Monadic reflection in Lax Logic Tristan Crolard LACL University of - PowerPoint PPT Presentation

Monadic reflection in Lax Logic Tristan Crolard LACL – University of East Paris TPDC’11 Novi Sad – May 29, 2011

Introduction • Revisit “ Representing monads ” [Filinski, 1994] from a logical standpoint. • Goal: understand the logical meaning of shift / reset in the restricted framework of a major application, i.e. implementing monadic reflection. • Through the formulas-as-types interpretation, a monad ♦ ⊔ corresponds to the modality from Lax logic [Curry, 1952]: ⊢ unit : ϕ ⇒ ♦ ϕ ⊢ bind : ( ϕ ⇒ ♦ ψ ) ⇒ ♦ ϕ ⇒ ♦ ψ • Monadic reflection is given by these logical rules: Γ ⊢ t : ϕ Γ ⊢ t : ♦ ϕ [ reify ] [ reflect ] Γ ⊢ [ t ] : ♦ ϕ Γ ⊢ µ ( t ) : ϕ • In this talk, we consider only provability .

Example: the exception monad � f a | ( inr b ) � inr b . : type . ε ♦ ϕ = ϕ ∨ ε . = λa : ϕ. inl a . unit = λt. case t of ( inl a ) bind f � h � a | ( inr e ) � h . Defining raise and handle in direct style: raise = µ ( inr t ) . t handle e = case [ t ] of ( inl a )

Moggi’s monadic translation (CBV) Translation of types ( σ atomic): σ ♦ ≡ σ • ( ϕ ⇒ ψ ) ♦ ≡ ϕ ♦ ⇒ ♦ ψ ♦ • ( ♦ ψ ) ♦ ≡ ♦ ϕ ♦ • Translation of terms: x ♦ ≡ unit x • ( λx.t ) ♦ ≡ unit λx.t ♦ • ( t 1 t 2 ) ♦ ≡ bind ♦ �� ♦ � � • bind f t 2 λf. t 1 µ ( t ) ♦ ≡ bind id t ♦ • [ t ] ♦ ≡ unit t ♦ • Lemma. If Γ ⊢ t : ϕ is derivable then Γ ♦ ⊢ t ♦ : ♦ ϕ ♦ is derivable. Proof. Γ ♦ ⊢ t ♦ : ♦ ϕ ♦ Γ ♦ ⊢ t ♦ : ♦♦ ϕ ♦ Γ ♦ ⊢ unit t ♦ : ♦♦ ϕ ♦ [ unit ] Γ ♦ ⊢ bind id t ♦ : ♦ ϕ ♦ [ join ]

Filinski’s CPS-translation (CBV) Define ∇ ϕ = ( ϕ ⇒♦ o ) ⇒♦ o where o is some universal answer type (not sound). Translation of types ( σ atomic): σ ∇ ≡ σ • ( ϕ ⇒ ψ ) ∇ ≡ ϕ ∇ ⇒ ∇ ψ ∇ • ( ♦ ψ ) ∇ ≡ ♦ ϕ ∇ • Translation of terms: x ∇ ≡ λk.k x • ( λx.t ) ∇ ≡ λk.k ( λx.t ∇ ) • ∇ ( λf.t 2 ∇ ( λa.f a k )) ( t 1 t 2 ) ∇ ≡ λk.t 1 • µ ( t ) ∇ ≡ λk.t ∇ ( bind k ) • [ t ] ∇ ≡ λk.k ( t ∇ unit ) •

Delimited control Reflect/reify are definable in direct style from shift / reset [Filinski, 1994] = reset ( unit t ) . [ t ] µ ( t ) = shift ( λk. bind k t ) . That is, these equations are valid: ( reset ( unit t )) ∇ = [ t ] ∇ • ( shift ( λk. bind k t )) ∇ = µ ( t ) ∇ • where: reset ∇ = λm.λc. ( c ( m id )) • shift ∇ = λh.λc. ( h ( λv.λc ′ .c ′ ( c v )) id ) •

Answer type polymorphism Footnote from “ Representing monads ”: “Alternatively, with a little more care, we can take ∇ ϕ = ∀ α ( ϕ →♦ α ) →♦ α ; it is straightforward to check that both the term translation and the operations defined in the following can in fact be typed according to this schema.” So, let us do it carefully : • Formalization in Twelf (work in progress) • Experimenting with TeXmacs as a front end (the Twelf source is generated from the slides)

Plan of the rest of the talk • Formalize System F in Twelf • Check that the operations are well-typed in direct style • Check that the CPS-translations of the operations are well-typed • Interpret the logical type of shift for the usual monads: − continuation monad − state monad − exception monad

System F (HOAS) Types : type . type ⊔ ⇒ ⊔ : type → type → type . � 2 ⊔ ∧ ⊔ : type → type → type . ⊔ ∨ ⊔ : type → type → type . ∀ ⊔ . ⊔ : ( type → type ) → type . binding 1 in ∀ ⊔ . ⊔ void = ∀ β.β .

Terms : type . term � 3 Abstraction λ ⊔ : ⊔ . ⊔ : type → ( term → term ) → term . binding in 1 λ ⊔ : ⊔ . ⊔ Application : term → term → term . ⊔ ⊔ � 2 Polymorphic abstraction Λ ⊔ . ⊔ : ( type → term ) → term . binding 1 in Λ ⊔ . ⊔

Instantiation ⊔ { ⊔ } : term → type → term . � 4 Derived syntax for let let ⊔ : ⊔ = ⊔ in ⊔ = [ τ ] [ u ] [ t ] ( λx : τ.t [ x ]) u . binding 1 in let ⊔ : ⊔ = ⊔ in ⊔ Pairing � ⊔ , ⊔ � : term → term → term . � 4 Pattern matching � 4 let � ⊔ , ⊔ � = ⊔ in ⊔ : term → ( term → term → term ) → term . binding 2 in let � ⊔ , ⊔ � = ⊔ in ⊔ binding 1 in let � ⊔ , ⊔ � = ⊔ in ⊔

Injections inl ⊔ : term → term . � ⊔ | ( inr ⊔ ) � ⊔ inr ⊔ : term → term . � 5 � ⊔ | ( inr ⊔ ) � ⊔ Pattern matching � 3 � ⊔ | ( inr ⊔ ) � ⊔ case ⊔ of ( inl ⊔ ) : term → ( term → term ) → ( term → term ) → term . binding 4 in case ⊔ of ( inl ⊔ ) binding 2 in case ⊔ of ( inl ⊔ ) Monadic constants unit : term . bind : term . Delimited control operators reset : term . shift : term .

Typing judgment ⊢ ⊔ : ⊔ : term → type → type . { x } ⊢ x : ϕ → ⊢ t [ x ] : ψ [ lam ] ⊢ λx : ϕ.t [ x ] : ϕ ⇒ ψ ⊢ t 1 : ϕ ⇒ ψ ⊢ t 2 : ϕ [ app ] ⊢ t 1 t 2 : ψ { α } ⊢ t [ α ] : ψ [ α ] [ abs ] ⊢ Λ α.t [ α ] : ∀ α.ψ [ α ] ⊢ t : ∀ α.ψ [ α ] [ inst ] ⊢ t { ϕ } : ψ [ ϕ ]

Typing judgment ⊢ t 1 : ϕ ⊢ t 2 : ψ [ pair ] ⊢ � t 1 , t 2 � : ϕ ∧ ψ { x } ⊢ x : ϕ → ( { y } ⊢ y : ψ → ⊢ u [ x ][ y ] : τ ) ⊢ t : ϕ ∧ ψ [ match ] ⊢ let � x, y � = t in u [ x ][ y ] : τ ⊢ t : ψ [ inr ] ⊢ inr t : ϕ ∨ ψ � u 1 [ x ] | ( inr y ) � u 2 [ y ] : φ ⊢ t : ϕ [ inl ] ⊢ inl t : ϕ ∨ ψ { x } ⊢ x : ϕ → ⊢ u 1 [ x ] : φ { y } ⊢ y : ψ → ⊢ u 2 [ y ] : φ ⊢ t : ϕ ∨ ψ [ case ] ⊢ case t of ( inl x )

Lax logic Primitive monad or lax modality [Curry, 1952] ♦ ⊔ : type → type . ⊢ unit : ϕ ⇒ ♦ ϕ [ unit ] ⊢ bind : ( ϕ ⇒ ♦ ψ ) ⇒ ♦ ϕ ⇒ ♦ ψ [ bind ]

Delimited control Fixed answer type : type . o ⊢ reset : ♦ ϕ ⇒ ♦ ϕ [ reset ] ⊢ shift : (( ϕ ⇒ ♦ o ) ⇒ ♦ o ) ⇒ ϕ [ shift ]

Monadic reflection Reflect/reify are definable from shift / reset : = reset ( unit t ) . [ t ] µ ( t ) = shift ( λk : ϕ ⇒ ♦ o. bind k t ) . Lemma. The following typing rules are derivable: ⊢ t : ϕ ⊢ t : ♦ ϕ [ reify ] [ reflect ] ⊢ [ t ] : ♦ ϕ ⊢ µ ( t ) : ϕ %solve ⊢ t : ϕ → ⊢ [ t ] : ♦ ϕ %solve ⊢ t : ♦ ϕ → ⊢ µ ( t ) : ϕ

Polymorphic monadic reflection Polymorphic type for shift : ⊢ shift : ∀ α. (( ϕ ⇒ ♦ α ) ⇒ ♦ α ) ⇒ ϕ [ shift ] Reflect is still definable from shift : µ ( t ) = shift (Λ α.λk : ϕ ⇒ ♦ α. bind k t ) . Lemma. The following typing rule is derivable: ⊢ t : ♦ ϕ [ reflect ] ⊢ µ ( t ) : ϕ %solve ⊢ t : ♦ ϕ → ⊢ µ ( t ) : ϕ Remark. ∀ α. (( ϕ ⇒ ♦ α ) ⇒ ♦ α ) is equivalent to ♦ ϕ . %solve ⊢ t : ♦ ϕ → ⊢ Λ α.λk : ϕ ⇒ ♦ α. bind k t : ∀ α. (( ϕ ⇒ ♦ α ) ⇒ ♦ α ) %solve ⊢ f : ∀ α. (( ϕ ⇒ ♦ α ) ⇒ ♦ α ) → ⊢ ( f { ϕ } unit ) : ♦ ϕ

� Derived typing rule for reify lemma ⊢ t : ϕ ⊢ [ t ] : ♦ ϕ � Proof. D of 1 [ unit ] � ⊢ t : ϕ ⊢ unit : ϕ ⇒ ♦ ϕ � [ reset ] [ app ] D of 1 ⊢ reset : ♦ ϕ ⇒ ♦ ϕ ⊢ unit t : ♦ ϕ [ app ] [&] � ⊢ t : ϕ ⊢ reset ( unit t ) : ♦ ϕ %mode + D of 1 − D of 2 %worlds () D of 1 D of 2 %total {} D of 1 D of 2

� Derived typing rule for reflect lemma ⊢ t : ♦ ϕ ⊢ µ ( t ) : ϕ Proof. D of k [ bind ] ⊢ k : ϕ ⇒ ♦ α D of t � [ app ] ⊢ bind k : ♦ ϕ ⇒ ♦ α ⊢ t : ♦ ϕ [ D of k ] [ app ] ⊢ bind k t : ♦ α [ k ] ⊢ k : ϕ ⇒ ♦ α → ⊢ bind k t : ♦ α { k } ⊢ k : ϕ ⇒ ♦ α → ⊢ bind k t : ♦ α [ α ] [ lam ] ⊢ λk : ϕ ⇒ ♦ α. bind k t : ( ϕ ⇒ ♦ α ) ⇒ ♦ α [ shift ] � [ abs ] D of t ⊢ Λ α.λk : ϕ ⇒ ♦ α. bind k t : ∀ α. (( ϕ ⇒ ♦ α ) ⇒ ♦ α ) [ app ] [& � ⊢ t : ♦ ϕ ⊢ shift (Λ α.λk : ϕ ⇒ ♦ α. bind k t ) : ϕ � %mode + D of 1 − D of 2 %worlds () D of 1 D of 2 %total {} D of 1 D of 2

Different continuation monads 1. Continuation monad ∇ ϕ = ( ϕ ⇒ o ) ⇒ o . 2. Modal continuation monad ∇ ϕ = ( ϕ ⇒ ♦ o ) ⇒ ♦ o . 3. Polymorphic continuation monad ∇ ϕ = ∀ α. ( ϕ ⇒ α ) ⇒ α . 4. Polymorphic modal continuation monad ∇ ϕ = ∀ α. ( ϕ ⇒ ♦ α ) ⇒ ♦ α . Remark. Cases 1 and 3 are obtained by taking ♦ as the identity monad.

Modal continuation monad ∇ ϕ = ( ϕ ⇒ ♦ o ) ⇒ ♦ o . = λt : ϕ.λk : ϕ ⇒ ♦ o. ( k t ) . unit ∇ = λk : ϕ ⇒ ∇ ψ.λm : ∇ ϕ.λc : ψ ⇒ ♦ o.m ( λv : ϕ.k v c ) . bind ∇ %solve ⊢ unit ∇ : ϕ ⇒ ∇ ϕ %solve ⊢ bind ∇ : ( ϕ ⇒ ∇ ψ ) ⇒ ∇ ϕ ⇒ ∇ ψ

Polymorphic modal continuation monad ∇ ϕ = ∀ α. ( ϕ ⇒ ♦ α ) ⇒ ♦ α . = λt : ϕ. Λ α.λk : ϕ ⇒ ♦ α. ( k t ) . unit ∇ = λm : ∇ ϕ.λk : ϕ ⇒ ∇ ψ. Λ α.λc : ψ ⇒ ♦ α.m { α } ( λv : ϕ. ( k v ) { α } c ) . bind ∇ %solve ⊢ unit ∇ : ϕ ⇒ ∇ ϕ %solve ⊢ bind ∇ : ∇ ϕ ⇒ ( ϕ ⇒ ∇ ψ ) ⇒ ∇ ψ