MMP.10:5.1 - Construct an unknown rule from requirements on its repetitions

A device has three labeled modes A, B and C. The required rule changes the mode on every use and returns to the starting mode after three uses. The question is to construct a deterministic rule, with no additional internal state. The rule itself is the unknown object.

Let S={A,B,C}. Choose one output variable p_i in S for each input i. The requirements become p_i != i and p_(p_(p_i)) = i for every i. Function composition gives the meaning of the repeated application. MATH.1 constructs composable paths; MATH.5 extends an interpretation of their elementary steps to the compounds.

To express the rule by selected pairs instead, choose r_ij in {0,1}. Add sum_j r_ij=1 for each row, r_ii=0, and, for all i,j,k, (r_ij=1 AND r_jk=1) implies r_ki=1. The row condition makes a function. The implication expresses the return after three uses: the first two selected transitions determine a required third.

Recover p by taking the unique selected column in each row. Conversely, p creates the table by selecting exactly its output pair in each row. These constructions are inverse. The triple-application requirement is therefore the same in both formulations, with MATH.7 carrying that relation. A rule A→B→C→A and its reverse both satisfy it.

There are precisely two such rules. From p^3=id, p is invertible with inverse p^2. Its cycles have lengths dividing three. Since a one-element cycle is forbidden, the three modes form one three-element cycle, with two possible orientations. This reasoning proves completeness; listing two examples alone would not.

Change the device to four modes, retaining the three-use return and no unchanged mode. A permutation of four elements cannot partition them into cycles all of length three, so no rule exists under these conditions. Change instead to a return after two uses. The conditions become p_(p_i)=i and p_i!=i; the indicator formulation requires symmetry r_ij=r_ji. It admits three pairings of four modes. Remove the former three-use implication: leaving it in the formulation would make the new, feasible requirement appear impossible.

The result is a rule that can be implemented and its stated scope: deterministic changes of the visible mode without hidden state. A proposal with additional state describes a different device and needs a new account of its operation.