Infinite-message distributed source coding for two-terminal interactive computing

A two-terminal interactive function computation problem with alternating messages is studied within the framework of distributed block source coding theory. For any arbitrary fixed number of messages, a single-letter characterization of the minimum sum-rate function was provided in previous work using traditional information-theoretic techniques. This, however, does not directly lead to a satisfactory characterization of the infinite-message limit, which is a new, unexplored dimension for asymptotic-analysis in distributed block source coding involving potentially infinitesimal-rate messages. This paper introduces a new convex-geometric approach to provide a blocklength-free single-letter characterization of the infinite-message minimum sum-rate function as a functional of the joint source pmf. This characterization is not obtained by taking a limit as the number of messages goes to infinity. Instead, it is in terms of the least element of a family of partially-ordered marginal-perturbations-concave functionals associated with the functions to be computed. For computing the Boolean AND function of two independent Bernoulli sources at one and both terminals, the respective infinite-message minimum sum-rates are characterized in closed analytic form. These sum-rates are achievable using infinitely many infinitesimal-rate messages. The convex-geometric functional viewpoint also suggests an iterative algorithm for evaluating any finite-message minimum sum-rate function.