Applied category theory @KenScambler The emerging science of - - PowerPoint PPT Presentation

Applied category theory


The emerging science of compositionality

@KenScambler

Category theory

From “Seven Sketches of Compositionality” Fong & Spivak (2018)

Applied category theory

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make lemon filling

fill crust make meringue

separate egg

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add meringue

Applied category theory

• Emerging field!
• Explosion of papers, research, development in last 5

years

• Builds a common vocabulary of composition across

disciplines

• Enables intuitive, precise visual representations

Brand new ACT Journal (2019?)

Brand new ACT Book (2018)

ACT Topics & papers

Q: Is functional programming applied category theory?

Q: Is functional programming applied category theory? A: Yes! (…arguably)

Plan

• I want to put applied category theory on your roadmap
• The big idea: compositionality
• Case study: Complex adaptive system design: military search & rescue
• Four concepts of composition:
• Monoids
• Categories
• Operads
• Monoidal categories
• Two graphical languages:
• String diagrams
• Wiring diagrams

Compositionalit y

The meaning of a complex expression is determined by:


• 1. The meaning of the constituent parts
• 2. The rules for how those parts are combined

Frege’s Principle of Compositionality

Making big things

Compositionality lets us

Making big things
 


• ut of little things

Compositionality lets us

Compositionality lets us

Making big things
 


• ut of little things


preserving guarantees

Compositionality applied

• Little programs that work 


BIG programs that work?


• Little electric circuits that work


BIG electric circuits that work?


• Little Feynman diagrams that describe real physics 


BIG Feynman diagrams that describe real physics?


• Little resource plans that are feasible 


BIG resource plans that are feasible?


• Little networks of agents that can solve problems 


BIG networks of agents that can solve problems?

Compositionality distilled - monoids

c d e f h j a b i g

a b

a b Compose any two

ab

Associativity – no need for brackets

a b c

Associativity – no need for brackets

a b c

Associativity – no need for brackets

a b c

Associativity – no need for brackets

a b c

Associativity – no need for brackets

abc

With associativity

abc

=

a b c Just the sum of its parts with

Without associativity

+

• r

abc

=

a b c Uh-oh! We need external data

Identity – doesn’t do anything when you compose

a

Identity – doesn’t do anything when you compose

a

Identity – doesn’t do anything when you compose

a

Compositionality distilled - categories

c d e f h j a b i g

a b The building blocks are called morphisms The colours are called objects

a b Can only compose when the colours match

ab The middle bit disappears

a b

Associativity – no need for brackets

c

a b

Associativity – no need for brackets

c

a b

Associativity – no need for brackets

c

a b

Associativity – no need for brackets

c

abc

Associativity – no need for brackets

a

Identities – don’t do anything when you compose

a

Identities – don’t do anything when you compose

a

Identities – don’t do anything when you compose

Case study: Complex Adaptive System Design

from 
 “Complex Adaptive System Design” blog posts by John C. Baez (2016-2019) “Network Models” by Baez, Foley, Moeller, Pollard (2017)

Military research: search & rescue

• Systems of systems
• Many different kinds of networks
• Complex realtime behaviour
• How to connect them all?

Helo 1

Target 1

Port 1

Helicopter knows the target and home base

In comms range In comms range My port Intent to rescue

Helo 1 UAV 1

Target 1

UAV knows the target, and the helicopter as a home base

In comms range Intent to rescue My port

Overlay the networks We now have a bigger network working together

Helo 1

Target 1

Port 1 My port Intent to rescue UAV 1 My port In comms range In comms range In comms range Intent to rescue

Helo network UAV network Overlaying is monoidal; you can always overlay

Helo network UAV network Overlaying is monoidal; you can always overlay

• verl

ay

Combined network Overlaying is monoidal; you can always overlay

How to sew together unrelated networks?

Helo 1 UAV 1 Target 1 UAV 2 UAV 3 UAV 5

Rescue-Scene

UAV 4 In comms range In comms range

UAV-Search-Net

Helo 2

Homebase

Intent to rescue My port Havana

1 2 3 4 5 6 7 8 9

How to sew together unrelated networks?

“Take 3 networks of 3, 4 and 2 vertices; return a network of 9 vertices”

Intent to rescue

Helo 1 UAV 1 Target 1 UAV 2 UAV 3 UAV 5 UAV 4 In comms range In comms range Helo 2 In comms range Intent to rescue My port Intent to rescue Havana

Is that a category?

Network-1 Network-2 Network-3

Nope! You can’t compose the networks themselves. Network-1 Network-2 Network-3

Morphisms with lots of inputs?

The ”type” is the number of vertices expected in subnetworks “Take 3 networks of 3, 4 and 2 vertices; return a network of 9 vertices”

3 4 2 9

Compositionality distilled – operads

a c b d

a c b

a c b

a c b

a c b

abc

a

a

a

a

Obvious example: types & functions

receiveInvoice

User Invoice Signature Receipt

storeReceipt

Receipt Shoebox Shoebox

login

Username Password User

receiveInvoice

Invoice Signature

storeReceipt

Shoebox Shoebox

login

Username Password

Invoice Signature

fullInvoiceFlow

Shoebox Shoebox Username Password


 def fullInvoiceFlow(username,
 password, invoice, signature, shoebox) {
 user = login(username, password) receipt = receiveInvoice(user, invoice, signature) return storeReceipt(receipt, shoebox) }


Mapping it to reality

Helicopter Port UAV Target my port Intent to rescue Comms channel

Havana

Rescue- Scene UAV-Search-Net Homebase

Mapping it to reality

Helicopter Port UAV Target my port Intent to rescue Comms channel

Havana

Rescue-Scene := ...
 UAV-Search-Net := ...
 Homebase := ... 
 joinNetworks(Rescue-Scene, 
 UAV-Search-Net, 
 Homebase) To actually use an operad, translate it to the operad of sets & functions, preserving composition & identity

From “Seven Sketches of Compositionality” Fong & Spivak (2018)

String diagrams

prepare lemon meringue pie

make lemon filling

fill crust make meringue

separate egg

prepared crust lemon butter sugar yolk egg white sugar meringue lemon filling unbaked lemon pie unbaked pie

add meringue

prepare lemon meringue pie

make lemon filling

fill crust make meringue

separate egg

prepared crust lemon butter sugar yolk egg white sugar meringue lemon filling unbaked lemon pie unbaked pie

add meringue

morphisms

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make lemon filling

fill crust make meringue

separate egg

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add meringue

• bjects

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separate egg

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composition

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make lemon filling

fill crust make meringue

separate egg

prepared crust lemon butter sugar yolk egg white sugar meringue lemon filling unbaked lemon pie unbaked pie

add meringue

tensoring

Tensoring forms a monoid (up to isomorphism)

lemon butter sugar make 
 lemon filling

… on the objects of the category

lemon ⨂ butter ⨂ sugar lemon filling

Monoidal categories

Category C
 Tensor operation 
 ⨂ : (object, object) -> object such that (a ⨂ b) ⨂ c ≃ a ⨂ (b ⨂ c) Unit object I : object such that a ⨂ I ≃ I ⨂ a ≃ a

Monoidal categories

f g

x y y z

f g

x y z

f g

x y y z x y y z

f ⨂ g

= Horizontal composition Vertical composition

x ⨂ y

f ⨂ g

=

y ⨂ z

Diagrams inside diagrams

prepare lemon meringue pie

make lemon filling

fill crust make meringue

separate egg

prepared crust lemon butter sugar yolk egg white sugar meringue lemon filling unbaked lemon pie unbaked pie

add meringue Make lemon filling

lemon butter sugar

Blend

yolk

Beat Combine

lemon filling

Diagrams inside diagrams

prepare lemon meringue pie fill crust make meringue

separate egg

prepared crust lemon butter sugar yolk egg white sugar meringue lemon filling unbaked lemon pie unbaked pie

add meringue

Blend Beat Combine

• ven

unbaked pie

baked pie

• ven

prepare lemon meringue pie bake pie

prepare lemon meringue pie

• ven

unbaked pie

baked pie

• ven

bake pie

Surprise! It’s another operad

Lemon Meringue Pie operad

prepare lemon meringue pie

sugar ⨂ white meringue prepared crust ⨂ lemon ⨂ butter ⨂ sugar ⨂ egg ⨂ sugar unbaked pie egg yolk ⨂ white lemon ⨂ butter ⨂ sugar ⨂ yolk lemon filling prepared crust ⨂ lemon filling unbaked lemon pie unbaked lemon pie ⨂ meringue unbaked pie

String diagrams in software architecture

• Fantastic for software design
• I’ve used them for years to train junior & senior developers
• Modularity, boundaries

Simon Brown

Level 1

Level 1

Level 2

Level 1

Level 2

Level 3

Level 1

Level 2

Level 3

Level 4 Operad! (maybe)

Operad of wiring diagrams

from 
 “The operad of wiring diagrams: formalizing a graphical language for databases, recursion and plug-and-play circuits” by David I. Spivak (2013)

Inputs x1, x2, x3

Input types {s,r,t} {w, x, y, z} {u, v} Output type {a,b,c,d,e}

Compositional SQL queries

SELECT L.student, L.address FROM attends a1, gender g1, attends a2, gender g2, lives L WHERE a1.student=g1.student AND a2.student=g2.student AND L.student=g1.student AND a1.course=a2.course AND g1.gender=‘male’ AND g2.gender=‘female’

SELECT L.student, L.address FROM attends a1, gender g1, attends a2, gender g2, lives L WHERE a1.student=g1.student AND a2.student=g2.student AND L.student=g1.student AND a1.course=a2.course AND g1.gender=‘male’ AND g2.gender=‘female’

SELECT L.student, L.address FROM attends a1, gender g1, attends a2, gender g2, lives L WHERE a1.student=g1.student AND a2.student=g2.student AND L.student=g1.student AND a1.course=a2.course AND g1.gender=‘male’ AND g2.gender=‘female’

Tables Constants

SELECT L.student, L.address FROM attends a1, gender g1, attends a2, gender g2, lives L WHERE a1.student=g1.student AND a2.student=g2.student AND L.student=g1.student AND a1.course=a2.course AND g1.gender=‘male’ AND g2.gender=‘female’

SELECT

SELECT L.student, L.address FROM attends a1, gender g1, attends a2, gender g2, lives L WHERE a1.student=g1.student AND a2.student=g2.student AND L.student=g1.student AND a1.course=a2.course AND g1.gender=‘male’ AND g2.gender=‘female’

WHERE 
 (wiring)

In conclusion

• Applied category theory is an exciting new field that seeks to

mine the benefits of compositionality wherever they may be found

• Supports marvelous graphical languages
• Early days!
• Watch this space

The future

• Anything where “making big things out of little things” is an

important part

• So far: category theorists discovering applications and

promoting

• Industry might eventually begin to lead the charge
• Computer science & programming will be at the forefront:

software has eaten the world

• Functional programming will be at the forefront: we are early

adopters

People to follow on Twitter

John C. Baez
 @johncarlosbae z Jules Hedges
 @_julesh_

• Dr. Eugenia Cheng


@DrEugeniaCheng Jade Master
 @JadeMasterMath Tai-Danae Bradley
 @math3ma Joe Moeller
 @CreeepyJoe Statebox
 @statebox Jelle Herold
 @wires_wires Fabrizio Remano Genovese
 @fabgenoves e

Further reading

• “Seven Sketches of Compositionality” by Brendan Fong, David I. Spivak

(2018)

• “Applied category theory course” by John C. Baez (2018)
• “What is applied category theory?” by Tai-Danae Bradley (2018)
• “Network models” by Baez, Foley, Moeller, Pollard (2017)
• “The operad of wiring diagrams: formalizing a graphical language for

databases, recursion and plug-and-play circuits” by David I. Spivak (2013)