Abstract

Recently, a group of theoretical computer scientists posted a paper on the Arxiv with the strange-looking title “MIP = RE”, surprising and impacting not only complexity theory but also some areas of math and physics. Specifically, it resolved, in the negative, the “Connes’ embedding conjecture” in the area of von-Neumann algebras, and the “Tsirelson problem” in quantum information theory. It further connects Turing’s seminal 1936 paper which defined algorithms to Einstein’s 1935 paper with Podolsky and Rosen which challenged quantum mechanics. You can find the paper here https://arxiv.org/abs/2001.04383

As it happens, both acronyms MIP and RE represent proof systems, of a very different nature. To explain them, we’ll take a meandering jurney through the classical and modern definitions of proof. I hope to explain how the methodology of computational complexity theory, especially modeling and classification (of both problems and proofs) by algorithmic efficiency, naturally leads to the genaration of new such notions and results (and more acronyms, like NP). A special focus will be on notions of proof which allow interaction, randomness, and errors, and their surprising power and magical properties.