A Study on the Joint Source-Channel Coding for Computing Functions: An Approach from a Dichotomy of Functions

In this paper, a problem of joint source-channel coding for computing functions of outputs from correlated sources is studied. In particular, the system of computing two-input functions where one of two outputs is available at the decoder as full-side information is investigated. Our result reveals that if the sensitivity of functions introduced by Ahlswede and Csisz\'ar and the smoothness of sources introduced by Kuzuoka and Watanabe are satisfied, then the condition that the function is computable over the channel (i.e., the value of the function is correctly decoded) coincides with that for the identity function (i.e., reproducing the entire source outputs).