Definition
Fix a source of pasts (a process W, with past W⃗_t) and a target future (a block Z⃗⁺ of some process Z). Two pasts are equivalent when they induce the same conditional law over the target:
w⃗ ∼ w⃗′ ⟺ P(Z⃗⁺ | W⃗_t = w⃗) = P(Z⃗⁺ | W⃗_t = w⃗′), at every block length.
A causal state is a class of this relation: a set of pasts, nothing more. The causal-state machine is a further object: the causal states together with the transitions between them that the process induces, and the symbols emitted on each. The states are the partition; the machine is the predictive mechanism they form. A13 gave the relation as h ∼_Z h′ ⟺ P(Z_future | h) = P(Z_future | h′), and A14 asked that the entry “state explicitly whether the histories are taken from the microscopic process or from the macro process. THEORY-820 depends on comparing those two predictive-state partitions.” So:
The two named machines differ in the source of pasts and the target:
- ε-machine, W = Z: a process’s own pasts for its own future (Crutchfield and Shalizi; THEORY-756).
- υ-machine, W = X and Z = g(X⃗): the micro pasts for a coarse-graining’s future (Rosas et al.; THEORY-820 (b)).
The partition is the coarsest prescient one: any function of the pasts that carries all they carry about the target future refines it (THEORY-820 (b), THEORY-756). For a coarse-graining Z, the two coincide exactly when Z is informationally closed (THEORY-820 (c); TERM-068, sense 1).
What it is not
- Not “predictive state” in a separate sense. A13’s name. The record’s word is causal state, and the title keeps it.
- Not any sufficient statistic. It is the coarsest one (THEORY-820 (b), THEORY-756).
- Not target-free. It is canonical only given the target, and the target only given the measuring instrument (CLAIM-152’s note of 2026-10-20, from Crutchfield and Shalizi’s logistic map).
- Not interventional. The classes are of conditional laws, so “causal” here is predictive, as in THEORY-820’s Granger sense.