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Subtyping

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  1. Subtyping Chapter 15 Benjamin Pierce Types and Programming Languages

  2. Varieties of Polymorphism • Parametric polymorphism A single piece of code is typed generically • Imperative or first-class polymorphism • ML-style or let-polymorphism • Ad-hoc polymorphism The same expression exhibit different behaviors when viewed in different types • Overloading • Multi-method dispatch • Intentional polymorphism • Subtype polymorphism A single term may have many types using the rule of subsumption allowing to selectively forget information

  3. Simple Typed Lambda Calculus t ::= terms x variable  x: T. t abstraction t t application T::= types  T  T types of functions

  4. x : T  (T-VAR)  x : T , x : T1 t2 : T2 (T-ABS)  x : T1. t2: T1 T2 Type Rules  t : T t ::= terms x variable  x: T. t abstraction  t2 : T11  t1 : T11T12 T::= types  T  T types of functions (T-APP)  t1 t2 : T12 ::= context   empty context , x : T term variable binding

  5. Records New syntactic forms New Evaluation Rules extends  t ::= …. Terms: {li=tii  1..n} recordt.l projection {li=vii  1..n }. lj  vj(E-ProjRCD)  t:{ li: Tii  1..n } For each i ti : Ti t  t’ (T-Tuple) (T-Proj) v ::= …. Values: {li=vi i  1..n} records  t.lj : Tj  {li=tii  1..n} : { li: Tii  1..n } (E-Proj} t.l t’.l T ::= …. types: {li:Tii  1..n } record type New typing rules tj  t’j (E-Tuple) {li=vii  1..j-1, li=tii  j..n}  {li=vii  1..j-1,lj=t’j,lk=tkk  j+1..n}

  6. Record Example (r : {x: Nat}. r.x) {x=0, y=0 } {x: Nat, y: Nat} <: {x: Nat} S <:T  t : S (T-SUB)  t : T  t2 : T11  t1 : T11T12 (T-APP)  t1 t2 : T12

  7. Healthiness of Subtypes S <: S (S-REFL) U <:T S <: U (S-TRANS) S <:T

  8. Width Record Subtyping {li:Tii 1..n+k } <: {li:Tii 1..n } (S-RCDWIDTH) {x: Nat} has at least the field x of type Nat {x: Nat, y : Nat} has at least the field x of type Nat and the field y of type Nat {x: Nat, y: Bool} has at least the field x of type Nat and the field y of type Bool {x: Nat, y: Nat} <: {x: Nat} {x: Nat, y: Bool} <: {x: Nat}

  9. Record Depth Subtyping For each i Si <: Ti (S-RCDEPTH) {li:Sii 1..n } <: {li:Tii 1..n }

  10. {x: {a: Nat, b: Nat}, y: {m:Nat}} <: {x: {a: Nat}, y: {}} {a:Nat, b: Nat} <: {a: Nat} (S-RCDWIDTH) {m: Nat} <: {} (S-RCDWIDTH) (S-RCDEPTH) {x: {a: Nat, b: Nat}, y: {m: Nat}} <: {x: {a: Nat}, y: {}}

  11. {x: {a: Nat, b: Nat}, y: Nat} <: {x: {a: Nat}, y: Nat}} {a:Nat, b: Nat} <: {a: Nat} (S-RCDWIDTH) Nat <: Nat (S-REFL) (S-RCDEPTH) {x: {a: Nat, b: Nat}, y: Nat} <: {x: {a: Nat}, y: {}}

  12. {x: {a: Nat, b: Nat}, y: {m: Nat}} <: {x: {a: Nat}} {x: {a:Nat, b: Nat}}, {y: {m: Nat}} <: {x: {a: Nat}, b:Nat}} (S-RCDWIDTH) {x: {a: Nat, b: Nat} <: {x: {a: Nat}} (S-RCDEPTH) {a: Nat, b: Nat} <: {a: Nat} (S-RCDWIDTH) (S-TRANS) {x: {a: Nat, b: Nat}, y: {m: Nat}} <: {x: {a: Nat}}

  13. Field Permutation {kj:Sjj  1..n } is a permutation of {li:Tii 1..n } (S-RCDPERM) {kj:Sjj  1..n } <: {li:Tii 1..n }

  14. Record Subtying • Forgetting fields (S-RCDWIDTH) • Forgetting subrecords (S-RCDEPTH) • Reordering fields (S-RCDPERM)

  15. S1 <: T1 (S-ARROWN) S1 S2 <: T1 T2 Naïve Handling of Functions S2 <: T2 n:{x: Nat, y: Nat}  n.x +1 : Nat n:{x: Nat, y: Nat}  n.y +2 : Nat} (T-RCD) n:{x: Nat, y: Nat}  {x: n.x +1 , y: n.y +2}) : {x: Nat, y: Nat} (S-ARROWN) {x:Nat, y: Nat}{x:Nat, y: Nat} <: {x: Nat} {x: Nat, y: Nat} {x:Nat, y: Nat} <: {x: Nat} (S-RCDWIDTH) {x:Nat, y: Nat} <: {x: Nat, y: Nat} (S-REFL) (T-ABS) (T-SUB)  n:{x: Nat, y: Nat}. {x: n.x +1 , y: n.y +2}) : {x: Nat, y: Nat}  {x: Nat, y: Nat}  n:{x: Nat, y: Nat}. {x: n.x +1 , y: n.y +2}) : {x: Nat}  {x: Nat, y: Nat}  1 : Nat (T-RECD) (T-APP)  {x: 1} : {x: Nat}  n:{x: Nat, y: Nat}. {x: n.x +1 , y: n.y +2}) {x:1} : {x: Nat, y: Nat}

  16. T1 <: S1 (S-ARROW) S1 S2 <: T1 T2 Handling of Functions S2 <: T2 • Arguments types are handled in reversed way (contravariant) • Result types are handled in the same direction (covariant)

  17. Top type T <: Top (S-TOP)

  18. Properties of the subtyping relation • Preorder on types

  19. t2 t’2 (E-APP2) v1 t2 v1t’2 SOS for Simple Typed Lambda Calculus t ::= terms x variable  x: T. t abstraction t t application t1 t2 t1 t’1 (E-APP1) t1 t2 t’1 t2 v::= values  x: T. t abstraction values ( x: T11. t12) v2 [x v2] t12 (E-APPABS) T::= types  Top maximum type T  T types of functions

  20. x : T  (T-VAR)  x : T , x : T1 t2 : T2 (T-ABS)  x : T1. t2: T1 T2 T1 <: S1 (S-ARROW) S1 S2 <: T1 T2 Type Rules  t : T S2 <: T2 S <: S (S-REFL) U <:T S <: U  t2 : T11  t1 : T11T12 (S-TRANS) (T-APP) S <:T  t : S S <:T (T-SUB)  t1 t2 : T12  t : T T <: Top (S-TOP)

  21. Records Evaluation Rules extends  syntactic forms {li=vii  1..n }. lj  vj(E-ProjRCD) t ::= …. Terms: {li=tii  1..n} recordt.l projection {kj:Sjj  1..n } is a permutation of {li:Tii 1..n } For each i Si <: Ti (S-RCDEPTH) (S-RCDPERM) {kj:Sjj  1..n } <: {li:Tii 1..n } {li:Sii 1..n } <: {li:Tii 1..n }  t:{ li: Tii  1..n } For each i ti : Ti t  t’ (T-Proj) (T-Tuple) v ::= …. Values: {li=vi i  1..n} records  {li=tii  1..n} : { li: Tii  1..n }  t.lj : Tj t.l t’.l (E-Proj} T ::= …. types: {li:Tii  1..n } record type New subtyping rules typing rules {li:Tii 1..n+k } <: {li:Tii 1..n } (S-RCDWIDTH) tj  t’j (E-Tuple) {li=vii  1..j-1, li=tii  j..n}  {li=vii  1..j-1,lj=t’j,lk=tkk  j+1..n}

  22. r : T’ = {x: T=Nat} (T-VAR) r :{x: Nat} r.x : T (T-ABS) Example {x:Nat, y:Nat} <: {x:Nat} (S-RCDWIDTH) r :{x: Nat}r : T’ = {… x: T …} 0: Nat 0: Nat (T-Tuple) (T-Proj)  {x=0, y=0} : S={ x: Nat, y: Nat} S <:{x : Nat} {x=0, y=0}: S (T-SUB)  r :{x: Nat}. r.x : {x: Nat} T  {x=0, y=0}: {x : Nat} (T-APP) (r : {x: Nat}. r.x) {x=0, y=0 } : T

  23. Properties of the type system(15.3) • Uniqueness of types • Linear time type checking • Type Safety • Well typed programs cannot go wrong • No undefined semantics • No runtime checks • If t is well typed then either t is a value or there exists an evaluation step t  t’ [Progress] • If t is well typed and there exists an evaluation step t  t’ then t’ is also well typed [Preservation]

  24. Upcasting (Information Hiding) • Special case of ascription S <: T t: S (T-SUB) t : T (T-ASCRIBE)  t as T: T

  25. Downcasting • Generate runtime assertion • But the typechecker can assume the right type t : S (T-DOWNCAST) t as T : T Progress is no longer guaranteed Can replace downcasts by dynamic type check v :T (E-DOWNCAST) v as T  v  t1 : S , x: T t2: U  t3: U (T-TYPETEST)  if t1in T then x  t2else t3: U v :T v :T (E-TYPETESTF) (E-TYPETESTT) if v in T then x  t2 else t3 [x v] t2 if v in T then x  t2 else t3 t3 (E-ASCRIBE) v as T  v

  26. Downcasting vs. Polymorphism • Downcasting is useful for reflection • Polymorphism saves the need for downcasts in most cases • reverse : X: list X  list X • Polymorphism leads to shorter and more secure code • Polymorphism can have better performance • Downcasting complicates the runtime system • Polymorphism complicates language definition • The interactions between polymorphism and subtyping complicates type checking/inference • Irrelevant for Java

  27. Variants Modified syntactic forms t ::= …. Terms: <l=t> as T tagging case t of <li = xi>  tii 1..n case v ::= …. Values: <l=v> as T tagged value T ::= …. types: <li = Tii 1..n> type of variants Evaluation Rules extends  ti  t’i (E-VARIANT) <l i=ti> as T <li=t’i>as T case (<lj = v> as T) of <li = xi>  tii 1..n  [xjv] tj (E-CaseVariant) t  t’ (E-CASE) case t of <li = xi>  tii 1..n  case t’ of <li = xi>  tii 1..n

  28. Modified Type rules for Variants New subtyping rules Modified syntactic forms t ::= …. Terms: <l=t> as T tagging case t of <li = xi>  tii 1..n case <li:Tii 1..n > <: <li:Tii 1..n +k> (S-VARIANTWIDTH) <kj:Sjj  1..n > is a permutation of <li:Tii 1..n > For each i Si <: Ti (S-VARIANTDEPTH) (S-VARIANTPERM) <li:Sii 1..n > <: <li:Tii 1..n > <kj:Sjj  1..n > <: <li:Tii 1..n > tj : Tj v ::= …. Values: <l=v> as T tagged value (T-VARIANT) <l j=tj>as <li = Tii 1..n> : < lj= Tj> T ::= …. types: <li = Tii 1..n> type of variants For each i , xi :Titi : T  t : <li = Tii 1..n> modified typing rules (T-CASE) case t of <li = xi>  tii 1..n : T

  29. S <: T (S-List) List S <: List T Lists

  30. S <: T T <: S (S-Ref) ref S <: ref T References

  31. S <: T T <: S S <: T (S-Array) (S-ArrayJava) Array S <: Array T Array S <: Array T Arrays

  32. Base Types Bool <: Nat (S-BoolNat)

  33. Coercion Semantics for Subtyping (15.6) • The compiler can perform conversions between types • Sometimes at compile time • Subtyping is no longer transparent • Improve performance

  34. Summary • Subtyping improves reuse • Tricky semantics • Complicates type checking/inference • Well understood for many language features