Recently, computational techniques that employ physical systems (physical computing systems) have been developed. To utilize physical computing systems, their design strategy is important. Although there are practical learning based methods and theoretical approaches, no general method exists that provides specific design guidelines for given systems with rigorous theoretical support. In this paper, we propose a novel algebraic design framework for a physical computing system for time-series generation, which is capable of extracting specific design guidelines. Our approach describes input-output relationships algebraically and relates them to this task. We present two theorems and the results of experiments. The first theorem offers a basic strategy for algebraic design. The second theorem explores the ``replaceability" of such systems.