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Relation between models and reference frames

Reference frame is way of foliating the multiway graph

Model is a way of coloring pieces of the multiway graph that are equivalent

Consider the non-accumulative case

In[]:=
MultiwayRulesNestGraph[x_∘y_<->(y_∘x_)∘y_,(a∘b)∘a,3,AspectRatio->1/2]
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A model tells us which are equal:

In[]:=
symbolifyPatterns[x_∘y_<->(y_∘x_)∘y_]/.TwoWayRule->Equal
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x∘y(y∘x)∘y
In[]:=
ResourceFunction["FindFiniteModels"][x∘y(y∘x)∘y,2]
Out[]=
{0}SmallCircle{{0,0},{0,0}},{1}SmallCircle{{0,0},{0,1}},{5}SmallCircle{{0,1},{0,1}},{7}SmallCircle{{0,1},{1,1}},{10}SmallCircle{{1,0},{1,0}},{15}SmallCircle{{1,1},{1,1}}
In[]:=
VertexList[%1410]
Out[]=
{(a∘b)∘a,b∘a,(a∘(a∘b))∘a,((b∘a)∘b)∘a,(a∘(a∘(a∘b)))∘a,(a∘((b∘a)∘b))∘a,(((a∘b)∘a)∘(a∘b))∘a,((b∘(b∘a))∘b)∘a,(((a∘b)∘a)∘b)∘a,(a∘(a∘(a∘(a∘b))))∘a,(a∘(a∘((b∘a)∘b)))∘a,(a∘(((a∘b)∘a)∘(a∘b)))∘a,(((a∘(a∘b))∘a)∘(a∘(a∘b)))∘a,(a∘((b∘(b∘a))∘b))∘a,(a∘(((a∘b)∘a)∘b))∘a,((((b∘a)∘b)∘a)∘((b∘a)∘b))∘a,((b∘(b∘(b∘a)))∘b)∘a,((b∘((a∘b)∘a))∘b)∘a,((((b∘a)∘b)∘(b∘a))∘b)∘a,(((a∘(a∘b))∘a)∘b)∘a,((((b∘a)∘b)∘a)∘b)∘a,((b∘a)∘(a∘b))∘a,(((a∘b)∘a)∘((b∘a)∘b))∘a,(((a∘b)∘((a∘b)∘a))∘(a∘b))∘a,(((a∘(a∘b))∘a)∘(a∘b))∘a,((((b∘a)∘b)∘a)∘(a∘b))∘a}
In[]:=
ApplyFiniteModel[#,SmallCircle{{0,1},{1,1}},{a,b}]&/@VertexList[%1410]
Out[]=
{{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}},{{0,1},{1,1}}}
In[]:=
ApplyFiniteModel[#,SmallCircle{{1,0},{0,0}},{a,b}]&/@VertexList[%1410]
Out[]=
{{{0,1},{0,0}},{{1,0},{0,0}},{{1,0},{0,0}},{{1,1},{0,0}},{{0,1},{0,0}},{{0,0},{0,0}},{{1,1},{0,0}},{{0,1},{0,0}},{{0,1},{0,0}},{{1,0},{0,0}},{{1,1},{0,0}},{{0,0},{0,0}},{{1,1},{0,0}},{{1,0},{0,0}},{{1,0},{0,0}},{{1,1},{0,0}},{{1,1},{0,0}},{{1,1},{0,0}},{{0,1},{0,0}},{{1,1},{0,0}},{{1,1},{0,0}},{{1,0},{0,0}},{{0,1},{0,0}},{{1,0},{0,0}},{{1,0},{0,0}},{{1,1},{0,0}}}
In[]:=
With[{g=EdgeList[MultiwayRulesNestGraph[x_∘y_<->(y_∘x_)∘y_,(a∘b)∘a,3,AspectRatio->1/2]]},Graph[g,VertexStyle->(#->Hue[FromDigits[Flatten[ApplyFiniteModel[#,SmallCircle{{1,0},{0,0}},{a,b}]],2]/20]&/@VertexList[g]),GraphLayout->"LayeredDigraphEmbedding"]]
Out[]=
In[]:=
With[{g=EdgeList[MultiwayRulesNestGraph[x_∘y_<->(y_∘x_)∘y_,(a∘b)∘a,3,AspectRatio->1/2]]},Graph[g,VertexStyle->(#->Hue[FromDigits[Flatten[ApplyFiniteModel[#,SmallCircle{{0,1},{1,1}},{a,b}]],2]/20]&/@VertexList[g]),GraphLayout->"LayeredDigraphEmbedding"]]
Out[]=
In[]:=
Function[mat,With[{g=EdgeList[MultiwayRulesNestGraph[x_∘y_<->(y_∘x_)∘y_,(a∘b)∘a,2,AspectRatio->1]]},Graph[g,AspectRatio->1,VertexSize->.2,VertexStyle->(#->Hue[FromDigits[Flatten[ApplyFiniteModel[#,SmallCirclemat,{a,b}]],2]/20]&/@VertexList[g]),GraphLayout->"LayeredDigraphEmbedding"]]]/@Tuples[{1,0},{2,2}]
Out[]=
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
In[]:=
ParallelMap[Function[mat,With[{g=EdgeList[MultiwayRulesNestGraph[x_∘y_<->(y_∘x_)∘y_,(a∘b)∘a,3,AspectRatio->1]]},Graph[g,AspectRatio->1,VertexSize->1,VertexStyle->(#->Hue[FromDigits[Flatten[ApplyFiniteModel[#,SmallCirclemat,{a,b}]],2]/20]&/@VertexList[g]),GraphLayout->"LayeredDigraphEmbedding"]]],Tuples[{1,0},{2,2}]]

Reference frames: foliations that determine what we do first in the entailment graph

E.g. make a foliation where one gets to small theorems fast

E.g. imagine a foliation where you’re always looking at larger expressions :: this doesn’t satisfy partial order

The observer is e.g. merely saying what color things are ... that is their coarse graining

The observer approximates the structure by a model, then wants to make an effective theory based on the model.
If the model is a slam dunk, then the effective theory is trivial.
​
If the model isn’t completely there, then the observer might still be able to make an effective theory.... [[ e.g. describe the dynamics as transitions between colors ]]

Models vs. proofs:

To verify a model with arity a one needs k^k^a cases [[[ one can imagine a “faithful model” that is a perfect “classification” ... all expressions that are equal have equal values and vice versa ]]]
Proof: only follow the proof path

In the expression equality case, everything is two-way

Accumulative case

In the accumulative case, there is a definite flow

We should draw the accumulative entailment cone ... then for a given model this cone will be colored in variegated way
Any correct model will confirm that an entailed theorem is correct....
This is an actual model....

To make a reference frame, you need a monotone function.....

E.g. from an expression, get a function that always increases for an entailment......

Imagine a model that maps expressions to integers

Say ∘ is Plus; say every variable is 1

In this case, FWIW every integer can eventually be generated.....
[[[ Relation to CH ? ]]]

A monotonic model would be a reference frame.....

E.g. A AA
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