimism while facing hard research problems. MIT is the place it is, primarily because of the amazing people around. I have gained tremen- dously by interacting with faculty at MIT and I would like to thank Scott Aaronson, Aleksander Mądry, Dana Moshkovitz and Vinod Vaikuntanathan. I would especially like to thank Costis Daskalakis for giving me advice on many occasions and also serving on my committee. Finally I’m very grateful to Yael Kalai for valuable guidance and being such an inspiring source of energy. I have had excellent and diverse TA opportunities at MIT which have served as valuable learning experiences and I’m grateful to Dana Moshkovitz, David Karger, Dina Katabi and Piotr Indyk for providing me the same. For getting me interested in theoretical research and trying to build some discipline into me, I would like to thank Supratik Chakraborty, Nutan Limaye and Neeraj Kayal. I would also like to 4 thank Csaba Szepesvári for a wonderful internship opportunity at Google DeepMind. For making the ToC group such a warm environment for discussions (technically and otherwise) I would like to thank all my fellow students. There are too many people to thank, so I’ll try to keep it brief: Michael Forbes, Mohammad Bavarian, Matt Coudron, Alan Guo, Henry Yuen, John Wright, Robin Kothari, Shalev Ben-David, Badih Ghazi, Gautam “G” Kamath, Maryam Aliak- barpour, Jerry Li, Mădălina Persu, Luke Schaeffer, Adam Sealfon, Katerina Sotiraki, Prashant Va- sudevan, Prabhanjan Ananth, Akshay Degwekar, Nishanth Dikkala, Govind Ramnarayan, Manolis Zampetakis, Sitan Chen ::: alas, that wasn’t brief and I have still missed so many people. Addi- tionally I would like to thank the admins Joanne, Rebecca and Debbie for cutting through all the bureaucracy with a pedestrian sense of ease, making ToC among the most fun groups at MIT! For making my visits to Harvard fun and enjoyable I would like to thank Mitali Bafna, Elad Haramaty and Preetum Nakkiran. For making MIT a home, I would like thank Yashovardhan Chati and Devendra Shelar for tolerating me as a roommate (and all the fun alley cricket at home, with apologies to people living downstairs!) and Vaibhav, Divya, Krithika, Rajan, Saurabh, Shilpa, Ranjitha and Parnika for the never ending conversations on life, universe and everything. Finally, I would like to thank my family who have raised me up to where I am today. This includes my parents, mom Urmila and aunt Naina, my brother Sudeep and sister-in-law Swarna (and a yet-to-arrive young one!). Sudeep has been a constant source of inspiration ever since I was tiny; he taught me all the important things in life very early on (including topics like combinatorics, trigonometry and calculus). Finally, I would like to thank my wife Apoorvaa, who brought along with her very inspiring in-laws Meenal and Chandrahas. I was incredibly lucky to have met Apoorvaa very early in life and I cannot imagine how life would have been otherwise. 5 Dedicated to the memory of my father, Uday Kamath, who inspired the mathematical spark in both of his sons. 6 Contents 1 Introduction 2 1.1 Complexity Theory .................................... 3 1.2 Outline of the thesis ................................... 4 1.3 Note on the content of this thesis ............................ 6 2 Complexity Theory Basics 8 2.1 Computational Models .................................. 8 2.2 Propositional Proof Complexity ............................. 11 2.3 Query Complexity .................................... 14 2.4 Communication Complexity ............................... 15 2.5 Query-to-Communication Lifting ............................ 17 2.6 Reductions in Query and Communication ........................ 19 3 Tools for Lifting Theorems 20 3.1 Rectangles are non-negative juntas ........................... 20 3.2 Proof of Full Support Lemma .............................. 22 4 Dag-like Models and Equivalences 26 4.1 Abstract dags ....................................... 26 4.2 Concrete dags ....................................... 28 4.2.1