Bidirectional coupling between a loop and an internal state

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The first transition allows the outer process L to depend on x, while the second allows x to depend on L. This notation makes the proposed two-way path explicit. Actual causal coupling requires those dependencies to be operative and testable; a function signature can include an argument that its implementation ignores.

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Original context

2.4. Recurrent information requires a causal connection

A generic coupled implementation could take the form

Lt+1=F(Lt,ut,xt),  xt+1=G(xt,Lt).

(2.6)

Here xt=πx(St) is the selected state of the inner process, which I call the center. The map πx identifies which part or function of the bookkeeping state is being studied. Lt is the state of the recurrent process coupled to that center, ut is input at update t, and F and G are the respective update functions. Thus x identifies the system, while xt describes a selected aspect of its changing state.

If G does not depend on Lt, the recurrent process cannot change the center through that channel. If the dependence exists, the outcome still depends on the update rule, retained information, inputs, and coupling. Circulation alone does not establish accumulation or a threshold crossing.

{L}_{t+1}=F({L}_{t},{u}_{t},{x}_{t}),  {x}_{t+1}=G({x}_{t},{L}_{t}).

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