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Probabilistically Checkable Proofs, Agreement Tests, and High Dimensional Expanders

Periodic Reporting for period 2 - PCPHDX (Probabilistically Checkable Proofs, Agreement Tests,and High Dimensional Expanders)

Reporting period: 2019-08-01 to 2021-01-31

"This project is focused on probabilistically checkable proofs (PCPs) and high dimensional expanders. The central objective of this project is to connect between the two terms: PCPs and HDXs, that come from very different worlds.
The first term, PCPs, is a well studied object in theoretical computer science and has to do with methods of proof that allow for interaction and randomness. It turns out that with randomness one can provide proofs that can be checked extremely efficiently. The verifier doesn't need to read the proof, only probe a tiny random part of it. PCPs are useful in various contexts, one important one is that of showing lower bounds for approximation. Certain problems are hard to compute, and PCPs show that they are even hard to approximate.
The second term, HDX, is a construction of a combinatorial object which is like a graph but with higher dimensions, namely a simplicial complex. Such objects are known to exist via deep number theoretic constructions, and more recently also more elementary group theoretic constructions have been described. The high dimensional aspect of the object means that it is a certain generalization of expander graphs which themselves are very important and central objects in multiple fields including computer science and group theory.
The goal of the project is to explore ways to bring HDXs into the study of PCPs, and to bring PCP insight and questions into the study of HDXs. One of the most promising directions is to construct new error correcting codes based on HDXs that have properties similar to codes that come from PCPs. Error correcting codes are ubiquitous in the information age, and the extra ""local-testability"" properties have applications for example in blockchains."
The project is placed between areas in theoretical computer science and math. As such, there are multiple communities interested in these topics and a major effort is to generate dialog and language enabling free discussion and collaboration between the different researchers.
Thus, we have been active in creating workshops to gather the community together. In the summer of 2019 the PI co-directed a 6 week program at the Simons Institute in Berkeley dedicated to codes and high dimensional expanders.

The main scientific results achieved so far are:
* agreement tests: these are a basic object that connects between PCPs and HDXs. We have published a paper that generalizes and improves all previously known agreement tests: This includes older tests from the PCP literature, as well as new tests based on high dimensional expanders.
* locally testable codes: WE have completed a project (not yet published) showing that agreement tests allow reducing the task of proving global LTCness to local LTCness. The major task ahead of us is to find a concrete code that satisfies the criteria of this work and will be an LTC placed on HDX.
* hardness: We have shown that high dimensional expanders provide a structure on which certain computational problems are genuinely hard. We showed that for a powerful proof system called sum-of-squares, a specific constraint satisfaction problem is very hard.
Our main goal is to be able to put forth new error correcting codes that are placed on high dimensional expanders, and carry very good local testability properties such as constant rate, distance, and number of queries.
So far we have some candidate codes but we are only able to prove some and not all of their properties.
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