Periodic Reporting for period 3 - Interactive (Coding for Interactive Communication and the Power of Adaptivity)
Reporting period: 2024-01-01 to 2025-06-30
Error-correcting codes allow for reliable transmission of data over unreliable channels.
The 70 years of development that went into the study of error correction following Shannon's ground-breaking paper in 1948 yielded complete far-reaching applications to other theoretical and practical fields. However, many modern communi-
cation settings are not simply about transmitting information but rather operate over
many rounds of interactive communication between different parties.
During our project, we studied how interacting affects regular communication. Can we perform regular communication tasks better if we allow communication? Shanon's theorem states that for the stochastic noise interaction will not be helpful. However, we had an astonishing result: if the noise is adversarial, we can break the barriers of what is possible without interaction.
We also studied bounded memory communication and its applications to shortcut circuits. Here our goal was to identify when two parties can perform interactive communication in the presence of noise when their memory is very small.
We also studied the best possible rates one can achieve during interactive communication.
The 70 years of development that went into the study of error correction following Shannon's ground-breaking paper in 1948 yielded complete far-reaching applications to other theoretical and practical fields. However, many modern communi-
cation settings are not simply about transmitting information but rather operate over
many rounds of interactive communication between different parties.
During our project, we studied how interacting affects regular communication. Can we perform regular communication tasks better if we allow communication? Shanon's theorem states that for the stochastic noise interaction will not be helpful. However, we had an astonishing result: if the noise is adversarial, we can break the barriers of what is possible without interaction.
We also studied bounded memory communication and its applications to shortcut circuits. Here our goal was to identify when two parties can perform interactive communication in the presence of noise when their memory is very small.
We also studied the best possible rates one can achieve during interactive communication.
During this period, we studied two closely related topics: interactive communication with bounded memory and the construction of circuits resilient to noise.
We had significant progress on both of these topics: in the paper "Optimal Short-Circuit Resilient Formulas." We studied formulas(it is a weaker version of circuits) and figured out maximal noise resilience, showing matching lower and upper bounds. Next, in the paper "Interactive Coding with Small Memory" we have shown how to perform interactive communication when both parties have bounded memory. This result in turn leads us to the first quasi-polynomial construction of circuits resilient to the constant fraction of noise, which was one of the main goals of the research project.
During our research, we got a very surprising result which we did not expect to be true when we started the project: In the reliable transmission problem, a sender, Alice, wishes to transmit a bit-string x to a remote receiver, Bob, over a binary channel with adversarial noise. The solution to this problem is to encode x using an error correcting code. As it is long known that the distance of binary codes is at most 1/2, reliable transmission is possible only if the channel corrupts (flips) at most a 1/4-fraction of the communicated bits. We revisit the reliable transmission problem in the two-way setting, where both Alice and Bob can send bits to each other. Our main result is the construction of two-way error correcting codes that are resilient to a constant fraction of corruptions strictly larger than 1/4. Moreover, our code has constant rate and requires Bob to only send one short message
We had significant progress on both of these topics: in the paper "Optimal Short-Circuit Resilient Formulas." We studied formulas(it is a weaker version of circuits) and figured out maximal noise resilience, showing matching lower and upper bounds. Next, in the paper "Interactive Coding with Small Memory" we have shown how to perform interactive communication when both parties have bounded memory. This result in turn leads us to the first quasi-polynomial construction of circuits resilient to the constant fraction of noise, which was one of the main goals of the research project.
During our research, we got a very surprising result which we did not expect to be true when we started the project: In the reliable transmission problem, a sender, Alice, wishes to transmit a bit-string x to a remote receiver, Bob, over a binary channel with adversarial noise. The solution to this problem is to encode x using an error correcting code. As it is long known that the distance of binary codes is at most 1/2, reliable transmission is possible only if the channel corrupts (flips) at most a 1/4-fraction of the communicated bits. We revisit the reliable transmission problem in the two-way setting, where both Alice and Bob can send bits to each other. Our main result is the construction of two-way error correcting codes that are resilient to a constant fraction of corruptions strictly larger than 1/4. Moreover, our code has constant rate and requires Bob to only send one short message
We expect to have a good understanding of rate distance tradeoffs for various models of interactive communication. As well as we discovered recently a big impact interactive communication has on the regular error-correcting codes. We want to make this connection deeper. We also had significant progress on constructing circuits resilient to errors. We hope by the end of this project, we will be able to have a practical solution to turn any given circuit into a circuit resilient to errors with only a small overhead.