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LIT-1306: Software in the natural world: A computational approach to hierarchical emergence

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StatusActive — read 2026-10-20 (NOTE-1030) in full; worth reading for three exact notions of a self-contained coarse-graining Z of a stationary discrete process X and how they relate: informational closure, I(X⃗_t; Z⃗ᴸ_{t+1} | Z⃗_t) = 0 for every horizon, which is Barnett and Seth's dynamical independence without an environment; causal closure, the ε-machine of Z equivalent to its υ-machine (the coarsest partition of micro pasts that predicts Z's future); and computational closure, Z's ε-machine a coarse-graining of X's. Theorem 1 proves informational ⟺ causal closure for any coarse-graining of the past; Theorem 2 proves causal ⇒ computational closure for spatial ones, and Counterexample 4 the converse false. Proposition 4 and Theorem 3 tie closure to Kemeny–Snell lumpability: strong lumpability gives closure, closure with a Markov Z gives strong lumpability, weak lumpability gives only computational closure. Read it knowing that the sublattice claims are asserted and, for informationally closed spatial coarse-grainings, false under common refinement (a three-bit counterexample in the NOTE); that the stated sufficient condition for the Ising energy to close (a permutation-invariant Hamiltonian) fails with a uniform field at coincident energies; that Proposition 3 is a telescoping identity whose closure hypothesis is idle; that (A.11) and Example 2 contain slips; that the modular random walk closes by symmetry of the weights and not by modularity; and that every result is exact, with nothing on approximate closure.
Version2
Tagscomplex-systems, information-theory, causality, mathematics
Cited as “paper” byNOTE-1030 — Rosas et al. — Software in the natural world
Cited as “source” byTHEORY-820 — For a stationary discrete-valued process X and any coarse-graining Z of its past, informational closure at every horizon is the same condition as zero one-step transfer entropy from the micro past, Barnett and Seth's dynamical independence, and as causal closure, Z's ε-machine coinciding with its υ-machine; for a coarse-graining of the present state it also makes Z's causal states a coarse-graining of X's, which does not imply closure, and the closed coarse-grainings are not closed under common refinement, THEORY-822 — For a Markov chain and a partition of its states, strong lumpability, every state of a block sending the same probability into each block, is the column-equitability of the master-equation operator and makes the block label informationally closed; when the chain is irreducible, an informationally closed label that is itself Markov is strongly lumpable, with no aperiodicity needed; weak lumpability gives a Markov label that need not be closed; and the same holds on the causal-state chain of any stationary process

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Rosas, F. E., Geiger, B. C., Luppi, A. I., Seth, A. K., Polani, D., Gastpar, M. & Mediano, P. A. M. (2024). Software in the natural world: A computational approach to hierarchical emergence. arXiv:2402.09090 [nlin.AO].

Read in full in NOTE-1030, from the arXiv v2 PDF.

Why this is here

The exact theory of closure that QUESTION-218 and QUESTION-221 lacked for discrete processes. It shows that informational closure, Barnett and Seth’s dynamical independence (THEORY-733), is the same condition as causal closure. For Markov chains it is Kemeny–Snell strong lumpability when the coarse process is Markov. That is the master-equation form of THEORY-755’s column-equitable partitions. The reading’s results are in THEORY-820 and THEORY-822.

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