COMPUTER SCIENCE DOMINIC VERITY CENTRE OF AUSTRALIAN CATEGORY THEORY - - PDF document

SLIDE 1

AN

EXPLORATION

OF

STRING

DIAGRAMS IN

COMPUTER SCIENCE

DOMINIC VERITY

CENTRE

OF AUSTRALIAN

CATEGORY THEORY

MACQUARIE UNIVERSITY

DRHE

WORKSHOP

UNIVERSITY OF NEWCASTLE 10k NOVEMBER 2018

SLIDE 2

VEIT

CATEGORY

OF

FINITE

DIMENSIONAL

IR

VECTOR SPACES

TENSOR

PRODUCT

A

VECTOR

space

• OCBASEVi.mg

ITS

DUAL Link

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isyT

v v

Yin

O

nzhm A

O

O

u v

Hi

9It

ImT

csu.it

v

IYiEuimM

SLIDE 3

V

nine i

mi

Imma

i

E

i i

wf

r

O

nE

Etnn

nTe aEa E

SLIDE 4

ANAXIOMATISATION

OFTRACE

G

I

SYMMETRC MONOIDAL CATEGORY

nine Time

Emma

MONOIDAL

ADD

AN

EXTRA

OPERATION ON ARROWS

t.tw

turns

xx

f

U W

V U

TraYuff

U

V

O

O

Tru.uitlomluxou.HU

Homl44uT

QUESTION

WHAT AXIOMS

SHOULD

THE

ABSTRACT TRACE OPERATION

SATISFY

SLIDE 5

ANSWER

AXIOMS

ARE

EASIER

TO

IMAGINE

IF

WE

THINK

DIAGRAMMATICALLY

got

f tru.VE

uMqvanou

t O

YANK

iotoo.mn

auuh

SLIDE 6

SUPERPOSITIONS

at t.IE I

w.xoxi.inos.ivmnnhL

TIGHTENINGI

I

natural

Tru

ulf.lg wD

TrjfftgyT

SLIDE 7

SUD

D NATURALITY

i

O

Truivlf.com

Tra

Y CVxoh f

yT

A

MONOIDAL

CATEGORY

WITH A

ABSTRACT

TRACE SATISFYING THESE

Atoms

Is

CALLED

A

TRACED MONOIDAL CATEGORY

SLIDE 8

ACACCULATIONEXAMPLET

i

EE't

SLIDE 9

SLIDING

it

to

PROOF

OF

COMPLETENESS

OF

THESE

AXIOMS

THE

Int CONSTRUCTION

A

GENERALISATION OF THE

CONSTRUCTION

OF

THE RATIONAL

NUMBERS

SLIDE 10

FLOWCHART

CONTROL

FLOW GRAPHS

t

T l

Google Blockly Flowchart View

SLIDE 11

ABACUSMAC

HINES

TURING

REVISITED

if

A

MACHINE

with

A COUNTABLY

INANITE SET OF

REGISTERS

EACH OF WHICH

CONTAINS A

NATURAL Number

• r

SET

OF NAMES

For

THEN

REGSTAS

ApiAbESuT

S Vor

IN

A

MACHINE STATE

STATES

INA

SET

• f

Possible

Mt me

STATES

SLIDE 12

ABAC.US

NSTRaCToNsX

2CtljGoTOT

REMENTuT

IFx

0Ge fix

c ijGoT

ANA

pray

9EHI

• mwtnhca.li

IF

2

0 Goto

3

ELSE

X

X

1

Coto 3

Z

Y

y

t 1

Goto

1

3

DEgERufIYfAf

SLIDE 13

NoN

DESTRUCTVEAD

1

IF

E

O

Goto

3

ELN

3

i

f

l j Gop 1

Z

IF

x

• Goto

5

ELSE

X

I

I j Goto 3

9

E

ft

I j

Got 4

4

Y

y

t I g Gop

3

S

IF

E

O

Goto 7

ELSE

2

i

f lj Goto 6

6

X

x

I j Got

S

F

i

This

is

ALREADY GETTING

A

LITTLE ONER

OUS

MAYBE

A

FLOWCHART

WILL

MAKE

THINGS

CLEARER

SLIDE 14

ti

i

MAKO

i

i.ee

nh

IFtq

z

c

• MOVE

K

z

EDE.to

i

i

c

fanion

4

i

u

Ox

I

1

E

i

E

O

i

s X

T

l

id

t

l

if

i

e

1

SLIDE 15

LA

OFFCOWGARTSBA.sk

BUILDING BLOCKS i

t

t

as

Slee

• yo

t

Oo

f

O

O

O

e.a.ir

1 EE.nuE

erhuuco

ia i

I

f

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I

tooo

f

O

CE.im

Iouuh.Eii7aEEIxsmTn

SLIDE 16

OPERATORS

11

Fso

E

i7nwM

ftp.oi

• n
• EnEoensT

i

• m.ca

e

no.mwn

SLIDE 17

Iliff

trees

FEF.EE

Enirmwinn

SLIDE 18

THIS

LANGUAGE

FREELY GENERANS

A

CATEGONI

LF

OBJECTS

NATURAL NUMBERS

n

a

Sloc

e

I

1

Cb

I

E

J

E

I

I

1

1

F

I

I

Gim

I ins

FjGin

r

F

I

I

G

I

aIi

FIG inHome

F

MI

turns

Trff

I

in

THIS IS SOME

KIND

OF

FORMAL

LANGUAGE OF

FLOWCHARTS

BUT

IT

LACKS SEMANTICS

WHAT

IS THE

FORMAL MEANING

OF THESE

LINGUISTIC

FLOWCHARTS

IN

TERMS

• f

MECHANICAL

COMPUTATION

ALGEBRA

WITH

WHICH

To PROVE THAT

Two

FLOWAUNTS DEANO

THE

SAME

COMPUTATION

SLIDE 19

THESEMANTICCIATEGORY

PAI

THE

CATEGORY OF

SETS

AND

PARTIAL FUNCTIONS 1

DIS JOINT

UNION

EMPTY

SET UNIT

for THIS

IS

A SYMMETRIC MONOIDAL CATEGORY

CRUCIALLY

Per

1

ADMITS A

COMPUTATIONALLY

IMPORTANT TRACE

f

Ut

W

Vtw Tra

f

a

v

IFF

F ko

an

5 I

U

Xo AND

V

xn

AND

Xi EU

eien

AND

f

Xi

Xiii

Osian

WE

CALL

Xo

pen

At Execution

trace oo7

f

EicTgI

SLIDE 20

f

n.si

t

igh

W

ITERATING

f

MAY

NEVER

LEAD

US TO V

EVEN

IF

f

IS

TOTAL

ITS

TRACE Tray f NEED NOT BE

ALSO

IF

f

IS

PARTIAL That

THE

Process

• f

ITERATING

IT

MAY GETStuck

SLIDE 21

PROPOSITI

THE ITERATION

TRACE

DEFINED

ABOVE

MAKES

Pad 1,0

INTO

A

SYMMETRIC

TRACED MONOIDAL CATEGORY I

THIS

TMC

IS

THE

UR

EXAMPLE

OF

AN

ITERATION

CATEGORY

SLIDE 22

THESEMANTICFUNCTOI

It

D

Lf

PAR

O

O

Ef

II.to

fE

w

IIe I

State

Stole

Stole

n

copies

Escoe

e 1

State

State

S

1

s

xi

EeDs

i

SLIDE 23

IICCHI

State

Stole

state

S

finds

IF 5lb

O in G

IF

s b

to

II

JD

State State

State

imacs

c

s

in Cs

1

S

IIF's GI

IIGIo

EFD

E

m

ai

n nu

IIF I GI

IIF I

t TIGA

II TRIFID

Triffema

IFI

WE SAY

THAT

This

SEMANTICS IS

DEFINED

COMPOSITIONALLY

SLIDE 24

Proposition

A

PARTIAL

FUNCTION

f

State

State

DESCRIBES THE INPUT OUTPUT BEHAVIOUR

OF

AN ABACUS

MACHINE

IFF THEN

EXISTS

SOME

F

I

1

IN Le

with

f LIFT

D

SLIDE 25

Proposition

ON

TAKING

EQUIVALENCE

CLASSES

OF

ARROWS

OF Lf

UNDER

THE

EQUIVALENCE

RELATION

F

n

G

EFI

EGI

WE

GET A CATEGORY Fter

THIS INHERITS

A TRACED

MONOIDAL

CATEGORY STRUCTURE

From Pad

WHICH

IS

COMPATIBLE

WITH

THE

INTENDED

MEANINGS

• f THE

combinations

FjG

FIG

AND Trt

D

THIS IS

THE TMC

OF

FLOWCHARTS

FOR

THIS

NOTION

OF COMPUTATION

SLIDE 26

WE

MIGHT

NOW

IDGITIFY

FLOWCHART

DIA

Arms

WITH

DIAartms

IN

THE

TMC Flow

MANY ALGEBRAIC

PROPERTIES

OF

FLOWCHARTS

MAY

BE

DERIVED From

THE

TMC

FORMALISM

Flowchthts

THAT ARE SHOWN

TO

BE

Earth

USING

TMC

DIAGRAMMATIC REASONING

DEFINE

THE SAME

MACHINE

SLIDE 27

EXAMPLE

t

An

ai

c

F

AND G Don'T

v

INVOLVE

Air

e

O

Y

SLIDE 28

SOME

DERIVABLE

COMPUTATONAL

PRINCIPLES

got

I ah 9 IE

i

O

SLIDE 29

t

II

COMPILER

WRITERS

CALL

THIS

LAST

EQUATION

DEAD

CODE ELIMINATION

SLIDE 30

FINALLY

ARGUMENT BY CASES

IF

1

HIEI is ionising

VALUE

OF

b

ftp.t.J

A

YEE

SLIDE 31

THE

i

I

IN

the

D

Te

SLIDE 32

whim

i

i

1

4

I

Iif

t

SLIDE 33

DUALLY

WHEN

b fo

OUR

FCowanest

REDUCES

T

THE

t

SO

APPLYING THE

ARGUMENT BY

CASES

RULE

OUR

ORIGINAL

FLOWCHART

REDUCES

To

i

c

NOTE

THE ORIGINAL

FLOWCHART

IN Vocab

t

U

GIVE

RISE TO A

J

ITERATIVE ALGORITHM

Lf

TALSETEEDBA.CI

SLIDE 34

ARMAN

MARTIN MATHIESEN TOLNA

USE

This FRAMEWORK

to

show THAT

FLOYD

HOARE

LOGS

ARISE From

CERTAIN TRACE

PRESERVING Funigors

A GENERAL FRAMEWORK FOR

SOUND

COMPLETE FLOYD HOARE Locks

ACM TRANSACTIONS

OCT2009

7

HYLAND PROVIDES AN

ABSTRACT

Proof

• f

KLEENE'S THEOREM

IN

TERMS

OF

A TRACE PRESERVING

FUNCTOR FROM

FINITE

STATE AUTOMATA

TO

REGULAR LANGUAGES

ABSTRACT

CONCRETE

MODELS

FOR RECURSION

8008 GHICA JUNG

VARIOUS

APPLICATIONS TO

THE MODELLING

OF

DIGITAL

CIRCUITS

THE GEOMETRY

OF SYNTHESIS