In the fair cake-cutting, two different computational models are used:
- A discrete model, in which the algorithm issues queries to the players and proceeds according to their replies;
- A continuous model, in which one or more "knives" move continuously over the cake under certain restrictions, until a player shouts "stop".
The continuous model is strictly stronger: there are problems that can be solved easily in the continuous model, but cannot be solved in finite time in the discrete model, e.g, envy-free cake-cutting with connected pieces.
So I wonder what other problems, besides cake-cutting, can benefit from a "moving knife" or a similar continuous model?
For example, is there a continuous variant of a Turing machine, that can easily solve problems that are unsolvable for a discrete Turing machine?