Effect systems revisited control-flow algebra and semantics (slides) - PowerPoint PPT Presentation

from “Type and effect systems” (Nielson, Nielson, 1999) “Now we turn to explaining the individual steps in the overall methodology of designing and using type and effect systems: – devise a semantics for the programming language, – develop a program analysis in the form of a type and effect system ... – prove the semantic correctness of the analysis, – develop an efficient inference algorithm, ...” ◮ An effect-directed semantics unifies the first three steps 9 / 20

from “Type and effect systems” (Nielson, Nielson, 1999) “Now we turn to explaining the individual steps in the overall methodology of designing and using type and effect systems: – devise a semantics for the programming language, – develop a program analysis in the form of a type and effect system ... – prove the semantic correctness of the analysis, – develop an efficient inference algorithm, ...” ◮ An effect-directed semantics unifies the first three steps ◮ We develop an effect-directed semantics for rich Nielson-Nielson-style effects 9 / 20

Type-directed semantics 10 / 20

Type-directed semantics untyped model: � e � : D D ∼ = Z + ( D → D ) + { wrong } 10 / 20

Type-directed semantics untyped model: � e � : D D ∼ = Z + ( D → D ) + { wrong } simply typed model : � Γ ⊢ e : τ � : D τ D int = Z D σ → τ = D σ → D τ 10 / 20

Type-directed semantics untyped model: � e � : D D ∼ = Z + ( D → D ) + { wrong } simply typed model : � Γ ⊢ e : τ � : D τ D int = Z D σ → τ = D σ → D τ effect-directed model : � Γ ⊢ e : τ, F � : D F τ 10 / 20

Type-directed semantics untyped model: � e � : D D ∼ = Z + ( D → D ) + { wrong } simply typed model : � Γ ⊢ e : τ � : D τ D int = Z D σ → τ = D σ → D τ effect-directed model : � Γ ⊢ e : τ, F � : D F τ Core idea: algebra-semantics homomorphism Algebraic structure of F determines structure on family D F τ . 10 / 20

� � � � � Effect analysis and semantics Lattice effects Richer structure on F (Gifford, Lucassen ’86) (Nielson, Nielson ’94, ’99) Γ ⊢ e : τ, F “Marriage of effects and monads” (Wadler, Thiemann ’03) � Γ ⊢ e : M F τ � : � Γ � → T � τ � Effect-directed semantics This paper (Katsumata ’14) Richer effect-directed � Γ ⊢ e : M F τ � : � Γ � → T F � τ � semantics (graded joinads) (graded monads) 11 / 20

� � � � � Effect analysis and semantics Lattice effects Richer structure on F (Gifford, Lucassen ’86) (Nielson, Nielson ’94, ’99) Γ ⊢ e : τ, F “Marriage of effects and monads” (Wadler, Thiemann ’03) � Γ ⊢ e : M F τ � : � Γ � → T � τ � Effect-directed semantics This paper (Katsumata ’14) Richer effect-directed � Γ ⊢ e : M F τ � : � Γ � → T F � τ � semantics (graded joinads) (graded monads) Operations on T F homomorphic to operations on F 11 / 20

Modelling effects with graded monads (Katsumata 2014) 12 / 20

Modelling effects with graded monads (Katsumata 2014) � Γ ⊢ e : τ, F � : � Γ � → T F � τ � 12 / 20

Modelling effects with graded monads (Katsumata 2014) � Γ ⊢ e : τ, F � : � Γ � → T F � τ � e.g. for state T A = S → ( A × S ). 12 / 20

Modelling effects with graded monads (Katsumata 2014) � Γ ⊢ e : τ, F � : � Γ � → T F � τ � e.g. for state T F A = ( reads ( F )) → ( A × writes ( F )). 12 / 20

Modelling effects with graded monads (Katsumata 2014) � Γ ⊢ e : τ, F � : � Γ � → T F � τ � e.g. for state T F A = ( reads ( F )) → ( A × writes ( F )). e.g. for partiality T A = ⊥ + A 12 / 20

Modelling effects with graded monads (Katsumata 2014) � Γ ⊢ e : τ, F � : � Γ � → T F � τ � e.g. for state T F A = ( reads ( F )) → ( A × writes ( F )). e.g. for partiality T ⊥ A = ⊥ , T ⊤ A = A , 12 / 20

Modelling effects with graded monads (Katsumata 2014) � Γ ⊢ e : τ, F � : � Γ � → T F � τ � e.g. for state T F A = ( reads ( F )) → ( A × writes ( F )). e.g. for partiality T ⊥ A = ⊥ , T ⊤ A = A , T ? A = A + 1

Modelling effects with graded monads (Katsumata 2014) � Γ ⊢ e : τ, F � : � Γ � → T F � τ � e.g. for state T F A = ( reads ( F )) → ( A × writes ( F )). e.g. for partiality T ⊥ A = ⊥ , T ⊤ A = A , T ? A = A + 1 Graded monads provide sequential composition

Modelling effects with graded monads (Katsumata 2014) � Γ ⊢ e : τ, F � : � Γ � → T F � τ � e.g. for state T F A = ( reads ( F )) → ( A × writes ( F )). e.g. for partiality T ⊥ A = ⊥ , T ⊤ A = A , T ? A = A + 1 Graded monads provide sequential composition Let ( F , • , 0) be an effect monoid.

Modelling effects with graded monads (Katsumata 2014) � Γ ⊢ e : τ, F � : � Γ � → T F � τ � e.g. for state T F A = ( reads ( F )) → ( A × writes ( F )). e.g. for partiality T ⊥ A = ⊥ , T ⊤ A = A , T ? A = A + 1 Graded monads provide sequential composition Let ( F , • , 0) be an effect monoid. Given d 1 : A → T F B and d 2 : B → T G C

Modelling effects with graded monads (Katsumata 2014) � Γ ⊢ e : τ, F � : � Γ � → T F � τ � e.g. for state T F A = ( reads ( F )) → ( A × writes ( F )). e.g. for partiality T ⊥ A = ⊥ , T ⊤ A = A , T ? A = A + 1 Graded monads provide sequential composition Let ( F , • , 0) be an effect monoid. Given d 1 : A → T F B and d 2 : B → T G C then d 2 ˆ ◦ d 1 : A → T F • G C

Modelling effects with graded monads (Katsumata 2014) � Γ ⊢ e : τ, F � : � Γ � → T F � τ � e.g. for state T F A = ( reads ( F )) → ( A × writes ( F )). e.g. for partiality T ⊥ A = ⊥ , T ⊤ A = A , T ? A = A + 1 Graded monads provide sequential composition Let ( F , • , 0) be an effect monoid. Given d 1 : A → T F B and d 2 : B → T G C ◦ d 1 : A → T F • G C with ˆ then d 2 ˆ id A : A → T 0 A . 12 / 20

Modelling effects with graded monads (Katsumata 2014) � Γ ⊢ e : τ, F � : � Γ � → T F � τ � e.g. for state T F A = ( reads ( F )) → ( A × writes ( F )). e.g. for partiality T ⊥ A = ⊥ , T ⊤ A = A , T ? A = A + 1 Graded monads provide sequential composition Let ( F , • , 0) be an effect monoid. Given d 1 : A → T F B and d 2 : B → T G C ◦ d 1 : A → T F • G C with ˆ then d 2 ˆ id A : A → T 0 A . With ordering Given partially ordered ( F , • , 0 , ⊑ ) then for all F ⊑ G then coercion: ι F , G , A : T F A → T G A 12 / 20

Semantics of branching: derived vs. parameterised 13 / 20

Semantics of branching: derived vs. parameterised � if e 0 then e 1 else e 2 � = COND( � e 0 � , � e 1 � , � e 2 � ) 13 / 20

Semantics of branching: derived vs. parameterised � if e 0 then e 1 else e 2 � = COND( � e 0 � , � e 1 � , � e 2 � ) ◮ COND definable via cond : B × A × A → A 13 / 20

Semantics of branching: derived vs. parameterised � if e 0 then e 1 else e 2 � = COND( � e 0 � , � e 1 � , � e 2 � ) ◮ COND definable via cond : B × A × A → A cond(true , x , y ) = x cond(false , x , y ) = y 13 / 20

Semantics of branching: derived vs. parameterised � if e 0 then e 1 else e 2 � = COND( � e 0 � , � e 1 � , � e 2 � ) ◮ COND definable via cond : B × A × A → A cond(true , x , y ) = x cond(false , x , y ) = y ◮ Restricts effect branching behaviour: 13 / 20

Semantics of branching: derived vs. parameterised � if e 0 then e 1 else e 2 � = COND( � e 0 � , � e 1 � , � e 2 � ) ◮ COND definable via cond : B × A × A → A cond(true , x , y ) = x cond(false , x , y ) = y ◮ Restricts effect branching behaviour: (if) Γ ⊢ e 0 : bool , F Γ ⊢ e 1 : τ, G Γ ⊢ e 2 : τ, H Γ ⊢ if e 0 then e 1 else e 2 : τ, F • ( G ⊔ H ) 13 / 20

Semantics of branching: derived vs. parameterised � if e 0 then e 1 else e 2 � = COND( � e 0 � , � e 1 � , � e 2 � ) ◮ COND definable via cond : B × A × A → A cond(true , x , y ) = x cond(false , x , y ) = y ◮ Restricts effect branching behaviour: (if) Γ ⊢ e 0 : bool , F Γ ⊢ e 1 : τ, G Γ ⊢ e 2 : τ, H Γ ⊢ if e 0 then e 1 else e 2 : τ, F • ( G ⊔ H ) cond : B × T G ⊔ H A × T G ⊔ H A → T G ⊔ H A 13 / 20

Semantics of branching: derived vs. parameterised � if e 0 then e 1 else e 2 � = COND( � e 0 � , � e 1 � , � e 2 � ) ◮ COND definable via cond : B × A × A → A cond(true , x , y ) = x cond(false , x , y ) = y ◮ Restricts effect branching behaviour: (if) Γ ⊢ e 0 : bool , F Γ ⊢ e 1 : τ, G Γ ⊢ e 2 : τ, H Γ ⊢ if e 0 then e 1 else e 2 : τ, F • ( G ⊔ H ) cond : B × T G ⊔ H A × T G ⊔ H A → T G ⊔ H A ◮ Towards Nielson-Nielson richer effects, but + may not be ⊔ . 13 / 20

Semantics of branching: derived vs. parameterised � if e 0 then e 1 else e 2 � = COND( � e 0 � , � e 1 � , � e 2 � ) ◮ COND definable via cond : B × A × A → A cond(true , x , y ) = x cond(false , x , y ) = y ◮ Restricts effect branching behaviour: (if) Γ ⊢ e 0 : bool , F Γ ⊢ e 1 : τ, G Γ ⊢ e 2 : τ, H Γ ⊢ if e 0 then e 1 else e 2 : τ, F • ( G ⊔ H ) cond : B × T G ⊔ H A × T G ⊔ H A → T G ⊔ H A ◮ Towards Nielson-Nielson richer effects, but + may not be ⊔ . ◮ Instead: parameterise semantics on COND : T F B × T G A × T H A → T ? +( F , G , H ) A 13 / 20

Definition: jonoid For a set of effects F then ( F , • , I , & , ? + , ⊑ ) is a joinoid control-flow algebra :

Definition: jonoid For a set of effects F then ( F , • , I , & , ? + , ⊑ ) is a joinoid control-flow algebra : ◮ ( F , • , I ) is a monoid, representing sequential composition and purity;

Definition: jonoid For a set of effects F then ( F , • , I , & , ? + , ⊑ ) is a joinoid control-flow algebra : ◮ ( F , • , I ) is a monoid, representing sequential composition and purity; ◮ ( F , & , I ) is a commutative monoid, representing parallel composition;

Definition: jonoid For a set of effects F then ( F , • , I , & , ? + , ⊑ ) is a joinoid control-flow algebra : ◮ ( F , • , I ) is a monoid, representing sequential composition and purity; ◮ ( F , & , I ) is a commutative monoid, representing parallel composition; ◮ letting F + G = ? +( I , F , G ) (pure guard)

Definition: jonoid For a set of effects F then ( F , • , I , & , ? + , ⊑ ) is a joinoid control-flow algebra : ◮ ( F , • , I ) is a monoid, representing sequential composition and purity; ◮ ( F , & , I ) is a commutative monoid, representing parallel composition; ◮ letting F + G = ? +( I , F , G ) (pure guard) ( F , +) is a semigroup, representing choice between branches

Definition: jonoid For a set of effects F then ( F , • , I , & , ? + , ⊑ ) is a joinoid control-flow algebra : ◮ ( F , • , I ) is a monoid, representing sequential composition and purity; ◮ ( F , & , I ) is a commutative monoid, representing parallel composition; ◮ letting F + G = ? +( I , F , G ) (pure guard) ( F , +) is a semigroup, representing choice between branches ◮ with right-distributivity axioms:

Definition: jonoid For a set of effects F then ( F , • , I , & , ? + , ⊑ ) is a joinoid control-flow algebra : ◮ ( F , • , I ) is a monoid, representing sequential composition and purity; ◮ ( F , & , I ) is a commutative monoid, representing parallel composition; ◮ letting F + G = ? +( I , F , G ) (pure guard) ( F , +) is a semigroup, representing choice between branches ◮ with right-distributivity axioms: ( F + G ) • H = ( F • H ) + ( G • H ) ( F + G ) & H = ( F & H ) + ( G & H )

Definition: jonoid For a set of effects F then ( F , • , I , & , ? + , ⊑ ) is a joinoid control-flow algebra : ◮ ( F , • , I ) is a monoid, representing sequential composition and purity; ◮ ( F , & , I ) is a commutative monoid, representing parallel composition; ◮ letting F + G = ? +( I , F , G ) (pure guard) ( F , +) is a semigroup, representing choice between branches ◮ with right-distributivity axioms: ( F + G ) • H = ( F • H ) + ( G • H ) ( F + G ) & H = ( F & H ) + ( G & H ) ◮ all operations are monotonic with respect to ⊑ . 14 / 20

Graded generalised joinad Definition: graded conditional joinad Given a jonoid ( F , • , I , & , ? + , ⊑ ):

Graded generalised joinad Definition: graded conditional joinad Given a jonoid ( F , • , I , & , ? + , ⊑ ): ◮ graded monad for the pre-ordered monoid ( F , • , I , ⊑ )

Graded generalised joinad Definition: graded conditional joinad Given a jonoid ( F , • , I , & , ? + , ⊑ ): ◮ graded monad for the pre-ordered monoid ( F , • , I , ⊑ ) ◮ additional operations:

Graded generalised joinad Definition: graded conditional joinad Given a jonoid ( F , • , I , & , ? + , ⊑ ): ◮ graded monad for the pre-ordered monoid ( F , • , I , ⊑ ) ◮ additional operations: cond F , G , H , A :T F B × T G A × T H A → T ? +( F , G , H ) A par F , G , A :T F A × T G A → T F & G A

Graded generalised joinad Definition: graded conditional joinad Given a jonoid ( F , • , I , & , ? + , ⊑ ): ◮ graded monad for the pre-ordered monoid ( F , • , I , ⊑ ) ◮ additional operations: cond F , G , H , A :T F B × T G A × T H A → T ? +( F , G , H ) A par F , G , A :T F A × T G A → T F & G A ◮ Satisfy jonoid axioms (modulo lifting to functors) 15 / 20

Graded generalised joinad Definition: graded conditional joinad Given a jonoid ( F , • , I , & , ? + , ⊑ ): ◮ graded monad for the pre-ordered monoid ( F , • , I , ⊑ ) ◮ additional operations: cond F , G , H , A :T F B × T G A × T H A → T ? +( F , G , H ) A par F , G , A :T F A × T G A → T F & G A ◮ Satisfy jonoid axioms (modulo lifting to functors) where ? + = cond and & = par Monoids are to (graded) monads as jonoids are to (graded) joinads 15 / 20

Theorem: soundness Given a graded joinadic semantics for the simply-effect-and- typed λ -calculus with if and par then, for all e , e ′ , Γ , τ, F :

Theorem: soundness Given a graded joinadic semantics for the simply-effect-and- typed λ -calculus with if and par then, for all e , e ′ , Γ , τ, F : Γ ⊢ e ≡ e ′ : τ, F � Γ ⊢ e : τ, F � = � Γ ⊢ e ′ : τ, F � ⇒

Theorem: soundness Given a graded joinadic semantics for the simply-effect-and- typed λ -calculus with if and par then, for all e , e ′ , Γ , τ, F : Γ ⊢ e ≡ e ′ : τ, F � Γ ⊢ e : τ, F � = � Γ ⊢ e ′ : τ, F � ⇒ wrt. CBV β - ≡ with additional equations:

Theorem: soundness Given a graded joinadic semantics for the simply-effect-and- typed λ -calculus with if and par then, for all e , e ′ , Γ , τ, F : Γ ⊢ e ≡ e ′ : τ, F � Γ ⊢ e : τ, F � = � Γ ⊢ e ′ : τ, F � ⇒ wrt. CBV β - ≡ with additional equations: if true then e else x ≡ e (if β 1’) if false then x else e ′ ≡ e ′ (if β 2’) (if b then e else e ′ ) par e ′′ (if-dist-par) ≡ if b then ( e par e ′′ ) else ( e ′ par e ′′ ) let x = (if e then e ′ else e ′′ ) in e ′′′ (if-dist-seq) ≡ if e then (let x = e ′ in e ′′′ ) else (let x = e ′′ in e ′′′ ) x par e ≡ ( x , e ) (par-pure) e par e ′ ≡ swap ( e ′ par e ) (par-sym) e par ( e ′ par e ′′ ) ≡ assoc (( e par e ′ ) par e ′′ ) (par-assoc) 16 / 20

Conclusions 17 / 20

Conclusions ◮ unify Hanne and Flemming’s rich effects with semantics 17 / 20

Conclusions ◮ unify Hanne and Flemming’s rich effects with semantics indexing non-seq. control monads − − − − − → graded monads − − − − − − − − − → graded joinads 17 / 20

Conclusions ◮ unify Hanne and Flemming’s rich effects with semantics indexing non-seq. control monads − − − − − → graded monads − − − − − − − − − → graded joinads ◮ Semantics for various kinds of parallel, concurrent (e.g. music) and speculative behaviour (e.g. prefetching) 17 / 20

Conclusions ◮ unify Hanne and Flemming’s rich effects with semantics indexing non-seq. control monads − − − − − → graded monads − − − − − − − − − → graded joinads ◮ Semantics for various kinds of parallel, concurrent (e.g. music) and speculative behaviour (e.g. prefetching) ◮ Considerable simplification to proofs 17 / 20

Conclusions ◮ unify Hanne and Flemming’s rich effects with semantics indexing non-seq. control monads − − − − − → graded monads − − − − − − − − − → graded joinads ◮ Semantics for various kinds of parallel, concurrent (e.g. music) and speculative behaviour (e.g. prefetching) ◮ Considerable simplification to proofs ◮ Represent effect-dependent optimisations more easily 17 / 20

Conclusions ◮ unify Hanne and Flemming’s rich effects with semantics indexing non-seq. control monads − − − − − → graded monads − − − − − − − − − → graded joinads ◮ Semantics for various kinds of parallel, concurrent (e.g. music) and speculative behaviour (e.g. prefetching) ◮ Considerable simplification to proofs ◮ Represent effect-dependent optimisations more easily ◮ Effect-directed semantics provides co-design approach 17 / 20

Conclusions ◮ unify Hanne and Flemming’s rich effects with semantics indexing non-seq. control monads − − − − − → graded monads − − − − − − − − − → graded joinads ◮ Semantics for various kinds of parallel, concurrent (e.g. music) and speculative behaviour (e.g. prefetching) ◮ Considerable simplification to proofs ◮ Represent effect-dependent optimisations more easily ◮ Effect-directed semantics provides co-design approach ◮ Equations of analysis carry over to semantics, and vice versa 17 / 20

Conclusions ◮ unify Hanne and Flemming’s rich effects with semantics indexing non-seq. control monads − − − − − → graded monads − − − − − − − − − → graded joinads ◮ Semantics for various kinds of parallel, concurrent (e.g. music) and speculative behaviour (e.g. prefetching) ◮ Considerable simplification to proofs ◮ Represent effect-dependent optimisations more easily ◮ Effect-directed semantics provides co-design approach ◮ Equations of analysis carry over to semantics, and vice versa ◮ Exposes which structure is needed in each direction 17 / 20

Thanks Hanne and Flemming for the inspiration. Happy Birthday! Thanks to Sam Aaron (Cambridge) for the Sonic Pi language used for the intro program 18 / 20

Backup slides 19 / 20

Modeling effects with monads Model effectful computations via some data type T 20 / 20

Modeling effects with monads Model effectful computations via some data type T � Γ ⊢ e : τ � : � Γ � → T � τ � 20 / 20

Modeling effects with monads Model effectful computations via some data type T � Γ ⊢ e : τ � : � Γ � → T � τ � e.g. for state T A = S → ( A × S ). 20 / 20

Modeling effects with monads Model effectful computations via some data type T � Γ ⊢ e : τ � : � Γ � → T � τ � e.g. for state T A = S → ( A × S ). e.g. for partiality T A = ⊥ + A 20 / 20

Modeling effects with monads Model effectful computations via some data type T � Γ ⊢ e : τ � : � Γ � → T � τ � e.g. for state T A = S → ( A × S ). e.g. for partiality T A = ⊥ + A Monads provide sequential composition Given f : A → T B and g : B → T C then