Coalgebra Love and Beauty in Science Ana Sokolova Logic Mentoring - - PowerPoint PPT Presentation

Coalgebra Love and Beauty in Science

Ana Sokolova

Logic Mentoring Workshop 2019

Do you know any coalgebra?

Ana Sokolova LMW 2019 22-6-19

Do you know any coalgebra?

Ana Sokolova

Yes, you know many coalgebras !

LMW 2019 22-6-19

Some coalgebras

X

c

Ñ FX

Ana Sokolova LMW 2019 22-6-19

Some coalgebras

X

c

Ñ FX

Ana Sokolova

NFA X ➝ 2 x (PX)A

x1

a

|

a

" x2 ✏ x3

b

k ˚

LMW 2019 22-6-19

Some coalgebras

X

c

Ñ FX

Ana Sokolova

NFA X ➝ 2 x (PX)A

x1

a

|

a

" x2 ✏ x3

b

k ˚

MC X ➝ DX + 1

x1

1 2



1 2

• x2

1 3 ✏ 2 3

/ x3

1

✏ x4 ✏ x5 ✏ ˚ ˚

LMW 2019 22-6-19

Some coalgebras

X

c

Ñ FX PA X ➝ (PDX)A

x1

a

|

a

"

b

• 2

3

|

1 3" 1 2" 1 2

| x2

a

3 x3

b

k x4

b

k

Ana Sokolova

NFA X ➝ 2 x (PX)A

x1

a

|

a

" x2 ✏ x3

b

k ˚

MC X ➝ DX + 1

x1

1 2



1 2

• x2

1 3 ✏ 2 3

/ x3

1

✏ x4 ✏ x5 ✏ ˚ ˚

LMW 2019 22-6-19

Some coalgebras

X

c

Ñ FX

Various transitions systems / automata are coalgebras

PA X ➝ (PDX)A

x1

a

|

a

"

b

• 2

3

|

1 3" 1 2" 1 2

| x2

a

3 x3

b

k x4

b

k

Ana Sokolova

NFA X ➝ 2 x (PX)A

x1

a

|

a

" x2 ✏ x3

b

k ˚

MC X ➝ DX + 1

x1

1 2



1 2

• x2

1 3 ✏ 2 3

/ x3

1

✏ x4 ✏ x5 ✏ ˚ ˚

LMW 2019 22-6-19

Where do they live ?

X

c

Ñ FX

Ana Sokolova LMW 2019 22-6-19

Where do they live ?

X

c

Ñ FX

Ana Sokolova

3D Organ Model 2D Tissue Model

LMW 2019 22-6-19

Where do they live ?

X

c

Ñ FX

Ana Sokolova

3D Organ Model 2D Tissue Model

LMW 2019 22-6-19

Bartocci et al. TCS09, CAV11

Where do they live ?

X

c

Ñ FX

Ana Sokolova

3D Organ Model 2D Tissue Model

Verification requires clear semantics

LMW 2019 22-6-19

Bartocci et al. TCS09, CAV11

Where do they live ?

X

c

Ñ FX

Ana Sokolova

3D Organ Model 2D Tissue Model

Verification requires clear semantics and suffers from state-space explosion

LMW 2019 22-6-19

Bartocci et al. TCS09, CAV11

Behavioural Equivalences

„ „

Ana Sokolova

NFA X ➝ 2 x (PX)A

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

NFA X ➝ 2 x (PX)A

language equivalence

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

NFA X ➝ 2 x (PX)A

Two states are equivalent iff the languages recognised from these two states are the same.

language equivalence

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

NFA X ➝ 2 x (PX)A

Two states are equivalent iff the languages recognised from these two states are the same.

language equivalence bisimilarity

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

NFA X ➝ 2 x (PX)A

Two states are equivalent iff the languages recognised from these two states are the same.

language equivalence bisimilarity

An equivalence relation R Ñ X ˆ X is a bisimulation of the NFA po, nq: X Ñ 2 ˆ pPXqA iff whenever px, yq P R, we have opxq “ opyq and for all a P A x

a

Ñ x1 ñ Dy1.y

a

Ñ y1 ^ px1, y1q P R. Bisimilarity, denoted by „, is the largest bisimulation.

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

NFA X ➝ 2 x (PX)A

Two states are equivalent iff the languages recognised from these two states are the same.

language equivalence bisimilarity

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

NFA X ➝ 2 x (PX)A

Two states are equivalent iff the languages recognised from these two states are the same.

language equivalence bisimilarity

R

bisimulation

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

NFA X ➝ 2 x (PX)A

Two states are equivalent iff the languages recognised from these two states are the same.

language equivalence bisimilarity

R

bisimulation

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

NFA X ➝ 2 x (PX)A

Two states are equivalent iff the languages recognised from these two states are the same.

language equivalence bisimilarity

R

bisimulation

a

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

NFA X ➝ 2 x (PX)A

Two states are equivalent iff the languages recognised from these two states are the same.

language equivalence bisimilarity

R

bisimulation

a a

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

NFA X ➝ 2 x (PX)A

Two states are equivalent iff the languages recognised from these two states are the same.

language equivalence bisimilarity

R

bisimulation

R

a a

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

NFA X ➝ 2 x (PX)A

Two states are equivalent iff the languages recognised from these two states are the same.

language equivalence bisimilarity

R

bisimulation largest bisimulation

R

a a

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

MC X ➝ DX + 1

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1

An equivalence relation R Ñ X ˆX is a bisimulation of the MC c: X Ñ DX `1 iff whenever px, yq P R, then either cpxq “ cpyq “ ˚ or for all R-equivalence classes C we have ÿ

zPC

cpxqpzq “ ÿ

zPC

cpyqpzq. Bisimilarity, denoted by „, is the largest bisimulation.

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1 Why are they both called bisimilarity ?

An equivalence relation R Ñ X ˆX is a bisimulation of the MC c: X Ñ DX `1 iff whenever px, yq P R, then either cpxq “ cpyq “ ˚ or for all R-equivalence classes C we have ÿ

zPC

cpxqpzq “ ÿ

zPC

cpyqpzq. Bisimilarity, denoted by „, is the largest bisimulation.

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1 What do they have in common ?

An equivalence relation R Ñ X ˆX is a bisimulation of the MC c: X Ñ DX `1 iff whenever px, yq P R, then either cpxq “ cpyq “ ˚ or for all R-equivalence classes C we have ÿ

zPC

cpxqpzq “ ÿ

zPC

cpyqpzq. Bisimilarity, denoted by „, is the largest bisimulation.

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1

R

bisimulation

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1

R

bisimulation

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1

μ

R

bisimulation

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1

μ ৵

R

bisimulation

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1

”R

μ ৵

R

bisimulation

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1

”R

μ ৵ lifting of R to distributions

R

bisimulation

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1

”R

μ ৵ lifting of R to distributions assign the same probability to “R-classes”

R

bisimulation

LMW 2019 22-6-19

Behavioural Equivalences

„ „

Ana Sokolova

bisimilarity

MC X ➝ DX + 1

”R

μ ৵ lifting of R to distributions assign the same probability to “R-classes”

R

bisimulation largest bisimulation

LMW 2019 22-6-19

Coalgebra

Uniform framework for dynamic transition systems, based on category theory.

Ana Sokolova

A coalgebra is generic transition system:

LMW 2019 22-6-19

Coalgebra

Uniform framework for dynamic transition systems, based on category theory.

X

c

Ñ FX

Ana Sokolova

A coalgebra is generic transition system:

LMW 2019 22-6-19

Coalgebra

Uniform framework for dynamic transition systems, based on category theory.

X

c

Ñ FX

states

Ana Sokolova

A coalgebra is generic transition system:

LMW 2019 22-6-19

Coalgebra

Uniform framework for dynamic transition systems, based on category theory.

X

c

Ñ FX

states

• bject in the base

category C

Ana Sokolova

A coalgebra is generic transition system:

LMW 2019 22-6-19

Coalgebra

Uniform framework for dynamic transition systems, based on category theory.

X

c

Ñ FX

states

• bject in the base

category C behaviour type

Ana Sokolova

A coalgebra is generic transition system:

LMW 2019 22-6-19

Coalgebra

Uniform framework for dynamic transition systems, based on category theory.

X

c

Ñ FX

states

• bject in the base

category C behaviour type functor on the base category C

Ana Sokolova

A coalgebra is generic transition system:

LMW 2019 22-6-19

Coalgebra

Uniform framework for dynamic transition systems, based on category theory.

X

c

Ñ FX

states

• bject in the base

category C behaviour type functor on the base category C form a category too

Ana Sokolova

A coalgebra is generic transition system:

LMW 2019 22-6-19

Coalgebra

Uniform framework for dynamic transition systems, based on category theory.

X

c

Ñ FX

states

• bject in the base

category C behaviour type functor on the base category C form a category too CoAlgCpFq

Ana Sokolova

A coalgebra is generic transition system:

LMW 2019 22-6-19

Coalgebra

Uniform framework for dynamic transition systems, based on category theory.

X

c

Ñ FX

generic notion of behavioural equivalence

«

states

• bject in the base

category C behaviour type functor on the base category C form a category too CoAlgCpFq

Ana Sokolova

A coalgebra is generic transition system:

LMW 2019 22-6-19

The category of F-coalgebras

Objects = coalgebras

Ana Sokolova

Arrows = coalgebra homomorphisms

CoAlgCpFq

Two states x, y P X are behaviourally equivalent, notation x « y iff there exists a coalgebra homomorphism h: X Ñ Y from c: X Ñ FX to some coalgebra d: Y Ñ FY such that hpxq “ hpyq.

LMW 2019 22-6-19

The category of F-coalgebras

Objects = coalgebras

X

c

Ñ FX

Ana Sokolova

Arrows = coalgebra homomorphisms

CoAlgCpFq

Two states x, y P X are behaviourally equivalent, notation x « y iff there exists a coalgebra homomorphism h: X Ñ Y from c: X Ñ FX to some coalgebra d: Y Ñ FY such that hpxq “ hpyq.

LMW 2019 22-6-19

The category of F-coalgebras

Objects = coalgebras

X

c

Ñ FX

Ana Sokolova

Arrows = coalgebra homomorphisms

CoAlgCpFq behaviour- preserving maps

Two states x, y P X are behaviourally equivalent, notation x « y iff there exists a coalgebra homomorphism h: X Ñ Y from c: X Ñ FX to some coalgebra d: Y Ñ FY such that hpxq “ hpyq.

LMW 2019 22-6-19

The category of F-coalgebras

Objects = coalgebras

X

c

Ñ FX

Ana Sokolova

Arrows = coalgebra homomorphisms

CoAlgCpFq

h: X Ñ Y

X

h

/

cX ✏

Y

cY

✏ FX

F h

/ FY behaviour- preserving maps

Two states x, y P X are behaviourally equivalent, notation x « y iff there exists a coalgebra homomorphism h: X Ñ Y from c: X Ñ FX to some coalgebra d: Y Ñ FY such that hpxq “ hpyq.

LMW 2019 22-6-19

The category of F-coalgebras

Objects = coalgebras

X

c

Ñ FX

Ana Sokolova

Arrows = coalgebra homomorphisms

CoAlgCpFq

h: X Ñ Y

X

h

/

cX ✏

Y

cY

✏ FX

F h

/ FY behaviour- preserving maps

Two states x, y P X are behaviourally equivalent, notation x « y iff there exists a coalgebra homomorphism h: X Ñ Y from c: X Ñ FX to some coalgebra d: Y Ñ FY such that hpxq “ hpyq.

LMW 2019 22-6-19

Behavioural equivalence is bisimilarity

X

c

Ñ FX

Ana Sokolova LMW 2019 22-6-19

Behavioural equivalence is bisimilarity

X

c

Ñ FX

Ana Sokolova

NFA X ➝ 2 x (PX)A

x1

a

|

a

" x2 ✏ x3

b

k ˚

LMW 2019 22-6-19

Behavioural equivalence is bisimilarity

X

c

Ñ FX

Ana Sokolova

NFA X ➝ 2 x (PX)A

x1

a

|

a

" x2 ✏ x3

b

k ˚

„ “ «

LMW 2019 22-6-19

Behavioural equivalence is bisimilarity

X

c

Ñ FX

Ana Sokolova

NFA X ➝ 2 x (PX)A

x1

a

|

a

" x2 ✏ x3

b

k ˚

MC X ➝ DX + 1

x1

1 2



1 2

• x2

1 3 ✏ 2 3

/ x3

1

✏ x4 ✏ x5 ✏ ˚ ˚

„ “ «

LMW 2019 22-6-19

Behavioural equivalence is bisimilarity

X

c

Ñ FX

Ana Sokolova

NFA X ➝ 2 x (PX)A

x1

a

|

a

" x2 ✏ x3

b

k ˚

MC X ➝ DX + 1

x1

1 2



1 2

• x2

1 3 ✏ 2 3

/ x3

1

✏ x4 ✏ x5 ✏ ˚ ˚

„ “ « „ “ «

LMW 2019 22-6-19

Behavioural equivalence is bisimilarity

X

c

Ñ FX PA X ➝ (PDX)A

x1

a

|

a

"

b

• 2

3

|

1 3" 1 2" 1 2

| x2

a

3 x3

b

k x4

b

k

Ana Sokolova

NFA X ➝ 2 x (PX)A

x1

a

|

a

" x2 ✏ x3

b

k ˚

MC X ➝ DX + 1

x1

1 2



1 2

• x2

1 3 ✏ 2 3

/ x3

1

✏ x4 ✏ x5 ✏ ˚ ˚

„ “ « „ “ «

LMW 2019 22-6-19

Behavioural equivalence is bisimilarity

X

c

Ñ FX PA X ➝ (PDX)A

x1

a

|

a

"

b

• 2

3

|

1 3" 1 2" 1 2

| x2

a

3 x3

b

k x4

b

k

Ana Sokolova

NFA X ➝ 2 x (PX)A

x1

a

|

a

" x2 ✏ x3

b

k ˚

MC X ➝ DX + 1

x1

1 2



1 2

• x2

1 3 ✏ 2 3

/ x3

1

✏ x4 ✏ x5 ✏ ˚ ˚

„ “ « „ “ « „ “ «

LMW 2019 22-6-19

Behavioural equivalence is bisimilarity

X

c

Ñ FX PA X ➝ (PDX)A

x1

a

|

a

"

b

• 2

3

|

1 3" 1 2" 1 2

| x2

a

3 x3

b

k x4

b

k

Ana Sokolova

NFA X ➝ 2 x (PX)A

x1

a

|

a

" x2 ✏ x3

b

k ˚

MC X ➝ DX + 1

x1

1 2



1 2

• x2

1 3 ✏ 2 3

/ x3

1

✏ x4 ✏ x5 ✏ ˚ ˚

• n

Sets

„ “ « „ “ « „ “ «

LMW 2019 22-6-19

Isn’t that beautiful ?

Ana Sokolova LMW 2019 22-6-19

Isn’t that beautiful ?

Ana Sokolova LMW 2019 22-6-19

if yes, read Rutten and Jacobs!

Isn’t that beautiful ?

Ana Sokolova LMW 2019 22-6-19

if yes, read Rutten and Jacobs! and come to my talk tomorrow at WiL

Isn’t that beautiful ?

Ana Sokolova LMW 2019 22-6-19

if yes, read Rutten and Jacobs! and come to my talk tomorrow at WiL and to the talk of

• ur LICS paper on

Wednesday

Ana Sokolova LMW 2019 23-6-19

Ana Sokolova LMW 2019 23-6-19

Ana Sokolova

Beyond coalgebra

LMW 2019 22-6-19

Ana Sokolova

What is the best about doing science ?

Beyond coalgebra

LMW 2019 22-6-19

What I love about doing science

Ana Sokolova

challenging creativity freedom novelty work with people work alone beauty elegance striving for perfection discovering explaining communicating it’s rewarding relevance joy meaningful

• pen

integrity community

LMW 2019 22-6-19

Let’s bring some order here

Ana Sokolova

freedom beauty

challenging

striving for perfection

joy communicating

with people

alone

meaningful

relevant

• pen

LMW 2019 22-6-19

Let’s bring some order here

Ana Sokolova

freedom beauty

challenging

striving for perfection

joy communicating

with people

alone

meaningful

relevant

• pen

LMW 2019 22-6-19

Let’s bring some order here

Ana Sokolova

freedom beauty

challenging

striving for perfection

joy communicating

with people

alone

meaningful

relevant

• pen

LMW 2019 22-6-19

Let’s bring some order here

Ana Sokolova

freedom beauty

challenging

striving for perfection

joy communicating

with people

alone

meaningful

relevant

• pen

LMW 2019 22-6-19

Let’s bring some order here

Ana Sokolova

freedom beauty

challenging

striving for perfection

joy communicating

with people

alone

meaningful

relevant

• pen

LMW 2019 22-6-19

Let’s bring some order here

Ana Sokolova

freedom beauty

challenging

striving for perfection

joy communicating

with people

alone

meaningful

relevant

• pen

LMW 2019 22-6-19

• Do what you do best and love
• Choose relevant topics
• Mix topics
• Learn from masters
• Dare to be independent
• Leave a trace
• There is plenty to learn: Exchange roles
• Do not worry much
• Have fun

How to survive the PhD time and further until tenure ?

Ana Sokolova LMW 2019 22-6-19

• Do what you do best and love
• Choose relevant topics
• Mix topics
• Learn from masters
• Dare to be independent
• Leave a trace
• There is plenty to learn: Exchange roles
• Do not worry much
• Have fun

How to survive the PhD time and further until tenure ?

Ana Sokolova LMW 2019 22-6-19

• Do what you do best and love
• Choose relevant topics
• Mix topics
• Learn from masters
• Dare to be independent
• Leave a trace
• There is plenty to learn: Exchange roles
• Do not worry much
• Have fun

How to survive the PhD time and further until tenure ?

Ana Sokolova LMW 2019 22-6-19

• Do what you do best and love
• Choose relevant topics
• Mix topics
• Learn from masters
• Dare to be independent
• Leave a trace
• There is plenty to learn: Exchange roles
• Do not worry much
• Have fun

How to survive the PhD time and further until tenure ?

Ana Sokolova LMW 2019 22-6-19

• Do what you do best and love
• Choose relevant topics
• Mix topics
• Learn from masters
• Dare to be independent
• Leave a trace
• There is plenty to learn: Exchange roles
• Do not worry much
• Have fun

How to survive the PhD time and further until tenure ?

Ana Sokolova LMW 2019 22-6-19

• Do what you do best and love
• Choose relevant topics
• Mix topics
• Learn from masters
• Dare to be independent
• Leave a trace
• There is plenty to learn: Exchange roles
• Do not worry much
• Have fun

How to survive the PhD time and further until tenure ?

Ana Sokolova LMW 2019 22-6-19