Definition
“Closure” in this record means one of the following, and a claim should say which.
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Informational closure, of a coarse-graining of a process. X is a stationary process with a discrete alphabet, X⃗_t its past up to t, and Z_t = g(X⃗_t) a coarse-graining, spatial if Z_t = f(X_t). Z is informationally closed when its own past carries everything the micro past carries about its future:
Δ_X(Z) = I(X⃗_t; Z_{t+1} | Z⃗_t) = 0.
This is transfer entropy from the micro past with Z’s whole history and no environment (TERM-057 with k and l unbounded). A12 wrote it as Δ_X(Z) = I(X_{≤t}; Z_{t+1} | Z_{≤t}) and Closed_X(Z) ⟺ Δ_X(Z) = 0. What the record holds of it (THEORY-820):
- Every horizon reduces to one step, so the definition is not relative to a horizon (THEORY-820 (a)).
- It is dynamical independence. That is Barnett and Seth’s condition (LIT-979; THEORY-733 for the linear Gaussian form).
- It is causal closure, Z’s ε-machine coinciding with its υ-machine (THEORY-820 (c); TERM-062).
- It is not preserved by refinement, nor by coarsening (CLAIM-409).
It is used, with no TERM until now, in THEORY-820, THEORY-822, THEORY-733, CLAIM-386, CLAIM-388, CLAIM-389, CLAIM-394, CLAIM-398, CASE-120, QUESTION-218, QUESTION-221 and QUESTION-222.
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Computational closure. Z’s causal states are a coarse-graining of X’s. It follows from 1 for spatial maps only, and does not imply it (THEORY-820 (e) and its Counterexample 4).
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Strong lumpability. Every state of a block sends equal probability into each block. It makes the label closed, and is equivalent to closure for a closed label that is Markov (THEORY-822).
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Vanishing Mori–Zwanzig memory. In a linear Gaussian VAR(1) it follows from 1 and does not imply it (CLAIM-389, CLAIM-397, CLAIM-398).
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Linear one-step closure defect. QA − ÂQ of the Simon–Ando aggregate (CLAIM-386), a matrix, not a condition.
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Organizational closure of constraints (TERM-070; THEORY-072, CLAIM-395). A set of constraints each produced by and productive of another in the set. Mossio, Saborido and Moreno’s 2009 closure, which a candle flame has (LIT-570, inside THEORY-072), is a weaker neighbour. THEORY-071’s “autonomy” and “self” mean this sense, and CLAIM-172’s “operational closure” is the enactive form.
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Homonyms, not the argument’s: the closure of a belief set under consequence (CLAIM-294), Galois closures (CLAIM-119), a downward-closed complex (CLAIM-347), and an observer’s feedback closure and semantic closure (CLAIM-341).
What it is not
- Not one property with several aspects. 1 and 6 concern different objects: a variable’s law, and a set of constraints. Noise has 1 (THEORY-820: “An i.i.d. noise component is closed”); nothing shows a flame has 6 under the 2015 definitions. QUESTION-093 asks which of them an agent needs.
- Not a mark of a system, in sense 1. Whether a closed variable is one system is QUESTION-218’s open half.
- “Computational closure” is not computational tractability. A11: “The similarity in names makes this an especially plausible source of future confusion.”
- “Causal closure” in sense 1 is closure in the Granger sense unless X is fully observed (THEORY-820: “Not interventional causation unless X is fully observed”).
- Informational closure and “causal-state closure” are not two senses. A11’s and A14’s tables listed them apart. THEORY-820 (a) and (c) prove they are one condition for stationary discrete processes.
- Common refinement is a meet in THEORY-820’s convention. A13 writes the common refinement of A and B as A ∨ B. A claim using either symbol should say which order it means.