Encoding Tasks and Rényi Entropy

A task is randomly drawn from a finite set of tasks and is described using a fixed number of bits. All the tasks that share its description must be performed. Upper and lower bounds on the minimum ρ-th moment of the number of performed tasks are derived. The case where a sequence of tasks is produced by a source and n tasks are jointly described using nR bits is considered. If R is larger than the Rényi entropy rate of the source of order 1/(1 + ρ) (provided it exists), then the ρ-th moment of the ratio of performed tasks to n can be driven to one as n tends to infinity. If R is smaller than the Rényi entropy rate, this moment tends to infinity. The results are generalized to account for the presence of side-information. In this more general setting, the key quantity is a conditional version of Rényi entropy that was introduced by Arimoto. A rate-distortion version of the problem is also solved, where, roughly speaking, at least one task within a specified distortion of the task produced by the source must be performed. Finally, a divergence that was identified by Sundaresan as a mismatch penalty in the Massey-Arikan guessing problem is shown to play a similar role here.