Coding for Interactive Communication with Small Memory and Applications to Robust Circuits

Classically, coding theory has been concerned with the problem of transmitting a single message in a format which is robust to noise. Recently, researchers have turned their attention to designing coding schemes to make two-way conversations robust to noise. That is, given an interactive communication protocol $Π$, an \emph{interactive coding scheme} converts $Π$ into another communication protocol $Π'$ such that, even if errors are introduced during the execution of $Π'$, the parties are able to determine what the outcome of running $Π$ would be in a noise-free setting. We consider the problem of designing interactive coding schemes which allow the parties to simulate the original protocol using little memory. Specifically, given any communication protocol $Π$ we construct robust simulating protocols which tolerate a constant noise rate and require the parties to use only $O(\log d \log s)$ memory, where $d$ is the depth of $Π$ and $s$ is a measure of the size of $Π$. Prior to this work, all known coding schemes required the parties to use at least $Ω(d)$ memory, as the parties were required to remember the transcript of the conversation thus far. Moreover, our coding scheme achieves a communication rate of $1-O(\sqrt{\varepsilon})$ over oblivious channels and $1-O(\sqrt{\varepsilon\log\log\tfrac{1}{\varepsilon}})$ over adaptive adversarial channels, matching the conjecturally optimal rates. Lastly, we point to connections between fault-tolerant circuits and coding for interactive communication with small memory.