Unreasonable Effectiveness of Compositional Thinking

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things to do

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LATER figure out a good example to work through the category theory intro (cracking eggs into an omelette?)

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functor is make it plant based?

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LATER llm semantic triples, neural networks?

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LATER map and territory in intro

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LATER forward email

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DONE email

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subject

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5/5 a convivial convening on composition

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email

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Some of you have heard about category theory, maybe from me, maybe from Gioele Zardini’s talks on co-design, or maybe from the person at INCOSE who said that “all systems engineering is a footnote on category theory” (or something like that, I wasn’t there).

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For anyone who is interested, a few of us are dedicating an hour next Monday, May 5th, to discussing this. I will bring some material to talk through, but it will mostly be an open discussion with the following light agenda:

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conception (immaculate) -> philosophical intuitions behind why composition may be a foundational concept

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construction (formal) -> a visual, not-at-all-mathy introduction to category theory

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conversation (dangling) -> is this a universal language of system dynamics, co-design, language, and logic? if so, so what?

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If interested, join us on Monday, May 5th at 5pm in 1-379. I’ll bring snacks so please RSVP here if you intend to join.

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Best,

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Deniz

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goals

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to communicate some of the intuitions why some people find category theory compelling

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to show why

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other name ideas

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unreasonable effectiveness of composition in human systems

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Composing and comparing: a convivial collaborative convention on (mathematical) categories and their (practical) consequences

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structure

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(immaculate) conception: an exploration of intuitions behind systems and thoughts

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(de)composition

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on intuition of composition

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some simple examples

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movement

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cooking

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any set of actions that can be tied together

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eg open the door, then walk into the room

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but if the door is open i dont need to open the door, so maybe it’s

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check if the door is open

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if it’s open, walk in the room

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if it’s not open, open the door, then walk in the room

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what i’m trying to do is to communicate some of the reason why i’m attracted to some of these ideas

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physics of systems?

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What is a system?

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a process that takes some inputs and produces some outputs

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transformer

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system dynamics example?

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A system is composed of components. A component is something you understand.

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functional and formal decomp

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physics of thought?

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let’s say thoughts and language are interchangeable for now

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semantic triples, resource description framework

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thoughts are essentially references to other thoughts

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why is a raven like a writing desk

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physics of logic?

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physics of causality?

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Space time requires us to think about movement as something that traverses both space and time and because of that that’s where composition comes from the idea that transformation happens and something stays invariant

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(formal) construction: intro to category theory

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what is a category?

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dots and arrows

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identity

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composition

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associativity

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what emerges from this structure?

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diagrams commute

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isomorphism

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universal properties

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terminal, initial objects

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“bestness” and uniqueness

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limits, products

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enriched categories

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morphisms with structure

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$\infty$-categories

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morphisms between morphisms (and beyond)

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(can anyone see the punchline?)

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categories of categories

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Cat: a category whose objects are categories

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functors

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maps between categories

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morphisms in Cat

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natural transformations

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morphisms between functors

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“translations between translations”

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some sketches: examples like logic, co-design, system dynamics, etc

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conversation

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concerns

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consequences

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examples

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meta-reflexivity and levels of abstraction

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Why Category Theory is Uniquely Meta-Reflective

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Category theory allows you to walk up and down abstraction levels using the same constructions.

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It’s self-similar:

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Categories have morphisms

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Categories themselves are objects in Cat

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Functors are morphisms in Cat

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Natural transformations are morphisms between functors

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This continues into 2-Cat, n-Cat, ∞-categories

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You can model structure about structure without leaving the language of categories.

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How other fields compare

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Set theory

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Foundational, but not self-reflective

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Can’t form the set of all sets → paradoxes

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Algebra

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Powerful, but “algebras of algebras” are not coherent as a general notion

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Analysis

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Doesn’t scale to describing itself

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Logic / Type theory

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Some self-reflection (e.g., universe levels), but often needs a meta-system

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Related to category theory via Curry–Howard–Lambek

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Conclusion

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Category theory is rare in that it uses its own tools to describe itself

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This makes it uniquely powerful as a mathematics of mathematics

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category, functor, natural transformation

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Category Theory Example: Applied to a Database

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Category = Schema

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Objects:

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Person

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City

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Morphism:

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$lives_in: \text{Person} \to \text{City}$ (a foreign key relation)

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This forms a category: objects and structure-preserving relationships.

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Functor = Dataset

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A functor $F: \mathcal{C} \to \mathbf{Set}$ assigns:

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$F(\text{Person}) = {\text{Alice},\ \text{Bob}}$

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$F(\text{City}) = {\text{Boston},\ \text{LA}}$

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$F(\text{lives_in}) = {(\text{Alice},\ \text{Boston}),\ (\text{Bob},\ \text{LA})}$

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This is one instance of data fitting the schema’s shape.

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Another Functor = Updated Dataset

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A second functor $G$ assigns:

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$G(\text{Person}) = {\text{Alice},\ \text{Bob}}$

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$G(\text{City}) = {\text{Boston},\ \text{LA}}$

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$G(\text{lives_in}) = {(\text{Alice},\ \text{LA}),\ (\text{Bob},\ \text{LA})}$

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Same schema, different data (e.g., Alice moved to LA).

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Natural Transformation = Data Migration

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A natural transformation $\eta: F \Rightarrow G$ maps data:

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$\eta_{\text{Person}}$: identity on people

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$\eta_{\text{City}}$: identity on cities

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But updates how $lives_in$ points

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This satisfies the naturality condition:

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$G(lives_in) \circ \eta_{\text{Person}} = \eta_{\text{City}} \circ F(lives_in)$

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So the structure (foreign key logic) is preserved even after the data update.

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just start writing

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composition is a consequence of causality

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i’d like to reference some dead white men (DWMs)

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i’d also like to reference some dead not-only-white not-only-men (DNNs)

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what are we asking

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in the original essay, wigner tries to answer the question of why mathematics seems to be the right, or best, language for physics

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here, we will try to answer a question that either assumes mathematics is the right, or best, language for physics, or if it does not assume it, it at least feels like a question that sibling trajectory

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why are composable steps at the core of every system we conceive of?

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it’s not obvious that processes must be made up of composable steps

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why is causality a chain? why is logic made up of steps?

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causality as a chain

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hume questioned the necessity of causal connection, saw it as a habit of mind

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david lewis causal connectivity as counterfactual dependence

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x causes y if, had x not happened, y wouldn’t have happened

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rooted in possible worlds logic

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judea pearl uses directed acyclic graphs explicitly

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logic made up of steps

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frege / bertrand russell / Whitehead, formal logic as stepwise derivation

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ludwig wittgenstein logical form as mirrored in the structure of reality, atomic facts compose into more complex facts

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brouwer in intuitionism, logic is a constructive mental activity, the steps come from the “mind’s unfolding”

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under the hood

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why is logic made up of steps

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identifying invariance across time makes it so that if we can get from a to b, and we know we can get from b to c, then we know that we can get from a to c. but sometimes things come up on our way from a to b. that’s fine, but that lies outside of the boundary we draw of what we are going to consider. when we draw a boundary, we decide what compositions we care about, and what invariances we will trust.

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this “invariance” is a lot like reducing side effects in functional programming

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is there a connection here to probabilistic thinking?

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a fundamental question

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is this “composition” something that we are discovering in the world, or something we’ve invented?

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much like the question of whether mathematics is discovered or invented

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what would it look like to have a language for this universal phenomenon we call composition

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things

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transformations

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to be

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first, this language must be able to state that things exist

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dot with an arrow pointing to itself

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to transform

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in order for the language to represent more than a universe in complete stasis, we must support transformations

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two dots with an arrow going from one dot to the other

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examples

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mixing paints

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deniz is alive

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two eggs becomes an omelette

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but there might be many paths here

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two transformations, one after the other

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composition

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three transformations, done in any order

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associativity

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what are some interesting findings in this language

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isomorphism

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