Comonadic notions of computation Tarmo Uustalu 1 Varmo Vene 2 1 - PowerPoint PPT Presentation

Comonadic notions of computation Tarmo Uustalu 1 Varmo Vene 2 1 Institute of Cybernetics, Tallinn 2 University of Tartu FMCS 2006, Kananaskis, 8 June 2006

Motivation Moggi and Wadler showed that effectful computations can be structured with monads. An effect-producing function from A to B is a map A → B in the Kleisli category, i.e., a map A → TB in the base category. Some examples applied in semantics: TA = A + 1 partiality TA = A + E exceptions A E = environment TA A ∗ = µ X . 1 + A × X TA = non-determinism ( A × S ) S = state TA Are all impure features captured by monads? What about comonads?

� � � � Comonads Definition A comonad on category C is given by a functor D : C → C a natural transformation ε A : DA → A counit of the comonad a natural transformation δ A : DA → D 2 A comultiplication of the comonad s.t. following diagrams commute δ A � D 2 A DA DA � � �������� � δ A � δ A � � δ A D δ A � � � � DA � D 3 A D 2 A D 2 A D 2 A ε DA D ε A δ DA

Comonads Comonads model notions of value in a context; DA is the type of contextually situated values of A . A context-relying function from A to B is a map A → B in the coKleisli category, i.e., a map DA → B in the base category. Product (environment) comonad Functor: DA = A × E Counit: ε A : A × E → A ( a , e ) �→ a Comultiplication: δ A : A × E → ( A × E ) × E ( a , e ) �→ (( a , e ) , e )

Comonads Streams comonad DA = A N = ν X . A × X Functor: Counit: A N → A : ε A α �→ α ( 0 ) Comultiplication: A N → ( A N ) N δ A : �→ λ n . ( λ m .α ( n + m )) α [ a 0 , a 1 , a 2 , . . . ] �→ [[ a 0 , a 1 , a 2 , . . . ] , [ a 1 , a 2 , a 3 . . . ] , . . . ]

Comonads for stream functions Dataflow computation = discrete-time signal transformations = stream functions. Example: simple dataflow programs = 0 fby ( pos + 1 ) pos sum x = x + ( 0 fby ( sum x )) fact = 1 fby ( fact ∗ ( pos + 1 )) fibo = 0 fby ( fibo + ( 1 fby fibo )) 0 1 2 3 4 5 6 . . . pos 0 1 3 6 10 15 21 . . . sum pos 1 1 2 6 24 120 720 . . . fact 0 1 1 2 3 5 8 fibo . . . Stream functions A N → B N are naturally isomorfic to A N × N → B

Comonads for stream functions General stream functions DA = A N × N Functor: Input streams with past/present/future: a 0 , a 1 , . . . , a n − 1 , a n , a n + 1 , a n + 2 , . . . Counit: A N × N → A ε A : ( α, n ) �→ α ( n ) Comultiplication: A N × N → ( A N × N ) N × N δ A : ( α, n ) �→ ( λ m . ( α, m ) , n )

Comonads for stream functions Causal stream functions ( ∼ DA = A + = A ∗ × A ) Functor: Input streams with past and present but no future Counit: A + → A : ε A [ a 0 , . . . , a n ] �→ a n Comultiplication: A + → ( A + ) + δ A : [ a 0 , . . . , a n ] �→ [[ a 0 ] , [ a 0 , a 1 ] , . . . , [ a 0 , . . . , a n ]] Anticausal stream functions Input streams with present and future but no past ( ∼ DA = A N = A × A N ) Functor:

Comonads for attribute grammars An attribute grammar is a CF grammar augmented with attributes and semantic equations. Example: preorder numbering of the nodes S ℓ − → E S b S b L S b − → R S b S b . numin + 1 L . numin = S b S b = L . numout + 1 R . numin S ℓ . numout S ℓ . numin = S b . numout S b = R . numout Tree functions where the output at a position depends on the input at that position and around it (synthesized, inherited attributes).

Comonads for attribute grammars Purely synthesized AG-s Functor: DA = Tree A = µ X . A × ( 1 + X × X ) Counit: ε A : Tree A → A ( a , s ) �→ a Comultiplication: : Tree A → Tree ( Tree A ) δ A � ( t , inl ( ∗ )) , if t = ( a , inl ( ∗ )) δ A ( t ) = ( t , inr ( δ A ( t 1 ) , δ A ( t 2 )) , if t = ( a , inr ( t 1 , t 2 ))

Comonads for attribute grammars General AG-s DA = ( 2 × Tree A ) ∗ × Tree A Functor: Path structure from the root to the focus and the local tree below the focus

Pre-[Cartesian closed] co-Kleisli categories Extending a pure language (the lambda calculus) with coeffect-constructs, we want the old constructs to remain and not to change their meaning too much. If D is a comonad on a Cartesian closed category C , how much of that structure carries over to CoKl ( D ) ? Products A × D B = df A × B π D = df π 0 ◦ ε 0 π D = df π 1 ◦ ε 1 � k 0 , k 1 � D = df � k 0 , k 1 �

� � � � Pre-[Cartesian closed] co-Kleisli categories For (pre-)exponents we need some extra structure on a comonad: � D π 0 , D π 1 � D (( DA ⇒ B ) × A ) D (( DA ⇒ B )) × DA ev D ( ε × id ) ev ( DA ⇒ B ) × DA B k � C ? k � D ( A × B ) � C D ( A × B ) DA × DB Λ D ( k ) � DB ⇒ C Λ( k ◦ ?) � DB ⇒ C DA DA

Pre-[Cartesian closed] co-Kleisli categories Definition A comonad D on a [symmetric] [semi]monoidal cat. C is said to be {lax/strong} [symmetric] [semi]monoidal, if it comes with a nat. {transf./iso.} m : DA ⊗ DB → D ( A ⊗ B ) [and a nat. {transf./iso.} e : I → DI ] behaving well wrt. α, [ l , r , ] [ γ, ] ε, δ . Pre-exponents Let D be a comonad on a Cartesian closed cat. C . Assuming that D that is a {lax/strong} [symmetric] [semi]monoidal wrt. the ( 1 , × ) symmetric monoidal structure on C , define this structure on A ⇒ D B CoKl ( D ) : = df DA ⇒ B ev D = df ev ◦ � ε ◦ D π 0 , D π 1 � Λ D ( k ) = df Λ( k ◦ m )

Pre-[Cartesian closed] co-Kleisli categories If D is strong monoidal, then C ⇒ D − is right adjoint to − × D C and hence ⇒ D is an exponent functor: D ( A × C ) → B DA × DC → B DA → DC ⇒ B However, this seems rare in computational applications, DA = A N being an atypical example. Strong symmetric monoidal structure on streams A N × B N → ( A × B ) N : m ( α, β ) �→ λ n . ( α ( n ) , β ( n ))

Pre-[Cartesian closed] co-Kleisli categories More common is that a comonad is lax symmetric semimonoidal, eg DA = A + , DA = A N × N . Lax symmetric semimonoidal structure on − N × N ( A N × N ) × ( B N × N ) → ( A × B ) N × N : m (( α, k 1 ) , ( β, k 2 ) �→ ( λ n . ( α ( n ) , β ( n )) , k 1 ) Then it suffices to have m satisfying m ◦ ∆ = D ∆ , where ∆ = � id , id � : A → A × A is the semicomonoid structure on the objects of C , to get that ⇒ D is a weak exponent functor.

Comonadic semantics Comonadic semantics is obtained by interpreting the lambda-calculus into CoKl ( D ) in the standard way. Comonadic semantics � A � D × D � B � D � A � D × � B � D � A × B � D = df = � A � D ⇒ D � B � D D � A � D ⇒ � B � D � A ⇒ B � D = df = � ( x ) x i � D π D = df = π i ◦ ε i 0 ◦ D � ( x ) t � D � ( x ) fst ( t ) � D π D π 0 ◦ � ( x ) t � D = df = 1 ◦ D � ( x ) t � D � ( x ) snd ( t ) � D π D π 1 ◦ � ( x ) t � D = df = � ( x )( t 0 , t 1 ) � D � � ( x ) t 0 � D , � ( x ) t 1 � D � D = df � � ( x ) t 0 � D , � ( x ) t 1 � D � = ev D ◦ D � � ( x ) t � D , � ( x ) u � D � D � ( x ) t u � D = df ev ◦ � � ( x ) t � D , ( � ( x ) u � D ) † � = Λ( � ( x , x ) t � D ◦ m ) � ( x ) λ xt � D Λ D ( � ( x , x ) t � D ) = df = Coeffect-specific constructs are interpreted specifically.

Comonadic semantics x : C ⊢ t : A implies � ( x ) t � D : � C � D → D � A � D , but not all equations of the lambda-calculus are validated. Closed terms: Type soundness for ⊢ t : A says that � t � D : 1 → D � A � D , i.e., D 1 → � A � D , so closed terms are evaluated relative to a coeffect over 1. In case of general or causal stream functions, this is a list over 1, the time from the start. If D is properly (symmetric) monoidal (e.g., ( − ) N ), we have a canonical choice e : 1 → D 1. Comonadic dataflow language semantics: The first-order language agrees perfectly with Lucid and Lustre by its semantics. The meaning of higher-order dataflow computation has been unclear. We get a neat semantics from mathematical considerations (cf. Colaço, Pouzet’s design with two flavors of function spaces).

� � � � � � � � � � � Distributive laws Definition A distributive law of a monad ( T , η, µ ) over a comonad ( D , ε, δ ) is a natural transformation λ A : DTA → TDA st. λ λ DTA TDA DTA TDA � � � � � T ε � δ T δ � ε � � D λ � DTDA λ � TD 2 A TA D 2 TA λ T λ � T 2 DA DT 2 A DA TDTA � � � η � � D η µ D µ � � � � λ � TDA λ � TDA DTA DTA

Distributive laws Clocked dataflow computation (partial-stream functions) TA = 1 + A A + = DA ( 1 + A ) + → 1 + A + λ : � inl ( ∗ ) if last ( as ) = inl ( ∗ ) as �→ inr ([ a i | inr ( a i ) ← as ]) otherwise

Distributive laws BiKleisli category Given a monad T and comonad D with a distributive law λ : DTA → TDA , the biKleisli category BiKl ( T , D ) is defined as: | BiKl ( T , D ) | = df |C| BiKl ( T , D )( A , B ) = df C ( DA , TB ) id D , T = df η ◦ ε ℓ ⋆ ◦ λ ◦ k † ℓ ◦ D , T k = df