Inductive Types for Free Representing Nested Inductive Types using - - PowerPoint PPT Presentation

Inductive Types for Free

Representing Nested Inductive Types using W-types

Michael Abbott (U. Leicester) Thorsten Altenkirch (U. Nottingham) Neil Ghani (U. Leicester)

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Ideology

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Ideology

David Turner: Elementary Strong Functional Programming

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Ideology

David Turner: Elementary Strong Functional Programming Martin-Löf Type Theory

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Ideology

David Turner: Elementary Strong Functional Programming Martin-Löf Type Theory Types = sets, programs = total functions.

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Ideology

David Turner: Elementary Strong Functional Programming Martin-Löf Type Theory Types = sets, programs = total functions. Dependent types to avoid accidental partiality (e.g. hd).

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Ideology

David Turner: Elementary Strong Functional Programming Martin-Löf Type Theory Types = sets, programs = total functions. Dependent types to avoid accidental partiality (e.g. hd). E.g.: Conor McBride’s Epigram system.

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Plan of the talk

Inductive and coinductive types. Container types for dummies. Properties of container types. W-types are sufficent for inductive types. Further work and applications Related work

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Inductive and coinductive types

Widespread in functional programming (e.g. Haskell)

Lam

Int

Lam Lam

Lam

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Inductive and coinductive types

Widespread in functional programming (e.g. Haskell)

Lam

Int

Lam Lam

Lam Categorically: Initial algebra of a functor Lam

• ✁

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vs.

• In a total setting
• ☎

(terminal coalgebras):

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vs.

• In a total setting
• ☎

(terminal coalgebras):

• ✁

finite lists

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vs.

• In a total setting
• ☎

(terminal coalgebras):

• ✁

finite lists

potentially infinite lists

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vs.

• In a total setting
• ☎

(terminal coalgebras):

• ✁

finite lists

potentially infinite lists

infinite lists

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vs.

• In a total setting
• ☎

(terminal coalgebras):

• ✁

finite lists

potentially infinite lists

infinite lists

• ✁

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vs.

• In a total setting
• ☎

(terminal coalgebras):

• ✁

finite lists

potentially infinite lists

infinite lists

• ✁

• Inductive Types for Free – p.5/22

vs.

• In a total setting
• ☎

(terminal coalgebras):

• ✁

finite lists

potentially infinite lists

infinite lists

• ✁

• ✁

can be nested

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vs.

• In a total setting
• ☎

(terminal coalgebras):

• ✁

finite lists

potentially infinite lists

infinite lists

• ✁

• ✁

can be nested

• ✁

• ✁
• ✁

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vs.

• In a total setting
• ☎

(terminal coalgebras):

• ✁

finite lists

potentially infinite lists

infinite lists

• ✁

• ✁

can be nested

• ✁

• ✁
• ✁

• ✁

Inductive Types for Free – p.5/22

vs.

• In a total setting
• ☎

(terminal coalgebras):

• ✁

finite lists

potentially infinite lists

infinite lists

• ✁

• ✁

can be nested

• ✁

• ✁
• ✁

• ✁

• ✆

• ✁

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Axiomatically?

How do we say? All inductive or coinductive datatypes

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Axiomatically?

How do we say? All inductive or coinductive datatypes Not every parametrized type has an initial algebra (or terminal coalgebra) in set theory, e.g.

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Axiomatically?

How do we say? All inductive or coinductive datatypes Not every parametrized type has an initial algebra (or terminal coalgebra) in set theory, e.g.

• ✁
• Inductive Types for Free – p.6/22

Axiomatically?

How do we say? All inductive or coinductive datatypes Not every parametrized type has an initial algebra (or terminal coalgebra) in set theory, e.g.

• ✁
• ✁
• ✁

• Inductive Types for Free – p.6/22

Axiomatically?

How do we say? All inductive or coinductive datatypes Not every parametrized type has an initial algebra (or terminal coalgebra) in set theory, e.g.

• ✁
• ✁
• ✁

• Strictly positive types, e.g.

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Axiomatically?

How do we say? All inductive or coinductive datatypes Not every parametrized type has an initial algebra (or terminal coalgebra) in set theory, e.g.

• ✁
• ✁
• ✁

• Strictly positive types, e.g.
• ✁

• rdinal notations

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Axiomatically?

How do we say? All inductive or coinductive datatypes Not every parametrized type has an initial algebra (or terminal coalgebra) in set theory, e.g.

• ✁
• ✁
• ✁

• Strictly positive types, e.g.
• ✁

• rdinal notations

How do we say? Strictly positive

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Containers and

• types

Container types are a syntax-free representation of strictly positive types.

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Containers and

• types

Container types are a syntax-free representation of strictly positive types. In an extensive LCCC: W-types (

• ☎
• ✁

• ✁

) are sufficent to show that container types are closed under

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Containers and

• types

Container types are a syntax-free representation of strictly positive types. In an extensive LCCC: W-types (

• ☎
• ✁

• ✁

) are sufficent to show that container types are closed under

• (here)

Inductive Types for Free – p.7/22

Containers and

• types

Container types are a syntax-free representation of strictly positive types. In an extensive LCCC: W-types (

• ☎
• ✁

• ✁

) are sufficent to show that container types are closed under

• (here)

(journal paper, submitted)

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Containers and

• types

Container types are a syntax-free representation of strictly positive types. In an extensive LCCC: W-types (

• ☎
• ✁

• ✁

) are sufficent to show that container types are closed under

• (here)

(journal paper, submitted) Application: Generic Programming e.g. see our work on derivatives of datatypes

Inductive Types for Free – p.7/22

Containers and

• types

Container types are a syntax-free representation of strictly positive types. In an extensive LCCC: W-types (

• ☎
• ✁

• ✁

) are sufficent to show that container types are closed under

• (here)

(journal paper, submitted) Application: Generic Programming e.g. see our work on derivatives of datatypes Application: Small trusted cores e.g. for Epigram

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Containers for dummies

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Containers for dummies

A container type is given by

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Containers for dummies

A container type is given by A collection of shapes

• , e.g.

} , , {

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Containers for dummies

A container type is given by A collection of shapes

• , e.g.

} , , {

An assignment of positions to shapes , e.g.

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Containers for dummies . . .

We can use a container by

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Containers for dummies . . .

We can use a container by Choosing a shape, e.g.

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Containers for dummies . . .

We can use a container by Choosing a shape, e.g. Filling the positions with payload (here natural numbers), e.g.

1 4 0

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Container for scientists

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Container for scientists

A container type

• ✁

is given by

• a set of shapes

a family of sets of position, e.g.

is a set for any

• .

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Container for scientists

A container type

• ✁

is given by

• a set of shapes

a family of sets of position, e.g.

is a set for any

• .

The extension

• ✁

• ✁

• f a container is the

endofunctor

• ✁

• ✁

• ✁

• ✁

• ✁

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Container for scientists

A container type

• ✁

is given by

• a set of shapes

a family of sets of position, e.g.

is a set for any

• .

The extension

• ✁

• ✁

• f a container is the

endofunctor

• ✁

• ✁

• ✁

• ✁

• ✁

Straightforward extension to

• ary containers

• ✁

• ✂

• ✁

• ✄

.

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Example: Lists

• ✁

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Example: Lists

• ✁

• ✂

• ✆

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Example: Lists

• ✁

• ✂

• ✆
• ☎

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Example: Lists

• ✁

• ✂

• ✆
• ☎

• ✆

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Morphisms of containers

Given containers

• and
• a morphism
• is given by

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Morphisms of containers

Given containers

• and
• a morphism
• is given by

• ✂

• ✁
• ✁

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Morphisms of containers

Given containers

• and
• a morphism
• is given by

• ✂

• ✁
• ✁

its extension is the natural transformation

• ✁

• ✁
• ✁
• ✁
• ✁
• ✁

• ✁
• ✁

• ✁

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Representation theorem

Theorem (AAG,FOSSACS 03) The extension functor

• ✁

is full and faithful.

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Representation theorem

Theorem (AAG,FOSSACS 03) The extension functor

• ✁

is full and faithful. Consequence: All polymorphic functions

• ✂

(i.e. natural transformations) are given by

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Representation theorem

Theorem (AAG,FOSSACS 03) The extension functor

• ✁

is full and faithful. Consequence: All polymorphic functions

• ✂

(i.e. natural transformations) are given by A length transformer

,

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Representation theorem

Theorem (AAG,FOSSACS 03) The extension functor

• ✁

is full and faithful. Consequence: All polymorphic functions

• ✂

(i.e. natural transformations) are given by A length transformer

, A where-did-you-come-from function

• ✂
• ✂

• ✂
• .

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Closure properties

Containers are closed under (*) Constant functors, Coproducts ( ) Products (

) Constant exponentation

Composition of functors initial algebras (

• ) [ICALP 04]

terminal coalgebras (

) [Journal paper]

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Closure properties

Containers are closed under (*) Constant functors, Coproducts ( ) Products (

) Constant exponentation

Composition of functors initial algebras (

• ) [ICALP 04]

terminal coalgebras (

) [Journal paper]

(*) In any Martin-Löf category = LCCC (locally cartesian closed category) + W-types.

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Coproducts of containers

Given containers

• ✁

• ✂

• Inductive Types for Free – p.15/22

Coproducts of containers

Given containers

• ✁

• ✂

• ✂

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Coproducts of containers

Given containers

• ✁

• ✂

• ✂

• ✂
• ✁
• ✁

• ☎

• ✁

• ✂

• ☎

• ✁

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Products of containers

Given containers

• ✁

• ✂

• Inductive Types for Free – p.16/22

Products of containers

Given containers

• ✁

• ✂

• ✄
• ✂

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Products of containers

Given containers

• ✁

• ✂

• ✄
• ✂

• ✁
• ✂

• ✄

• ✁

• ✂

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Initial algebras ( )

Given a 2-ary container

• ✁

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Initial algebras ( )

Given a 2-ary container

• ✁

• ✁

• ✂

• Inductive Types for Free – p.17/22

Initial algebras ( )

Given a 2-ary container

• ✁

• ✁

• ✂

• ☎
• ✁

• ✁
• ✁

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Initial algebras ( )

Given a 2-ary container

• ✁

• ✁

• ✂

• ☎
• ✁

• ✁
• ✁

• Inductive Types for Free – p.17/22

Initial algebras ( )

Given a 2-ary container

• ✁

• ✁

• ✂

• ☎
• ✁

• ✁
• ✁

• ✁
• ✂
• ✁
• ✂

• ✂

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Reply to referee comment

. . . Now, the above is a strictly positive defi nition so should have a least as well as a greatest solution which are not in general isomorphic. Thus the corollary mentioned in the proof of 4.1 would be wrong and as a result the entire argument collapses. I thus fear that the paper must be rejected; . . .

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Reply to referee comment

. . . Now, the above is a strictly positive defi nition so should have a least as well as a greatest solution which are not in general isomorphic. Thus the corollary mentioned in the proof of 4.1 would be wrong and as a result the entire argument collapses. I thus fear that the paper must be rejected; . . . Reply: Since there are no infinite paths in a finite tree, there is only one solution to this isomorphism, the initial

• ne.

This is reflected in the proof of proposition 4.1!

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Constructing the iso

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Constructing the iso

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Constructing the iso

• ✁

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Terminal coalgebras [Journal submission]

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Terminal coalgebras [Journal submission]

The same construction works for the

• case.

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Terminal coalgebras [Journal submission]

The same construction works for the

• case.

Now we have to show that the solution is initial!

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Terminal coalgebras [Journal submission]

The same construction works for the

• case.

Now we have to show that the solution is initial! We need

• types, the dual of
• types.

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Terminal coalgebras [Journal submission]

The same construction works for the

• case.

Now we have to show that the solution is initial! We need

• types, the dual of
• types.

However,

• types can be constructed from
• types.

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Terminal coalgebras [Journal submission]

The same construction works for the

• case.

Now we have to show that the solution is initial! We need

• types, the dual of
• types.

However,

• types can be constructed from
• types.

See: Containers - Constructing Strictly Positive Types on my publication page.

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Further work

Quotient containers to model types like bags. First steps, see our MPC paper. Constructing Polymorphic Programs with Quotient Types Dependent containers Work in progress.

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Related work

Joyal 86 Foncteurs Analytiques et Espèces de Structures Jay 95 A semantics for shape Dybjer 97 Representing inductively defined sets by

wellorderings in Martin-Löf’s type theory

Hoogendijk and de Moor 00 Container Types Categorically Moerdijk and Palmgren 00 Wellfounded Trees in Categories Hasegawa 02 Two applications of analytic functors Gambino and Hyland 03 Wellfounded Trees and Dependent

Polynomial Functors

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