This reminds me of the classic problem in computation, where the simplest form of computation, the lookup table, input -> output, is limited to a finite domain. Turing modified the computation to have a finite internal state and infinite external environment (tape), so it becomes a transition function (state, stimulus) -> (new state, response), applied recursively in a feedback loop, allowing it to operate on infinite domains.

Famously a simple lookup table for the transition function then suffices to compute any computable function.