Parametric Compositional Data Types Patrick Bahr Tom Hvitved - PowerPoint PPT Presentation

Parametric Compositional Data Types Patrick Bahr Tom Hvitved University of Copenhagen, Department of Computer Science { paba , hvitved } @ diku.dk Mathematically Structured Functional Programming 2012, Tallinn, Estonia, March 25th, 2012

Outline Motivation 1 Compositional Data Types 2 Higher-Order Abstract Syntax 3 2

The Issue Implementation/Prototyping of DSLs Rule Language UI Contract Language Language ERP Runtime Ontology Report System Language Language ... ... 3

The Issue Implementation/Prototyping of DSLs Rule Language UI Contract Language Language ERP Runtime Ontology Report System Language Language ... ... 3

The Issue Implementation/Prototyping of DSLs Rule Language UI Contract Language Language ERP Runtime Ontology Report System Language Language ... ... 3

The Issue Implementation/Prototyping of DSLs The abstract picture Rule Language We have a number of UI Contract domain-specific languages. Language Language Each pair of DSLs shares some ERP common sublanguage. Runtime Ontology Report System All of them share a common Language Language language of values. We have the same situation on ... ... the type level! 3

The Issue Implementation/Prototyping of DSLs The abstract picture Rule Language We have a number of UI Contract domain-specific languages. Language Language Each pair of DSLs shares some ERP common sublanguage. Runtime Ontology Report System All of them share a common Language Language language of values. We have the same situation on ... ... the type level! How do we implement this system without duplicating code?! 3

More General Application Even with only one language to implement this issue appears! 4

More General Application Even with only one language to implement this issue appears! Different stages of a compiler work on different languages. Desugaring: FullExp → CoreExp Evaluation: Exp → Value . . . 4

More General Application Even with only one language to implement this issue appears! Different stages of a compiler work on different languages. Desugaring: FullExp → CoreExp Evaluation: Exp → Value . . . Manipulating/extending syntax trees annotating syntax trees adding/removing type annotations 4

Compositional Data Types data Exp = Lit Int | Add Exp Exp | Mult Exp Exp 5

Compositional Data Types data Exp = Lit Int | Add Exp Exp s recursion i g | Mult Exp Exp n a t u r e decompose data Sig e = Lit Int data Term f = | Add e e In ( f ( Term f )) | Mult e e 5

Compositional Data Types data Exp = Lit Int | Add Exp Exp s recursion i g | Mult Exp Exp n a t u r e decompose data Sig e = Lit Int data Term f = | Add e e In ( f ( Term f )) | Mult e e combine type Exp = Term Sig 5

Compositional Data Types data Exp = Lit Int | Add Exp Exp s n i o g | Mult Exp Exp n i s a r u t u c r e e r decompose data Sig e = Lit Int data Term f = | Add e e In ( f ( Term f )) | Mult e e combine type Exp = Term Sig 5

Compositional Data Types data Lit e = Lit Int data Exp = Lit Int | Add Exp Exp s n i o g | Mult Exp Exp n i s a r u t u c r e e r decompose data Sig e = Lit Int data Term f = | Add e e In ( f ( Term f )) | Mult e e combine type Exp = Term Sig data Ops e = Add e e | Mult e e 5

Compositional Data Types data Lit e = Lit Int data Exp = Lit Int | Add Exp Exp s n i o g | Mult Exp Exp n i s a r u t u c r e e r decompose data Sig e = Lit Int data Term f = Lit ⊕ Ops | Add e e In ( f ( Term f )) | Mult e e combine type Exp = Term Sig data Ops e = Add e e | Mult e e 5

Compositional Data Types data Lit e = Lit Int data Exp = Lit Int | Add Exp Exp s n i o g | Mult Exp Exp n i s a r u t u c r e e r decompose data Sig e = Lit Int data Term f = Lit ⊕ Ops ⊕ ... | Add e e In ( f ( Term f )) | Mult e e combine type Exp = Term Sig data Ops e = Add e e | Mult e e 5

Compositional Data Types data Lit e = Lit Int data Exp = Lit Int | Add Exp Exp s n i o g | Mult Exp Exp n i s a r u t u c r e e r decompose annotations data Sig e = Lit Int data Term f = Lit ⊕ Ops ⊕ ... | Add e e In ( f ( Term f )) | Mult e e combine type Exp = Term Sig data Ops e = Add e e | Mult e e 5

Variable Binding A straightforward solution type Name = String data Lam e = Lam Name e data Var e = Var Name data App e = App e e 6

Variable Binding A straightforward solution type Name = String data Lam e = Lam Name e type Sig = Lam ⊕ Var ⊕ App data Var e = Var Name type Lambda = Term Lam data App e = App e e 6

Variable Binding A straightforward solution type Name = String data Lam e = Lam Name e type Sig = Lam ⊕ Var ⊕ App data Var e = Var Name type Lambda = Term Lam data App e = App e e Issues Definitions modulo α -equivalence Capture-avoiding substitutions Implementing embedded languages 6

Variable Binding A straightforward solution type Name = String data Lam e = Lam Name e type Sig = Lam ⊕ Var ⊕ App data Var e = Var Name type Lambda = Term Lam data App e = App e e Issues Definitions modulo α -equivalence Capture-avoiding substitutions Implementing embedded languages Goal Use higher-order abstract syntax to leverage the variable binding mechanism of the host language. 6

Higher-Order Abstract Syntax Explicit Variables type Name = String data Lam e = Lam Name e data Var e = Var Name data App e = App e e 7

Higher-Order Abstract Syntax Explicit Variables Higher-Order Abstract Syntax type Name = String data Lam e = Lam Name e data Lam e = Lam ( e → e ) data Var e = Var Name data App e = App e e data App e = App e e 7

Higher-Order Abstract Syntax Explicit Variables Higher-Order Abstract Syntax type Name = String data Lam e = Lam Name e data Lam e = Lam ( e → e ) data Var e = Var Name data App e = App e e data App e = App e e 7

Higher-Order Abstract Syntax Lam "x" ( ... Var "x" ... ) Lam ( λ x → ... x ... ) Explicit Variables Higher-Order Abstract Syntax type Name = String data Lam e = Lam Name e data Lam e = Lam ( e → e ) data Var e = Var Name data App e = App e e data App e = App e e 7

Higher-Order Abstract Syntax Lam "x" ( ... Var "x" ... ) Lam ( λ x → ... x ... ) Explicit Variables Higher-Order Abstract Syntax type Name = String data Lam e = Lam Name e data Lam e = Lam ( e → e ) data Var e = Var Name data App e = App e e data App e = App e e Issues inefficient and cumbersome recursion schemes (catamorphism needs an algebra and the inverse coalgebra) Full function space allows for exotic terms 7

Higher-Order Abstract Syntax Lam "x" ( ... Var "x" ... ) Lam ( λ x → ... x ... ) Explicit Variables Higher-Order Abstract Syntax type Name = String data Lam e = Lam Name e data Lam e = Lam ( e → e ) data Var e = Var Name data App e = App e e data App e = App e e Issues inefficient and cumbersome recursion schemes (catamorphism needs an algebra and the inverse coalgebra) � Fegaras & Sheard (1996): parametric functions space Full function space allows for exotic terms 7

Higher-Order Abstract Syntax Lam "x" ( ... Var "x" ... ) Lam ( λ x → ... x ... ) Explicit Variables Higher-Order Abstract Syntax type Name = String data Lam e = Lam Name e data Lam e = Lam ( e → e ) data Var e = Var Name data App e = App e e data App e = App e e Issues inefficient and cumbersome recursion schemes (catamorphism needs an algebra and the inverse coalgebra) � Fegaras & Sheard (1996): parametric functions space Full function space allows for exotic terms � Washburn & Weirich (2008): polymorphism & abstract type of terms 7

Parametric Higher-Order Abstract Syntax [Chlipala 2008] Idea Signature for lambda bindings: data Lam a e = Lam ( a → e ) 8

Parametric Higher-Order Abstract Syntax [Chlipala 2008] Idea Signature for lambda bindings: data Lam a e = Lam ( a → e ) Recursive construction of terms: data Trm f a = In ( f a ( Trm f a )) 8

Parametric Higher-Order Abstract Syntax [Chlipala 2008] Idea Signature for lambda bindings: data Lam a e = Lam ( a → e ) Recursive construction of terms: data Trm f a = In ( f a ( Trm f a )) type Term f = ∀ a . Trm f a 8

Parametric Higher-Order Abstract Syntax [Chlipala 2008] Idea Signature for lambda bindings: data Lam a e = Lam ( a → e ) Recursive construction of terms: data Trm f a = In ( f a ( Trm f a )) | Var a type Term f = ∀ a . Trm f a 8

Parametric Higher-Order Abstract Syntax [Chlipala 2008] Idea Signature for lambda bindings: data Lam a e = Lam ( a → e ) Recursive construction of terms: data Trm f a = In ( f a ( Trm f a )) | Var a newtype Term f = Term ( ∀ a . Trm f a ) 8

Parametric Higher-Order Abstract Syntax [Chlipala 2008] Idea Signature for lambda bindings: data Lam a e = Lam ( a → e ) Recursive construction of terms: data Trm f a = In ( f a ( Trm f a )) | Var a newtype Term f = Term ( ∀ a . Trm f a ) Example data Sig a e = Lam ( a → e ) | App e e 8

Parametric Higher-Order Abstract Syntax [Chlipala 2008] Idea Signature for lambda bindings: data Lam a e = Lam ( a → e ) Recursive construction of terms: data Trm f a = In ( f a ( Trm f a )) | Var a newtype Term f = Term ( ∀ a . Trm f a ) Example data Sig a e = Lam ( a → e ) | App e e e :: Term Sig e = Term $ Lam ( λ x → Var x ‘ App ‘ Var x ) 8