PPT-Unprovability of strong complexity lower bounds in bounded arithmetic

Dagstuhl Workshop March 2023 Igor Carboni Oliveira University of Warwick 1 Join work with Jiatu Li Tsinghua 2 Context Goals of Complexity Theory include separating complexity classes. cmuedu Joshua Brody IIIS ITCS Tsinghua University Beijing China joshuaebrodygmailcom Kevin Matulef IIIS ITCS Tsinghua University Beijing China matulefgmailcom Abstract We develop a new technique for proving lower bounds in property testing by showing The heart of this technique is a complex ity measure for multivariate polynomials based on the linear span of their partial derivatives We use the technique to obtain new lower bounds for computing sym metric polynomials which hold over 64257elds of Indeed developing bounds on the per formance of procedures can give complementary insights By exhibiting fundamental limits of performance perhaps over restricted classes of estimators it is possible to guarantee that an a lgorithm we have developed Program Analysis and Verification . Nikolaj Bj. ø. rner. Microsoft Research. Lecture 3. Overview of the lectures. Day. Topics. Lab. 1. Overview of SMT and applications. . SAT solving,. Z3. Encoding combinatorial problems with Z3. :. . The Basics, Accomplishments, Connections and Open problems. Toniann. . Pitassi. University of Toronto. Overview. P. roof systems we will cover. Propositional, Algebraic, Semi-Algebraic. Lower bound methods. 2 - . Calculations. www.waldomaths.com. Copyright © . Waldomaths.com. 2010, all rights reserved. Two ropes, . A. and . B. , have lengths:. A = . 36m to the nearest metre . B = . 23m to the nearest metre.. approximate membership. dynamic data structures. Shachar. Lovett. IAS. Ely . Porat. Bar-. Ilan. University. Synergies in lower bounds, June 2011. Information theoretic lower bounds. Information theory. relaxations. via statistical query complexity. Based on:. V. F.. , Will Perkins, Santosh . Vempala. . . On the Complexity of Random Satisfiability Problems with Planted . Solutions.. STOC 2015. V. F.. A combinatorial approach to P . vs. NP. Shachar. Lovett. Computation. Input. Memory. Program . Code. Program code is . constant. Input has . variable length (n). Run time, memory – grow with input length. Hrubeš . &. . Iddo Tzameret. Proofs of Polynomial Identities . 1. IAS, Princeton. ASCR, Prague. The Problem. How . to solve it by hand . ?. Use the . polynomial-ring axioms . !. associativity. , . relaxations. via statistical query complexity. Based on:. V. F.. , Will Perkins, Santosh . Vempala. . . On the Complexity of Random Satisfiability Problems with Planted . Solutions.. STOC 2015. V. F.. Searching. : Given a large set of distinct keys, preprocess them so searches can be performed as quickly as possible. 1. CS 840 Unit 1: Models, Lower Bounds and getting around Lower bounds. Searching. dynamic data structures. Shachar. Lovett. IAS. Ely . Porat. Bar-. Ilan. University. Synergies in lower bounds, June 2011. Information theoretic lower bounds. Information theory. is a powerful tool to prove lower bounds, e.g. in data structures. Alexander A. . Razborov. University of . Chicago. Steklov. . Mathematical Institute . IAS, . Avi. is 60 conference, October 5, 2016. Subtitle: on my . (and others’) largely unsuccessful . attempts to make .