Row polymorphism 1/ 25 Record operations 2. Extend a record with a - PowerPoint PPT Presentation

Row polymorphism 1/ 25

Record operations 2. Extend a record with a field ( extend ) 3. Access the contents of a field ( access ) 2/ 25 1. An empty record ( empty )

Presence variables : ∀𝛽 : ∗ . ∀𝛾 : ∗ . ∀𝛿 : ∗ . ∀𝜀 : ∗ . ∀𝜒 : ∗ ⇒ ∗ . 𝜀 → Record 𝛽 𝛾 𝛿 → Record ( 𝜒 𝜀 ) 𝛾 𝛿 where 𝜒 is a type constructor variable that can be instantiated with: 3/ 25 extend ���� � Present � ( 𝜇𝛽 : ∗ . Absent)

Ill-formed records Polymorphic record types allow some ill-formed type expressions: These are prevented using the kind system by creating a new kind presence such that: Absent : presence Present : ∗ ⇒ presence Record : presence ⇒ presence ⇒ presence ⇒ ∗ 4/ 25 � Record Int (Present String) (Present String) � List (Present Int)

Infinite records What if we had infinite record types: { . . . ; foo : bar ; . . . } 5/ 25

Infinite records . . . . . . ; l : 𝛿 � ; . . . } : ∀ 𝛽 : ∗ . . . . empty { . . . ; m : Present 𝛽 ; . . . ; l : 𝛾 � ; . . . } → 𝛽 ; m : 𝜒 𝛽 ; { . . . . . . } → ; : { . . . ; l : Absent ; . . . } : ∀ 𝛽 : ∗ . ∀ 𝛾 : presence . . . . . . . ∀ 𝜒 : ∗ ⇒ presence . 𝛽 → { . . . ; m : 𝛾 ; . . . ; l : 𝛿 � 6/ 25 extend � . ∀ 𝛿 � : presence . access � . ∀ 𝛾 � : presence .

Infinite records Each record type appearing above can be divided into two parts: 1. A finite part 2. A co-finite part where either every type parameter is a free variable or every type parameter is Absent . 7/ 25

Infinite records . . . Co-finite . . . } ; Absent : l ; { { . . . Finite { } . . . } ; Absent : l ; 8/ 25

Infinite records . . . Co-finite . . . } ; Absent : l ; { { . . . Finite { } . . . } ; Absent : l ; 8/ 25

Infinite records Finite Co-finite . . . } ; 𝛿 � : l ; { . . . {m : 𝛾 } { . . . . . . } ; 𝛿 � : l ; . . . ; m : 𝛾 ; 9/ 25

Infinite records Finite Co-finite . . . } ; 𝛿 � : l ; { . . . {m : 𝛾 } { . . . . . . } ; 𝛿 � : l ; . . . ; m : 𝛾 ; 9/ 25

Infinite records Finite Co-finite . . . } ; 𝛿 � : l ; { . . . {m : 𝜒 𝛽 } { . . . . . . } ; 𝛿 � : l ; . . . ; m : 𝜒 𝛽 ; 10/ 25

Infinite records Finite Co-finite . . . } ; 𝛿 � : l ; { . . . {m : 𝜒 𝛽 } { . . . . . . } ; 𝛿 � : l ; . . . ; m : 𝜒 𝛽 ; 10/ 25

Infinite records Present 𝛽 } Co-finite . . . } ; : 𝛾 � l ; { . . . Finite { m : { . . . . . . } ; : 𝛾 � l ; . . . Present 𝛽 ; ; m : 11/ 25

Infinite records Present 𝛽 } Co-finite . . . } ; : 𝛾 � l ; { . . . Finite { m : { . . . . . . } ; : 𝛾 � l ; . . . Present 𝛽 ; ; m : 11/ 25

Row variables { . . . ; l : Absent ; . . . } ⇓ { } 12/ 25

Row variables { . . . ; l : Absent ; . . . } ⇓ { } 12/ 25

Row variables { . . . ; m : Present 𝛽 ; . . . ; l : 𝛾 � ; . . . } ⇓ {m : Present 𝛽 | 𝜍 } 13/ 25

Row variables { . . . ; m : Present 𝛽 ; . . . ; l : 𝛾 � ; . . . } ⇓ {m : Present 𝛽 | 𝜍 } 13/ 25

Row variables empty : {} : ∀𝛽 : ∗ . ∀ 𝛾 : presence . ∀ 𝜍 : row (m) . ∀ 𝜒 : ∗ ⇒ presence . 𝛽 → {m : 𝛾 | 𝜍 } → {m : 𝜒 𝛽 | 𝜍 } : ∀𝛽 : ∗ . ∀ 𝜍 : row (m) . {m : Present 𝛽 | 𝜍 } → 𝛽 14/ 25 extend � access �

Variant operations 2. Extend a match with a variant constructor ( extend_match ) 3. Use a variant constructor ( create ) 15/ 25 1. Match a variant with no constructors ( match_empty )

Variant operations l e t ( square () = print_constant l e t match_empty )) s ) s − > p r i n t _ s t r i n g f ) f − > p r i n t _ f l o a t i ) i − > p r i n t _ i n t 16/ 25 print_constant = l e t match_empty ) f )) *. i )) * square = extend_match ��� ( fun i − > c r e a t e ��� ( i ( extend_match ����� ( fun f − > c r e a t e ����� ( f extend_match ��� ( fun ( extend_match ����� ( fun ( extend_match ������ ( fun ( c r e a t e ��� 5))

Variant operations match_empty : ∀ 𝛽 : ∗ . 𝜍 ] Present 𝛽 | 𝛽 → [M : : ∀𝛽 : ∗ . ∀ 𝜍 : row (M) . 𝜍 ] → 𝛿 [M : 𝜒 𝛽 | 𝜍 ] → 𝛿 ) → ( 𝛽 → 𝛿 ) → ( [M : 𝛾 | ∗ ⇒ presence . ∀ 𝜍 : row (M) . ∀ 𝜒 : ∀ 𝛽 : ∗ . ∀ 𝛾 : presence . ∀ 𝛿 : ∗ . ] → 𝛽 [ 17/ 25 extend_match � : c r e a t e �

Object types < foo : i n t ; bar : f l o a t > An object type where the method foo has type int and the method bar has type float . Both methods are present, and all other methods are absent. 18/ 25

Object types < foo : i n t ; bar : f l o a t ; . . > The object may contain other methods besides foo and bar . In other words, the .. represents an unnamed row variable. 19/ 25

Object limitations v a l : ' a − > ' b − > ' c − > < l : ' a ; m : ' b ; n : ' c > 20/ 25 Instead of extend � we have: c r e a t e �����

Polymorphic variant types [ ‘ Foo of i n t | ‘ Bar of f l o a t ] Represents a variant type where the constructor `Foo has type int and the constructor `Bar has type float . Both constructors are definitely present. 21/ 25

Polymorphic variant types [< ‘ Foo of i n t | ‘ Bar of f l o a t ] The variant is polymorphic in the presence of both constructors. In other words, the < represents two unnamed presence variables. 22/ 25

Polymorphic variant types [< ‘ Foo of i n t | ‘ Bar of f l o a t > ‘ Bar ] The variant is only polymorphic in the presence of the `Foo constructor – the `Bar constructor is definitely present. In other words, the < represents a single unnamed presence variable associated with `Foo . 23/ 25

Polymorphic variant types [> ‘ Foo of i n t | ‘ Bar of f l o a t ] The variant may contain more constructors than just `Foo and `Bar . In other words, the > represents an unnamed row variable. Constructors `Foo and `Bar are definitely present. 24/ 25

Variant limitations v a l match ����� : ( ' a − > ' d) − > ( ' b − > ' d) − > ( ' c − > ' d) − > ' a | `M of ' b | `N of ' c ] − > ' d 25/ 25 Instead of extend_match � we have: [< `L of