## Pure Set Equivalencing

Pure Set Equivalencing

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## Evolution vs. Equivalencing

Evolution vs. Equivalencing

Given computational boundedness, only a limiting geodesic ball of equivalences

#### Persistence in time ⇔ equivalencing in time

Persistence in time ⇔ equivalencing in time

#### Any dynamics generates equivalencing ... but if run for finite time, only a certain amount

Any dynamics generates equivalencing ... but if run for finite time, only a certain amount

In our thoughts, these things are equivalent [ decompose into piston elements ]

#### Look at causal graph for piston [ shrinking one part ]

Look at causal graph for piston [ shrinking one part ]

### A “function”

A “function”

#### Start with multiway graph, then equivalence parts [cf ensemble]

Start with multiway graph, then equivalence parts [cf ensemble]

### Some parts of the graph shrink because they’re “fast dynamics”

Some parts of the graph shrink because they’re “fast dynamics”

Could explicitly represent e.g graph isomorphism steps ... but we’re contracting those parts ....

Cf category theory ; contraction to “keep multiple things in mind” ; how does this happen in an ANN?

### We are not universal observers

We are not universal observers

Because they are not computationally bounded

## Equivalencing e.g. to make group elements

Equivalencing e.g. to make group elements

## Minimal Model for Observer

Minimal Model for Observer

Given a set of string rewrites, how do we characterize the space of canonical results and/or the mapping?

[ We just want to characterize the equivalence classes, perhaps labeling them ... but we don’t need evolution to a canonical form ]

[ Odd vs even number of Bs ]

Imagine that an observer “makes its decision” at a particular event ; the past light cone contributes to that decision

Persistence of observer in time means the observer can look back [because those events can be “part of the same observer”]

####

#### Pure equivalencing

Pure equivalencing

#### DFA acceptor

DFA acceptor

## Constructed Observers

Constructed Observers

### StringLength counter

StringLength counter

E.g. an observer where all they look at is the total length of strings

[[ Is this actually correct? ]]

#### The canonical form version:

The canonical form version:

#### The attractor version:

The attractor version:

#### Theorem-proving analogy

Theorem-proving analogy

You have equivalences, but you want to drive everything to a canonical form [ for this, you also need an ordering function ]

#### Canonical form is an attractor (but not a fixed point)

Canonical form is an attractor (but not a fixed point)

In language design, this would be like evaluating to x, where x=y,y=x

### Reaching an attractor ...

Reaching an attractor ...

There are many representatives inside the attractor; the claim is that once you’re bubbling around in the attractor all the places you get to are considered “nearby” ; things outside the attractor are far away.... [in other words, coarse graining of the “near events” will be useful/successful]

#### The observer makes rapid transitions between “equivalent” states; much faster than the system itself makes transitions between states

The observer makes rapid transitions between “equivalent” states; much faster than the system itself makes transitions between states

#### By conflation etc. an observer can make itself atomic

By conflation etc. an observer can make itself atomic

## Multiscale String System

Multiscale String System

[ Intense “equivalence” interactions ; with a feeder that is less intense ]

This is where we input data, then we wait for it to get to an attractor where it bubbles around in an equivalence class.

In the piston example, we’re continually taking data, and seeing the effect [like a generative neural net instead of a classifier net]

In the piston example, we’re continually taking data, and seeing the effect [like a generative neural net instead of a classifier net]

Is a “thought” a particular configuration of a neural net, or an equivalence class of configurations?

[ It’s also like a phase transition ... ]

## The Making of Equivalence Classes

The Making of Equivalence Classes

The equivalencing transformations have to occur at high frequency than the other things that are going on....

#### Things that are rattling around in our brains we consider as a “atomic thoughts”

Things that are rattling around in our brains we consider as a “atomic thoughts”

Transformations between equivalent states isn’t the full story; because to say “it’s a single thought” requires ergodicity

#### cf. motion of the piston as “atomic” (i.e. as a single entity)

cf. motion of the piston as “atomic” (i.e. as a single entity)

#### “Observer approximation” : the fast interactions within the measuring device are all conflated to a single event

“Observer approximation” : the fast interactions within the measuring device are all conflated to a single event

E.g. for the piston all we need do is trace the molecule-wall interaction, not what happens inside the wall

[[ Want this graph, together with a collection of events that are “inside the piston” ... and then we can highlight the gas-piston interface events ]]

Some events simply “knit together the piston” ....

[[ How to make a piston? :

mass of spheres; size of spheres; packing density ]]

mass of spheres; size of spheres; packing density ]]

[ With hard squares, we can make a fully dense piston ; but the speed of sound is then infinite ]

Make the piston like a brick wall ; with hard squares offset in strips

[[ Does the piston need “cement” ... or like a Roman arch ]]

Need causal

## “Observation Process”

“Observation Process”

All sorts of inputs come in .... then the observation process leads to representing all of them just in terms of equivalence classes

The “atomic thought” is an attractor

## Length Measurement by Time of Flight

Length Measurement by Time of Flight

The photon is “flying” ... but meanwhile the clock is ticking, must faster than the flight time...

[Like in the piston the molecules are knocking around inside the solid much faster than the gas molecules are moving]

[Like in the piston the molecules are knocking around inside the solid much faster than the gas molecules are moving]

## Piston Physics

Piston Physics

## Fundamental Units [SI Base Units]

Fundamental Units [SI Base Units]

### Mass

Mass

Most of these use gravity to determine mass ; perhaps easier to determine energy

### Length

Length

[ Number of elementary lengths ]

### Time

Time

[ Number of elementary times ]

### Temperature

Temperature

[ Energy per degree of freedom ]

[ requires thermal equilibrium ]

E.g. mercury thermometer: collectively atoms move further apart

E.g. semiconductor thermometer: lots of electrons make it into the conduction band

E.g. semiconductor thermometer: lots of electrons make it into the conduction band

### Electric current

Electric current

Counting electrons

### Amount of substance

Amount of substance

Counting molecules

### Luminous intensity

Luminous intensity

Counting photons

### Other stuff....

Other stuff....

#### Spin + other quantum numbers

Spin + other quantum numbers

# [ To dos ]

[ To dos ]

[ QM multiway graph in 2nd law ]

Hard sphere gas with piston [ possibly hard squares instead ]

Fast + slow string rewrite system

Time of flight length measurement minimal model [ can this be integrated with relativistic causal graphs ]

[[ causal graph for a neural net ]]