From Parametricity to Conservation Laws,via Noether's Theorem

Written by Robert Atkeybob.atkey@gmail.com

Presented by Ben Carriel & Ben Greenman2014-04-07

PHYSICS

Distance is invariant under translation and change of coordinate representation.

Noether's Theorem

Noether's Theoremn (1915) "Any differentiable symmetry of the action

of a physical system has a corresponding conservation law"

Quick Example

invariant under translation

invariant under translation

invariant under translation

invariant under translation

(Noether's Theorem)

invariant under translation

(Noether's Theorem)

Conservation of Momentum

Pretty cool, right?

𝜏TYPES

λ f : (int -> int) ref . λ n : int . f := (λ acc : int ref . λ m : int . case (n = m : bool) of (acc := (mul !acc m); acc) : int (!f (acc := (mul !acc m); acc) (m+1)) : int ) (ref 1) 1) (ref λ x : int . x))

π ≜ ΛT1 . ΛT2 . ΛT3 . λ v : T1 × T2 × T3 . v[T1] (λ x : T1 . λ y : T2 . λz : T3 . x)

(λ x : unit . 42 : int)

Atkey (2014)

Atkey (2014)

n Define a type system for Lagrangian Mechanics.

Atkey (2014)

n Define a type system for Lagrangian Mechanics.

n Derive conservation laws as "free theorems" by parametricity.

Lagrangian:

Lagrangian:

Type:

Lagrangian:

Type:

Reference:

Lagrangian:

Type:

Reference:

Lagrangian:

Type:

Reference:for all translations y in one-dimensional space

Lagrangian:

Type:

Reference:for all translations y in one-dimensional space

Lagrangian:

Type:

Reference:for all translations y in one-dimensional spacetype for smooth functions between spaces

Lagrangian:

Type:

Reference:for all translations y in one-dimensional spacetype for smooth functions between spaces

Lagrangian:

Type:

Reference:for all translations y in one-dimensional spacetype for smooth functions between spacesreal numbers that vary with linear transformation g and translation f.

type for smooth functions between spaces

Lagrangian:

Type:

Reference:for all translations y in one-dimensional spacetype for smooth functions between spacesreal numbers that vary with linear transformation g and translation f.

type for smooth functions between spaces

Lagrangian:

Type:

Reference:for all translations y in one-dimensional spacetype for smooth functions between spacesreal numbers that vary with linear transformation g and translation f.

type for smooth functions between spaces

Lagrangian:

Type:

Reference:for all translations y in one-dimensional spacetype for smooth functions between spacesreal numbers that vary with linear transformation g and translation f.

type for smooth functions between spaces

Lagrangian:

Type:

Reference:for all translations y in one-dimensional spacetype for smooth functions between spacesreal numbers that vary with linear transformation g and translation f.

type for smooth functions between spaces

What's the type for (-)?

What's the type for (-)?

Type:

Reference:

What's the type for (-)?

Type:

Reference:

What's the type for (-)?

Type:

Reference:group of invertible real n×n matrices

What's the type for (-)?

Type:

Reference:group of invertible real n×n matrices(symmetries in )

What's the type for (-)?

Type:

Reference:group of invertible real n×n matrices(symmetries in )

What's the type for (-)?

Type:

Reference:group of invertible real n×n matrices

translations in n-dimensional space(symmetries in )

What's the type for (-)?

Type:

Reference:group of invertible real n×n matrices

translations in n-dimensional space(symmetries in )

What's the type for (-)?

Type:

Reference:group of invertible real n×n matrices

translations in n-dimensional space(symmetries in )

What's the type for (-)?

Type:

Reference:group of invertible real n×n matrices

translations in n-dimensional spacen-dimensional vectors of real numbers that vary with linear transformation g and translation f.

(symmetries in )

Why GL(n)?

Why GL(n)?

Why GL(n)?

"Give me a Lagrangian and a group action satisfying these constraints, I'll give you a conservation law."

Why GL(n)?

"Give me a Lagrangian and a group action satisfying these constraints, I'll give you a conservation law."

Key point: we need an automorphism (i.e. symmetry) to start with

Why GL(n)?

What does this mean?

Reynolds: types are relations

Reynolds: types are relations

Wadler: relations are free theorems

Reynolds: types are relations

Wadler: relations are free theorems

Reynolds: types are relations

Wadler: relations are free theorems

Atkey: free theorems are symmetries

Atkey gives us a geometric interpretation

of types

Atkey gives us a geometric interpretation

of typesWe'll argue: Atkey subsumes Reynolds + Wadler

Kinds are reflexive graphs

Kinds are reflexive graphs

n Reynolds: types are sets, parametricity comes from the relations between them.

Kinds are reflexive graphs

n Reynolds: types are sets, parametricity comes from the relations between them.

n Basic relation between Reynolds' types is the subset relation ( ⊆ ).

Kinds are reflexive graphs

n Reynolds: types are sets, parametricity comes from the relations between them.

n Basic relation between Reynolds' types is the subset relation ( ⊆ ).

n Form a graph where the objects are types and the edges order types by ⊆ .

Kinds are reflexive graphs

Example: bool

Example: bool

Example: bool

True False

Example: nat

Example: nat

0

Example: nat

0 1 2 3

Example: cartesian space ( )

Example: cartesian space ( )

Example: cartesian space ( )

Example: cartesian space ( )

Each arrow represents a family of diffeomorphisms

Example: cartesian space

Example: cartesian space

Example: cartesian space

Example: cartesian space

Example: cartesian space

Example: cartesian space

Free Theorems

Free Theorems

Give me a function

Free Theorems

Give me a function

Free Theorems

Give me a function

Then for every function

Free Theorems

Give me a function

Then for every function

Free Theorems

Give me a function

Then for every function

We have the free theorem

Free Theorems

Give me a function

Then for every function

We have the free theorem

Free Theorems

Free Theorems

Free Theorems

Free Theorems

Free Theorems

Free Theorems

Atkey: Main Points

n Extend System Fω with type system encoding geometric invariances.

n Interpret kinds as reflexive graphs, types as reflexive graph morphisms.

n Connect free theorems of Wadler/Reynolds with Noether's theorem via symmetries of these reflexive graphs.

Atkey: Takeawaysn Types as geometries is a powerful new way of

manipulating our "syntactic discipline".

n Visual intuition, connections to group theory.

n Physics is only one potential application!

:𝜏

:𝜏The End