Coding for Interactive Communication with Small Memory and Applications to Robust Circuits
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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 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( d 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(√(ε)) over oblivious channels and 1-O(√(ε1ε)) 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.
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