That’s the right question to ask. Conant & Ashby intentionally leave both the type signature and the causal structure of the regulator undefined—they have a whole spiel about how it can apply to multiple different setups (though they fail to mention that in some of those setups—e.g. feedback control—the content of the theorem is trivial).
For purposes of my version of the theorem, the types of the variables themselves don’t particularly matter, as long as the causal structure applies. The proofs implicitly assumed that the variables have finitely many values, but of course we can get around that by taking limits, as long as we’re consistent about our notion of “minimal information”.
That’s the right question to ask. Conant & Ashby intentionally leave both the type signature and the causal structure of the regulator undefined—they have a whole spiel about how it can apply to multiple different setups (though they fail to mention that in some of those setups—e.g. feedback control—the content of the theorem is trivial).
For purposes of my version of the theorem, the types of the variables themselves don’t particularly matter, as long as the causal structure applies. The proofs implicitly assumed that the variables have finitely many values, but of course we can get around that by taking limits, as long as we’re consistent about our notion of “minimal information”.