Coarse-Grained Computation Between Dynamical Systems

Computation is a relationship between coarse-grained models, not a primitive property of physical systems — and this paper makes that precise.

The core idea is simple: a physical system A "computes" another system B if you can squint at both of them in the right way and see the same dynamics. More formally, if you can find a coarse-graining of A (grouping its microstates into macrostates) and a coarse-graining of B such that the resulting compressed systems evolve identically — step for step, state for state — then A computes B. The technical term for "evolve identically" here is dynamical isomorphism: there's a relabeling of macrostates that makes the two update rules interchangeable.

This matters because it dissolves a longstanding ambiguity about what it means for a physical system to compute something. The usual story is vague: a computer "implements" an algorithm somehow. This paper replaces that vagueness with a structural condition. The computer's logical states (registers, memory, program counter) form a coarse-grained dynamical system; the target process — say, a discretized ocean simulation — forms another. When those two coarse-grained systems are isomorphic, computation is happening, by definition. The ocean example in the paper makes this concrete: the ocean's continuous fluid dynamics get discretized onto a grid (one coarse-graining), and the digital computer's hardware gets abstracted to logical machine states (another coarse-graining), and when the simulation is correct, these two compressed systems are structurally identical.

The definition connects cleanly to three existing bodies of work. Wolpert & Korbel's framework already defines when a physical system "emulates" an abstract machine via a decoding map; this paper specializes that to the case where the target machine is itself a coarse description of another physical system. Hoel's causal emergence work suggests that macroscale descriptions can actually be more causally informative than microscale ones, which justifies why coarse-graining isn't just a lossy approximation but can be the right level of description. And Piccinini's mechanistic account of computation — which says computers manipulate medium-independent vehicles according to rules — maps naturally onto the macrostates and update rules of the coarse-grained system.

The paper also notes, honestly, that exact isomorphism is an idealization. Real simulations accumulate errors, so an approximate version of the definition allows trajectories to diverge by at most ε over time. The source does not develop this approximate case in depth, but flags it as the realistic regime. Overall this is a short, tightly argued theoretical note — more a precise definition and its justification than an empirical or computational result — and it should be read as such.