Multiway Systems
The rule can usually fire in more than one place. Instead of choosing, follow every choice at once — and you get a branching tree of all possible histories.
In Causal Graphs we quietly made a choice: when the rule could fire in several places, we picked an order. But what if we refuse to choose — and instead follow every possibility at once? That’s a multiway system, and it’s the Wolfram model’s path toward quantum mechanics.
As before we use a string of letters to keep it simple; the same idea applies to hypergraphs.
Don’t pick — branch
Same little rule, BA → AB. Starting from BABA, the rule can fire in two different
spots, giving two different next states. So we keep both, and do the same for each
of those, and so on. The result is a graph of every history the system could have:
BA → AB — wherever you see BA, you may replace it with AB. Green-outlined states are final (no move left).Try this: play it through. Watch the system split into two states at the first step — then watch those two branches merge back into the same state a step later. Every path eventually arrives at the same green “final” state,
AABB.
Branches that split and merge
That re-merging is the interesting part. The system genuinely branches — there are real, different intermediate histories — but they reconverge to one outcome. When a rule has this “all roads lead to the same place” property, lots of the model’s physics (a consistent shared reality, relativity, well-defined measurement) is argued to follow from it.
The precise version
A multiway system is built by applying the rule in all possible ways at each step, rather than committing to one. Identical states are merged into a single node, so branches that reach the same state reconverge. [setup] The branching structure is the model’s representation of all possible histories at once.
The “all roads merge” behaviour you saw is the visual face of causal invariance / confluence — different orders of rewriting can always reconverge. [setup]
Grounded in the technical paper and Wolfram’s 2020 announcement.
Where this is heading: quantum mechanics
Wolfram proposes that this branching is the origin of quantum behaviour: the different branches act like superposed histories, and a second structure built from the multiway graph — branchial space — is proposed to encode entanglement. [conjecture] This is the most speculative part of the project, and is treated far more carefully (path integral, Bell/CHSH) in Gorard’s quantum paper than in the popular essays. We’ll lean on that paper — and keep the skeptics’ view in mind — when we write the quantum notes.
Sources for this page: Technical paper · 2020 announcement · Gorard — quantum · 2021 update