ResourceFunction["MultiwaySystem"][{"A""AB","BB""A"},{"A"},7,"StatesGraph"]

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With[{g=ResourceFunction["MultiwaySystem"][{"A""AB","BB""A"},{"A"},7,"StatesGraph"]},HighlightGraph[g,Subgraph[g,PathGraph[FindShortestPath[g,"A","AABA"]]]]]

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With[{g=ResourceFunction["MultiwaySystem"][{"A""AB","BB""A","AB""A","A""BB"},{"A"},7,"StatesGraph"]},HighlightGraph[g,Subgraph[g,PathGraph[FindShortestPath[g,"A","AABA"],DirectedEdgesTrue]]]]//LayeredGraphPlot

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With[{g=ResourceFunction["FindStringProof"][{"A""AABA"},{"A""AB","BB""A"}]["ProofGraph"]},Graph[g,VertexLabels(#[[1]]#[[2,1,2,1,2]]&/@Options[g,"AnnotationRules"][[1,2]])]]

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Every substitution X = Y has an evolution edge and a causal edge

After transitive closure, the causal edges in the proof graph are the same as in the MW graph

TransitiveClosureGraph[ResourceFunction["MultiwaySystem"][{"A""AB","BB""A"},{"A"},3,"StatesGraph"]]

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ResourceFunction["MultiwaySystem"][{"A""AB","BB""A","A""AAB"},{"A"},4,"StatesGraph"]//LayeredGraphPlot

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[ semantic entailment ]

[ rules of inference ]

proposition: a = b

Case 1: every node is a proposition; a true theorem is a path from a proposition node to the True node

Case 2: every node is a word, and theorems are paths between “equivalent” words

#### Generational states ↔ theorems “in the bag”

Generational states ↔ theorems “in the bag”

A jump ahead event is causally connected to all its component events. [Though in the proof graph, only one causal edge is shown here] [ Because the proof graph is essentially a transitive reduction ]

Looking at centrality of lemmas:

## Multiway to Equational

Multiway to Equational

#### Can have unidirectional string case ... but involves lots of infrastructure

Can have unidirectional string case ... but involves lots of infrastructure

### Causal edges: making use of previous expr, and also previous jump-ahead theorem

Causal edges: making use of previous expr, and also previous jump-ahead theorem

Inevitably “positive mass”, because previous results just increase number of different paths.

#### Critical fact: you put the whole proof in the bag, and use it as a one-step theorem [univalence]

Critical fact: you put the whole proof in the bag, and use it as a one-step theorem [univalence]

#### Assumption : metamath is explored on geodesics ; therefore you don’t get to see all the theorems of this axiom system

Assumption : metamath is explored on geodesics ; therefore you don’t get to see all the theorems of this axiom system

Dynamical theory of epistemology

#### To be “about something” is to make everything deterministic; i.e. a total order

To be “about something” is to make everything deterministic; i.e. a total order

#### Transportability of category theory : uniformity of metamathematical space

Transportability of category theory : uniformity of metamathematical space

#### Expansion of MM universe: all the computer experiments you could do

Expansion of MM universe: all the computer experiments you could do

#### Expansion of rulial space : NKS is necessary; math can’t reach far enough [ NKS : quantum computer ; math : classical computer ]

Expansion of rulial space : NKS is necessary; math can’t reach far enough [ NKS : quantum computer ; math : classical computer ]