Are there languages that emphasize the single-datum <=> collection-of-data duality? E.g. making loop-with-

if

<=>

filter

-on-the-whole look very similar? But that's just a programming curiousity; I'm more interested if there are any mind bicycles that help one think about local vs. global rules? Physics is rich with examples where seeing both perspectives is insightful: • local F = ma <=> conservation of potential energy independent of specific path (for certain forces) • Gauss's laws relating single-point "differential" <=> volume/surface "integral" formulations of Maxwell equations • Noether's theorem is certainly up there, relating shapes of laws with conserved stuff, but I'll admit over my head... Example task: For a while now I've dreamed of making a model of weather that's simple enough for paper+pen+tokens, or generally something "board game-ish". Lies-to-children are fine, but would like it to demonstrate at least basic mechanics: (A) wind caused by hotter air rising and leaving a vacuum; (B) evaporation over sea => rain over land (C) "rain shadow" beyond a mountain (my country has this). It's not easy to discretize continuous-quantity equations into something like a cellular automaton... I'm leaning towards representing conserved quantities like water with tokens you can move, rather than per-cell state. And I don't want some huge "simulation"—I'd love to find rules that can demonstrate these processes in few enough steps to follow by hand. Hmm say I have air that wants to go up + right, and it holds just 1 water token—which way do I move it? Well I suppose I want a bit larger numbers so quantization won't matter that much... And I probably want alternating horizontal vs. vertical game phases (also dealing with 1D slice at a time can reduce previous-vs-next-state confusions). Any advice for tools to help think about rules/mechanics and what can emerge from them, beyond "try and see what what happens"? (But I guess I really should play with large "sand" style models, then try to scale down! try-and-see is more powerful than merely keeping in my head.)

Technicalities first: • Maude comes to mind as a language in which items are treated as equivalent to a list of one item. Howver, this happens by convention rather than as a language features. Lists are not built-in constructs. Still worth a look for examples of how this equivalence plays out in practice. • The integral version of F = m a is Lagrangian mechanics, which roughly says "a mechanical system follows the path that minimizes the action integral". The equivalence with Newton's version is very much in the spirit of Gauss' laws. • Noether's theorem is one level of abstraction above, it's about relations between invariants.

I'll pitch in Emoji Simulator as a CA prototyping tool that really likes for you to make rules that are 'small enough to follow by hand'. Exactly, there's enough design space for pattern-forming behaviors that it's hard to characterize without 'playing around', even on a smallish lattice. https://ncase.me/sim/ NetLogo is absolutely the next step. It's got enough built-in plotting tools to see if you're really reproducing the behavior of the global model. I did this with predator-prey at one point, and learned why the modelling approach splits into either metapopulation modelling (so that local extinctions are recoverable) or modelling the time-to-extinction (i.e. accidentally encountering the absorbing steady state). If you go into implementing a lattice gas automata I'm sure you'll get into similar nuances of advection models pretty quickly.