Cubical Indexed Induc ve Types Evan Cavallo Carnegie Mellon - - PowerPoint PPT Presentation

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Cubical Indexed Induc ve Types Evan Cavallo Carnegie Mellon - - PowerPoint PPT Presentation

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Cubical Indexed Induc ve Types Evan Cavallo Carnegie Mellon University jww Robert Harper HoTT-UF 2019 0 Higher induc ve types HoTT-UF 2019 1 Higher induc ve types roll quo ents and induc ve types into one HoTT-UF 2019 1

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HoTT-UF 2019

Cubical Indexed Inducve Types

Evan Cavallo

Carnegie Mellon University jww Robert Harper

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HoTT-UF 2019 1

Higher inducve types

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HoTT-UF 2019 1

Higher inducve types

roll quoents and inducve types into one

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HoTT-UF 2019 1

Higher inducve types

roll quoents and inducve types into one

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HoTT-UF 2019 1

Higher inducve types

roll quoents and inducve types into one Useful generality beyond ordinary quoents

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HoTT-UF 2019 1

Higher inducve types

roll quoents and inducve types into one Useful generality beyond ordinary quoents In ordinary TT: Cauchy reals, QIITs, . . .

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HoTT-UF 2019 1

Higher inducve types

roll quoents and inducve types into one Useful generality beyond ordinary quoents In ordinary TT: Cauchy reals, QIITs, . . . In higher-d TT: truncaons, . . .

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HoTT-UF 2019 2

Higher inducve types: what are they?

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HoTT-UF 2019 2

Higher inducve types: what are they?

HoTT: idea, many examples circles, pushouts, . . . ("ordinary" quoents) truncaons, localizaons (higher inducve types) Cauchy reals (higher inducve-inducve types)

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HoTT-UF 2019 2

Higher inducve types: what are they?

HoTT: idea, many examples circles, pushouts, . . . ("ordinary" quoents) truncaons, localizaons (higher inducve types) Cauchy reals (higher inducve-inducve types) no precise syntacc schema what forms can constructors take? deriving eliminators

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HoTT-UF 2019 2

Higher inducve types: what are they?

HoTT: idea, many examples circles, pushouts, . . . ("ordinary" quoents) truncaons, localizaons (higher inducve types) Cauchy reals (higher inducve-inducve types) no precise syntacc schema what forms can constructors take? deriving eliminators missing semancs for many how do programs with HITs compute? which HITs exist in denotaonal models? (pushouts?)

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HoTT-UF 2019 3

Higher inducve types: what are they?

advances in syntax Sojakova 2014: W-suspensions Basold, Geuvers, & van der Weide 2017: 1-d HITs Dybjer & Moeneclaey 2017: 2-d HITs Kaposi & Kovács 2018: n-d HIITs

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HoTT-UF 2019 4

Higher inducve types: what are they?

advances in semancs Lumsdaine & Shulman 2017: simplicial model cats syntax? HITs with parameters? Dybjer & Moeneclaey 2017: groupoid model Altenkirch, Kaposi, & Kovács 2019: w/ UIP

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HoTT-UF 2019 4

Higher inducve types: what are they?

advances in semancs Lumsdaine & Shulman 2017: simplicial model cats syntax? HITs with parameters? Dybjer & Moeneclaey 2017: groupoid model Altenkirch, Kaposi, & Kovács 2019: w/ UIP the "edge of understanding": syntax + semancs DM, AKK (truncated) cubical type theories

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HoTT-UF 2019 5

Cubical type theories

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HoTT-UF 2019 5

Cubical type theories

Move: higher type theories w/ computaonal meaning

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HoTT-UF 2019 5

Cubical type theories

Move: higher type theories w/ computaonal meaning Several variees: De Morgan cubes Cohen, Coquand, Huber, & Mörtberg 2015 Cartesian cubes Angiuli, Favonia, & Harper 2018 Angiuli, Brunerie, Coquand, Favonia, Licata, & Harper 2019 Substructural cubes Bezem, Coquand, & Huber 2013&2017

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HoTT-UF 2019 6

Move: higher type theories w/ computaonal meaning

Cubical type theories

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HoTT-UF 2019 6

Move: higher type theories w/ computaonal meaning

Cubical type theories

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HoTT-UF 2019 7

Cubical type theories

Move: higher type theories w/ computaonal meaning

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HoTT-UF 2019 8

Cubical type theories: why?

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HoTT-UF 2019 8

Cubical type theories: why?

Developed to give computaonal meaning to HoTT

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HoTT-UF 2019 8

Cubical type theories: why?

Developed to give computaonal meaning to HoTT also: construcve denotaonal models

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HoTT-UF 2019 8

Cubical type theories: why?

Developed to give computaonal meaning to HoTT also: construcve denotaonal models also: any models for many HITs

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HoTT-UF 2019 8

Cubical type theories: why?

Developed to give computaonal meaning to HoTT also: construcve denotaonal models also: any models for many HITs Useful for praccal formalizaon early movaon: Licata & Brunerie 2015

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HoTT-UF 2019 8

Cubical type theories: why?

Developed to give computaonal meaning to HoTT also: construcve denotaonal models also: any models for many HITs Useful for praccal formalizaon early movaon: Licata & Brunerie 2015 For HoTT purists: laboratory of higher ideas

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HoTT-UF 2019 9

Cubical inducve types: syntax

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HoTT-UF 2019 9

Cubical inducve types: syntax

Higher constructors via dimension arguments

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HoTT-UF 2019 9

Cubical inducve types: syntax

Higher constructors via dimension arguments Path constructors live in the inducve type

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HoTT-UF 2019 9

Cubical inducve types: syntax

Higher constructors via dimension arguments Path constructors live in the inducve type Scales well to >1-d (spheres, torus, . . .)

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HoTT-UF 2019 10

Cubical inducve types: syntax

Eliminaon: paern matching + coherence reqs

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HoTT-UF 2019 10

Cubical inducve types: syntax

Eliminaon: paern matching + coherence reqs

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HoTT-UF 2019 10

Cubical inducve types: syntax

Eliminaon: paern matching + coherence reqs

) (

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HoTT-UF 2019 10

Cubical inducve types: syntax

Eliminaon: paern matching + coherence reqs

) (

reduces when applied to a constructor

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HoTT-UF 2019 11

Cubical inducve types: syntax

Dealing with higher dimensions

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HoTT-UF 2019 11

Cubical inducve types: syntax

Dealing with higher dimensions HITs with >1-d constructors

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HoTT-UF 2019 11

Cubical inducve types: syntax

Dealing with higher dimensions HITs with >1-d constructors paern-matching on two HITs at once

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HoTT-UF 2019 12

Cubical inducve types: syntax

C & Harper: schema for indexed cubical HITs

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HoTT-UF 2019 12

Cubical inducve types: syntax

C & Harper: schema for indexed cubical HITs

← non-recursive ← recursive

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HoTT-UF 2019 12

Cubical inducve types: syntax

C & Harper: schema for indexed cubical HITs

← non-recursive ← recursive

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HoTT-UF 2019 12

Cubical inducve types: syntax

C & Harper: schema for indexed cubical HITs

← non-recursive ← recursive ← dimensions ← boundary

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HoTT-UF 2019 12

Cubical inducve types: syntax

C & Harper: schema for indexed cubical HITs

← non-recursive ← recursive ← dimensions ← boundary

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HoTT-UF 2019 13

Cubical inducve types: semancs

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HoTT-UF 2019 13

Cubical inducve types: semancs

Ingredients of a cubical type

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HoTT-UF 2019 13

Cubical inducve types: semancs

Ingredients of a cubical type Values

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HoTT-UF 2019 13

Cubical inducve types: semancs

Ingredients of a cubical type Values

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HoTT-UF 2019 13

Cubical inducve types: semancs

Ingredients of a cubical type Values Coercion and composion operaons

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HoTT-UF 2019 14

Interlude: semancs of cubical type theory

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HoTT-UF 2019 14

Interlude: semancs of cubical type theory

Coercion: all construcons respect paths

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HoTT-UF 2019 14

Interlude: semancs of cubical type theory

Coercion: all construcons respect paths

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HoTT-UF 2019 14

Interlude: semancs of cubical type theory

Coercion: all construcons respect paths

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HoTT-UF 2019 14

Interlude: semancs of cubical type theory

Coercion: all construcons respect paths

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HoTT-UF 2019 15

Interlude: semancs of cubical type theory

Coercion: all construcons respect paths Evaluate by cases on the type line

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HoTT-UF 2019 15

Interlude: semancs of cubical type theory

Coercion: all construcons respect paths Evaluate by cases on the type line

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Interlude: semancs of cubical type theory

Coercion: all construcons respect paths Evaluate by cases on the type line

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HoTT-UF 2019 16

Interlude: semancs of cubical type theory

Coercion: all construcons respect paths Evaluate by cases on the type line

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HoTT-UF 2019 16

Interlude: semancs of cubical type theory

Coercion: all construcons respect paths Evaluate by cases on the type line

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HoTT-UF 2019 16

Interlude: semancs of cubical type theory

Coercion: all construcons respect paths Evaluate by cases on the type line

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HoTT-UF 2019 16

Interlude: semancs of cubical type theory

Coercion: all construcons respect paths Evaluate by cases on the type line

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HoTT-UF 2019 17

Interlude: semancs of cubical type theory

Composion: strengthening the inducon hypothesis

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HoTT-UF 2019 17

Interlude: semancs of cubical type theory

Composion: strengthening the inducon hypothesis add homogeneous composion

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HoTT-UF 2019 17

Interlude: semancs of cubical type theory

Composion: strengthening the inducon hypothesis add homogeneous composion

✓

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HoTT-UF 2019 17

Interlude: semancs of cubical type theory

Composion: strengthening the inducon hypothesis add homogeneous composion

✓

generalize coe to heterogeneous composion

( )

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HoTT-UF 2019 18

Cubical inducve types: semancs

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HoTT-UF 2019 18

Cubical inducve types: semancs

What are the values of a higher inducve type?

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HoTT-UF 2019 18

Cubical inducve types: semancs

What are the values of a higher inducve type?

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HoTT-UF 2019 18

Cubical inducve types: semancs

What are the values of a higher inducve type?

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Cubical inducve types: semancs

What are the values of a higher inducve type?

( (

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HoTT-UF 2019 19

Cubical inducve types: semancs

What are the values of a higher inducve type?

( (

Can we implement coercion and composion?

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HoTT-UF 2019 20

Cubical inducve types: semancs

What are the values of a higher inducve type?

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HoTT-UF 2019 20

Cubical inducve types: semancs

What are the values of a higher inducve type? Unless R is symmetric and transive, ? may not exist

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HoTT-UF 2019 20

Cubical inducve types: semancs

What are the values of a higher inducve type? Unless R is symmetric and transive, ? may not exist Must revise our choice of values

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HoTT-UF 2019 21

Cubical inducve types: semancs

Idea: freely add homogeneous composion values

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HoTT-UF 2019 21

Cubical inducve types: semancs

Idea: freely add homogeneous composion values

( (

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HoTT-UF 2019 21

Cubical inducve types: semancs

Idea: freely add homogeneous composion values

( (

( + boundary reducons )

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HoTT-UF 2019 21

Cubical inducve types: semancs

Idea: freely add homogeneous composion values

( (

( + boundary reducons ) Eliminator maps hcom values to hcoms in the target

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HoTT-UF 2019 22

Cubical inducve types: semancs

Implement coercion by cases

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HoTT-UF 2019 22

Cubical inducve types: semancs

Implement coercion by cases

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HoTT-UF 2019 22

Cubical inducve types: semancs

Implement coercion by cases

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HoTT-UF 2019 22

Cubical inducve types: semancs

Implement coercion by cases

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HoTT-UF 2019 22

Cubical inducve types: semancs

Implement coercion by cases Why not free coercion values?

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HoTT-UF 2019 22

Cubical inducve types: semancs

Implement coercion by cases Why not free coercion values?

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HoTT-UF 2019 22

Cubical inducve types: semancs

Implement coercion by cases Why not free coercion values?

  • cf. Lumsdaine & Shulman
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HoTT-UF 2019 23

Cubical inducve types: semancs

Theorem (Canonicity). Any term in a HIT evaluates to a value.

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HoTT-UF 2019 23

Cubical inducve types: semancs

Theorem (Canonicity). Any term in a HIT evaluates to a value. Are we sasfied with our choice of values?

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HoTT-UF 2019 23

Cubical inducve types: semancs

Theorem (Canonicity). Any term in a HIT evaluates to a value. Are we sasfied with our choice of values? any int evaluates to pos, neg, seg, or hcom

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HoTT-UF 2019 23

Cubical inducve types: semancs

Theorem (Canonicity). Any term in a HIT evaluates to a value. Are we sasfied with our choice of values? any int evaluates to pos, neg, seg, or hcom Angiuli, Favonia, & Harper: zero-d composions can be excluded

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HoTT-UF 2019 23

Cubical inducve types: semancs

Theorem (Canonicity). Any term in a HIT evaluates to a value. Are we sasfied with our choice of values? any int evaluates to pos, neg, seg, or hcom Angiuli, Favonia, & Harper: zero-d composions can be excluded any zero-d int evaluates to pos or neg ✓

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HoTT-UF 2019 24

Indexed inducve types

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HoTT-UF 2019 24

Indexed inducve types

Simultaneously inducvely defined family of types

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HoTT-UF 2019 24

Indexed inducve types

Simultaneously inducvely defined family of types Vectors of a given length

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HoTT-UF 2019 24

Indexed inducve types

Simultaneously inducvely defined family of types Vectors of a given length Identy types

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HoTT-UF 2019 25

Indexed inducve types

What are the values of an indexed inducve type?

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HoTT-UF 2019 25

Indexed inducve types

What are the values of an indexed inducve type? Not only refl...

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HoTT-UF 2019 25

Indexed inducve types

What are the values of an indexed inducve type? Not only refl... Add free coercion values for coercion between indices

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HoTT-UF 2019 25

Indexed inducve types

What are the values of an indexed inducve type? Not only refl... Add free coercion values for coercion between indices Coercion in parameters sll reduces

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HoTT-UF 2019 25

Indexed inducve types

What are the values of an indexed inducve type? Not only refl... Add free coercion values for coercion between indices Coercion in parameters sll reduces Size depends on size of indices, but not parameters

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HoTT-UF 2019 26

Cubical type theory and the future of HITs

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HoTT-UF 2019 26

Cubical type theory and the future of HITs

Gracefully scaling to higher-d is essenal for pracce

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HoTT-UF 2019 26

Cubical type theory and the future of HITs

Gracefully scaling to higher-d is essenal for pracce where does cubical type theory have models?

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HoTT-UF 2019 26

Cubical type theory and the future of HITs

Gracefully scaling to higher-d is essenal for pracce where does cubical type theory have models? Going further inducve-inducve types (Hugunin 2019) implementaon ( , Cubical Agda) more flexible schemata