Phantom Types and Subtyping

1
Phantom Types and Subtyping

• Matthew Fluet Riccardo Pucella
• Dept. of Computer Science
• Cornell University

2
The Setting (I)

• Modern strongly typed functional language
• Parametric polymorphism
• val id ? ? ? fn x gt x
• Type constraints
• val idInt int ? int fn x gt x
• Datatypes and type constructors
• datatype ? tree
• Leaf of ?
• Node of ? tree ? ? tree

3
The Setting (II)

• Modern strongly typed functional language
• Limited expressibility at foreign function
  interfaces
• No polymorphic types
• No user defined datatypes
• No primitive notion of subtyping

4
The Problem

• datatype atom I of Int B of bool
• fun mkI (iint)atom I(i)
• fun mkB (bbool)atom B(b)
• fun toString (vatom)string ...
• fun double (vatom)atom ...
• fun conj (v1atom, v2atom)atom ...

toString (mkI 1) ? 1 toString (mkB
false) ? false
double (mkB true) ? run-time
error conj (mkI 3, mkB true) ? run-time error
5
Wish List

• Raise compile-time type errors on domain
  violations rather than run-time errors
• toString should apply to all atoms
• double should only apply to integer atoms
• conj should only apply to boolean atoms
• Preserve the implementation
• Would like to treat integer and boolean atoms as
  subtypes of all atoms

6
A First Solution (I)

• type All, Int, Bool
• datatype ? atom I of int B of bool
• fun mkI (iint)Int atom ...
• fun mkB (bbool)Bool atom ...
• fun toString (vAll atom)string ...
• fun double (vInt atom)Int atom ...
• fun conj (v1Bool atom, v2Bool atom)Bool atom
  ...

double (mkB true) ? compile-time type error
Int atom ? Bool atom conj (mkI 3, mkB true) ?
compile-time type error Bool atom ? Int atom
toString (mkI 1) ? compile-time type
error Int atom ? All atom toString (mkB false)
? compile-time type error Bool atom ? All atom
7
Phantom Types

• type All, Int, Bool
• datatype ? atom I of int B of bool
• fun mkI (iint)Int atom ...
• fun mkB (bbool)Bool atom ...
• Phantom types
• Abstract types that need not have any
  corresponding run-time values
• Phantom type variables
• Type instantiations of ? in ? atom do not
  contribute to the run-time representation of atoms

8
A First Solution (II)

• type All, Int, Bool
• datatype ? atom I of int B of bool
• fun mkI (iint)Int atom ...
• fun mkB (bbool)Bool atom ...
• fun toString (vAll atom)string ...
• fun double (vInt atom)Int atom ...
• fun conj (v1Bool atom, v2Bool atom)Bool atom
  ...
• fun intToAll (vInt atom)All atom v
• fun boolToAll (vBool atom)All atom v

toString (intToAll (mkI 1)) ? 1
toString (boolToAll (mkB false)) ? false
9
A Better Solution

• type All, Int, Bool
• datatype ? atom I of int B of bool
• fun mkI (iint)Int atom ...
• fun mkB (bbool)Bool atom ...
• fun toString (v? atom)string ...
• fun double (vInt atom)Int atom ...
• fun conj (v1Bool atom, v2Bool atom)Bool atom
  ...

double (mkB true) ? compile-time type error
Int atom ? Bool atom conj (mkI 3, mkB true) ?
compile-time type error Bool atom ? Int atom
toString (mkI 1) ? well typed Int atom
unifies with ? atom toString (mkB false) ? well
typed Bool atom unifies with ? atom
10
The Phantom Types Technique

• Use a superfluous type variable and type
  constraints to encode extra information
• Underlies many interesting uses of type systems
• Foreign function interfaces
• Embedded languages
• Uncaught exception analysis
• A folklore technique

11
Contributions

• A general encoding of subtyping hierarchies into
  phantom types
• A formalization of one use of the phantom types
  technique

12
Outline

• A recipe for interfaces and implementations
• Encoding subtyping hierarchies
• Bounded polymorphism
• Formalization

13
From Subtyping to Polymorphism

• Features of the example
• An underlying primitive type of values
• A set of operations
• A hierarchy of implicit subtypes

• mkI 1
• Int
• int atom
• toString
• All ? string
• ? atom ? string

14
The Recipe

• Given
• A primitive type ?p
• An implicit subtyping hierarchy ?1,,?n
• An implementation of ?p and its operations
• Derive
• A safe interface (the types)
• A safe implementation (the code)
• Restrictions
• Shared representation and operations

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Applying the Recipe

• Given
• A primitive type atom
• An implicit subtyping hierarchy All, Int, Bool
  
• An implementation structure Atom
• Derive
• A safe interface signature SAFE_ATOM
• A safe implementation structure SafeAtom

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Deriving the Interface (I)

• ??1?C unifies with ??2?A iff ?1 ? ?2

• Introduce type ? ?
• Encode each implicit type ? as ??? ?
• ??1? unifies with ??2? iff ?1 ? ?2

• Example
• ?All?C unit ?All?A ?
• ?Int?C int ?Int?A int
• ?Bool?C bool ?Bool?A bool

17
Deriving the Interface (II)

• Use concrete encodings in all covariant type
  positions
• Use abstract encodings in most contravariant type
  positions

18
Deriving the Interface (III)

• signature ATOM sig
• type atom
• val mkI int -gt atom
• val mkB bool -gt atom
• val toString atom -gt string
• val double atom -gt atom
• val conj atom atom -gt atom
• end
• ?
• signature SAFE_ATOM sig
• type ? atom
• val mkI int -gt ?Int?C atom
• val mkB bool -gt ?Bool?C atom
• val toString ?All?A atom -gt string
• val double ?Int?A atom -gt ?Int?C atom
• val conj ?Bool?A atom ?Bool?A atom -gt
  ?Bool?C atom
• end

19
Applying the Recipe

• Given
• An abstract type atom ?p
• An implicit subtyping hierarchy All, Int, Bool
  ?p
• An implementation structure Atom ?p
• Derive
• A safe interface signature SAFE_ATOM
• A safe implementation structure SafeAtom

?
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Deriving the Implementation (I)

• Need a type ? ? isomorphic to ?p
• the type system should consider ?1 ? and ?2 ?
  equivalent iff ?1 and ?2 are equivalent
• Opaque signature constraint
• Hides all type implementation details

21
Deriving the Implementation (II)

• structure SafeAtom1gt SAFE_ATOM struct
• type ? atom Atom.atom
• val mkI Atom.mkI
• val mkB Atom.mkB
• val toString Atom.toString
• val double Atom.double
• val conj Atom.conj
• end

22
Applying the Recipe

• Given
• An abstract type atom ?p
• An implicit subtyping hierarchy All, Int, Bool
  ?p
• An implementation structure Atom ?p
• Derive
• A safe interface signature SAFE_ATOM
• A safe implementation structure SafeAtom

?
?
23
Encoding Subtyping Hierarchies (I)

• Powerset lattice encoding
• S s1,,sn is a finite set
• Ordered by inclusion
• X ? S
• ?X?C t1 ? ? tn where ti unit if si ? X
• unit z otherwise
• ?X?A t1 ? ? tn where ti ?i if si ? X
• ?i z otherwise

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Encoding Subtyping Hierarchies (II)

• ?All?C unit ? unit
• ?Int?C unit ? unit z
• ?Bool?C unit z ? unit
• ?None?C unit z ? unit z
• ?All?A ?1 ? ?2
• ?Int?A ?1 ? ?2 z
• ?Bool?A ?1 z ? ?2
• ?None?A ?1 z ? ?2 z

25
Encoding Subtyping Hierarchies (III)

• Any finite hierarchy can be embedded in the
  powerset lattice of a set S
• Better encodings for specific classes of
  hierarchies

26
Bounded Polymorphism

• Extends both parametric polymorphism and
  subtyping
• double ???Int.? ? ?
• toString ???All.? ? string
• plus ???Int.(? ? ?) ? ?
• Provides a connection between type instantiation
  and subtyping
• We can safely encode a restricted form of bounded
  polymorphism using a simple extension of our
  recipe

27
Formalization

• Translation
• From a language with a restricted form of bounded
  polymorphism
• To a language with parametric polymorphism
• Using the recipe given earlier
• See paper for details

28
Conclusion

• Use type equivalence to encode information in a
  free type variable
• Use unification to enforce a particular relation
  on the information
• Practical issues
• complexity of types

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The Problem (II)

• datatype atom I of int B of bool
• fun mkI (iint)atom I(i)
• fun mkB (bbool)atom B(b)
• fun toString (vatom)string
• case v of
• I(i) gt Int.toString(i)
• B(b) gt Bool.toString(b)
• fun double (vatom)atom
• case v of
• I(i) gt I (2 i)
• _ gt raise (Fail type mismatch)
• fun conj (v1atom, v2atom)atom
• case (v1,v2) of
• (B(b1),B(b2)) gt B (b1 andalso b2)
• _ gt raise (Fail type mismatch)

31
A Better Solution (II)

• type All Int Bool unit
• datatype ? atom I of int B of bool
• fun mkI (iint)Int atom I(i)
• fun mkB (bbool)Bool atom B(b)
• fun toString (v? atom)string
• case v of
• I(i) gt Int.toString(i)
• B(b) gt Bool.toString(b)
• fun double (vInt atom)Int atom
• case v of
• I(i) gt I (2 i)
• _ gt raise (Fail type mismatch)
• fun conj (v1Bool atom, v2Bool atom)Bool atom
• case (v1,v2) of
• (B(b1),B(b2)) gt B (b1 andalso b2)
• _ gt raise (Fail type mismatch)

32
Bounded Polymorphism (I)

• Extends both parametric polymorphism and
  subtyping
• ????.???
• ????.(???)??

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Bounded Polymorphism (II)

• ?Int?A ? ? ?Int?C ?

• Example Nat ? Int
• double ???Int.? ? ?

• ?Int?A ? ? ?Int?A ?

• ?? ? ?? where ?? ?Int?A ?

• plus ???Int.(? ? ?)??
• ?? ? ?? ? ?? where ?? ?Int?A ?
• plus (mkI 1, natToInt (mkN 2))

34
Bounded polymorphism (III)

• Limitations
• Type variable bounds
• ????.????.(???) ? ?
• ?? ? ?? ? ?? where ?? ???A ? and ?? ???A ?

• First-class polymorphism

• Functional subtyping
• ???(?1 ? ?2).? ? ?2
• ?? ? ??2?C ? where ?? ??1?C ? ? ??2?A ?

35
Formalization (II)

• let f1 ??1??1 . ?x?1. c1 x in
• let fn ??n??n . ?x?n. cn x in
• ?
• safe interface types