Comodules over relative comonads for streams and infinite matrices - PowerPoint PPT Presentation

Comodules over relative comonads for streams and infinite matrices R´ egis Spadotti joint work with Benedikt Ahrens Institut de Recherche en Informatique de Toulouse Universit´ e Paul Sabatier types 2014 B. Ahrens and R. Spadotti Comodules over relative comonads 1/28

Goal • Category-theoretic semantics of co inductive data types in intensional Martin-L¨ of type theory (IMLTT) � Develop a notion of “coalgebra” for the signature of a codata type • Incorporate canonical cosubstitution B. Ahrens and R. Spadotti Comodules over relative comonads 2/28

Outline 1 Syntax: inductives and substitution 2 Homogeneous cosyntax: streams 3 Heterogeneous cosyntax: infinite triangular matrices B. Ahrens and R. Spadotti Comodules over relative comonads 3/28

Outline 1 Syntax: inductives and substitution 2 Homogeneous cosyntax: streams 3 Heterogeneous cosyntax: infinite triangular matrices B. Ahrens and R. Spadotti Comodules over relative comonads 4/28

Heterogeneous data types Motivation: binding syntax in MLTT Lc : Type → Type s , t : Lc X t : Lc( X + 1) x : X var( x ) : Lc( X ) app( s , t ) : Lc X abs( t ) : Lc X Heterogeneity of abs : recursive argument with bigger parameter X + 1 Substitution: � � subst X , Y : X → Lc Y → Lc X → Lc Y Avoiding capture: � � shift X , Y : X → Lc Y → X + 1 → Lc( Y + 1) B. Ahrens and R. Spadotti Comodules over relative comonads 5/28

Initial semantics for binding syntax Initial semantics for lambda calculus: Fiore, Plotkin & Turi ’99 • characterizes not only data type but also substitution • reformulated using monads by Hirschowitz & Maggesi ’07 Basis for this reformulation: Lemma (Substitution is monadic: Altenkirch & Reus ’99) (Lc , var , subst) forms a monad (in Kleisli form) B. Ahrens and R. Spadotti Comodules over relative comonads 6/28

Initial semantics for λ -calculus using monads Definition (Algebra for signature of Lc , H & M ’07) • a monad ( T , unit , bind) on Type • two morphisms of modules over T , App : T × T → T Abs : T ( + 1) → T B. Ahrens and R. Spadotti Comodules over relative comonads 7/28

Initial semantics for λ -calculus using monads Definition (Algebra for signature of Lc , H & M ’07) • a monad ( T , unit , bind) on Type • two morphisms of modules over T , App : T × T → T Abs : T ( + 1) → T “Module morphism” expresses commutativity with bind : App ◦ (bind f ) 2 bind f ◦ App = bind f ◦ Abs = Abs ◦ bind (shift f ) B. Ahrens and R. Spadotti Comodules over relative comonads 7/28

Initial semantics for λ -calculus using monads Definition (Algebra for signature of Lc , H & M ’07) • a monad ( T , unit , bind) on Type • two morphisms of modules over T , App : T × T → T Abs : T ( + 1) → T “Module morphism” expresses commutativity with bind : App ◦ (bind f ) 2 bind f ◦ App = bind f ◦ Abs = Abs ◦ bind (shift f ) Lemma (Initial semantics for Lc , H & M ’07) (Lc , app , abs) is the initial algebra, where Lc = (Lc , var , subst) B. Ahrens and R. Spadotti Comodules over relative comonads 7/28

Goal: characterize co data types with co substitution Goal dualize techniques of H & M to characterize • co data types in intensional ML type theory with • co substitution as final object In this talk • streams • infinite triangular matrices B. Ahrens and R. Spadotti Comodules over relative comonads 8/28

Outline 1 Syntax: inductives and substitution 2 Homogeneous cosyntax: streams 3 Heterogeneous cosyntax: infinite triangular matrices B. Ahrens and R. Spadotti Comodules over relative comonads 9/28

Streams over a base type • Streams = infinite lists over some base type A . . . a 0 a 1 a 2 head tail • Specified by destructors s : Stream A s : Stream A head A ( s ) : A tail A ( s ) : Stream A • Propositional equality is not adequate for infinite object � bisimulation B. Ahrens and R. Spadotti Comodules over relative comonads 10/28

Axioms for streams • Formation rules Stream : Type → Type ∼ : Stream A → Stream A → Prop • Destructors � head , tail � : Stream A → A × Stream A ∼ head : s 1 ∼ s 2 → head s 1 = head s 2 ∼ tail : s 1 ∼ s 2 → tail s 1 ∼ tail s 2 coiter T : ( T → A × T ) → T → Stream A • Coiterator bisim R : R ⊆ R on � head , tail � → R ⊆ ∼ head(coiter T f t ) = π 1 ( f t ) • Computation tail(coiter T f t ) ∼ coiter T f ( π 2 ( f t )) B. Ahrens and R. Spadotti Comodules over relative comonads 11/28

Axioms for streams • Formation rules Stream : Type → Type ∼ : Stream A → Stream A → Prop • Destructors � head , tail � : Stream A → A × Stream A ∼ head : s 1 ∼ s 2 → head s 1 = head s 2 ∼ tail : s 1 ∼ s 2 → tail s 1 ∼ tail s 2 coiter T : ( T → A × T ) → T → Stream A • Coiterator bisim R : R ⊆ R on � head , tail � → R ⊆ ∼ head(coiter T f t ) = π 1 ( f t ) • Computation tail(coiter T f t ) ∼ coiter T f ( π 2 ( f t )) Provable with Coq coinductive type definitions B. Ahrens and R. Spadotti Comodules over relative comonads 11/28

Structure on streams • (Stream A , ∼ ) is a setoid , Stream : Type → Setoid • Canonical cosubstitution cosubst A , B : (Stream A → B ) → Stream A → Stream B is compatible with bisimilarity: cosubst A , B : Setoid(Stream A , eq B ) → Setoid(Stream A , Stream B ) with eq : Type → Setoid eq ⊣ forget B. Ahrens and R. Spadotti Comodules over relative comonads 12/28

Structure on streams Lemma (Stream , head , cosubst) is a comonad relative to eq : Type → Setoid . Definition ( Relative (co)monad, Alten., Chapm. & Uust. ’10) • underlying functor is not necessarily endo • needs “mediating” functor (above: eq ) B. Ahrens and R. Spadotti Comodules over relative comonads 13/28

About the destructor tail Morphisms of modules over monads characterize commutativity of substitution with constructors app : Lc × Lc → Lc subst f ◦ app = app ◦ (subst f ) 2 Morphisms of co modules over relative co monads characterize commutativity of co substitution with destructors tail : Stream → Stream tail ◦ cosubst f = cosubst f ◦ tail B. Ahrens and R. Spadotti Comodules over relative comonads 14/28

Final semantics for Stream Definition (Category of coalgebras) A coalgebra for the signature of Stream is given by a pair ( S , t ) : • a comonad S relative to eq : Type → Setoid • a morphism of comodules over S t : S → S Morphisms: . . . Lemma (Stream , tail) is the final object in the above category. B. Ahrens and R. Spadotti Comodules over relative comonads 15/28

Outline 1 Syntax: inductives and substitution 2 Homogeneous cosyntax: streams 3 Heterogeneous cosyntax: infinite triangular matrices B. Ahrens and R. Spadotti Comodules over relative comonads 16/28

An example of cosyntax: infinite triangular matrices Tri : the codata type of infinite triangular matrices • omit redundant information below the diagonal • have a variable type A of diagonal elements • e.g. invertible elements • a fixed type E of elements for rest of matrix • usage: Pascal matrices (binomial coefficients), mathematical physics (infinite-dim. problems) A E E E . . . A E E A E A B. Ahrens and R. Spadotti Comodules over relative comonads 17/28

Matrices through trapezia: the destructors of Tri A E E E t : Tri A . . . A E E top A ( t ) : A A E A E E E E t : Tri A A E E E rest A ( t ) : Tri( E × A ) . . . A E E A E A B. Ahrens and R. Spadotti Comodules over relative comonads 18/28

Matrices through trapezia: the destructors of Tri A E E E t : Tri A . . . A E E top A ( t ) : A A E A E × A E E E t : Tri A E × A E E rest A ( t ) : Tri( E × A ) . . . E × A E E × A B. Ahrens and R. Spadotti Comodules over relative comonads 18/28

Redecoration redec A , B : (Tri A → B ) → (Tri A → Tri B ) f : Tri A → B A E E E B E E E . . . . . . A E E B E E A E B E A B top ◦ redec f := f and rest ◦ redec f := redec (lift f ) ◦ rest with lift f : Tri( E × A ) → E × B B. Ahrens and R. Spadotti Comodules over relative comonads 19/28

Tri is a weak constructive comonad Sameness = bisimilarity Bisimilarity ∼ coinductively defined via destructors t ∼ t ′ t ∼ t ′ top( t ) = top( t ′ ) rest( t ) ∼ rest( t ′ ) Lemma (Matthes and Picard ’11) (Tri : Type → Type , top , redec) forms a “weak constructive comonad”. � “weak constructive” refers to compatibility conditions with bisimilarity B. Ahrens and R. Spadotti Comodules over relative comonads 20/28

Tri is a relative comonad Alternatively, Tri A is a setoid rather than a (plain) type top A : Setoid(Tri A , eq A ) redec A , B : Setoid(Tri A , eq B ) → Setoid(Tri A , Tri B ) with eq : Type → Setoid B. Ahrens and R. Spadotti Comodules over relative comonads 21/28

Tri is a relative comonad Alternatively, Tri A is a setoid rather than a (plain) type top A : Setoid(Tri A , eq A ) redec A , B : Setoid(Tri A , eq B ) → Setoid(Tri A , Tri B ) with eq : Type → Setoid Lemma (Reformulation of Matthes and Picard ’11) (Tri : Type → Setoid , top , redec) forms a comonad relative to eq : Type → Setoid . B. Ahrens and R. Spadotti Comodules over relative comonads 21/28