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Subtyping

A motivating example

In the simply typed lamdba-calculus with records from the last chapter, the term

    (\r:{y:nat}. r.y + 1) {x=10,y=11}

is not typable: it involves an application of a function that wants a one-field record to an argument that actually provides two fields, while the T_App rule demands that the domain type of the function being applied must match the type of the argument precisely.

But this is silly: we're passing the function a _better_ argument than it needs! The only thing the body of the function can possibly do with its record argument r is project the field y from it: nothing else is allowed by the type. So the presence or absence of an extra x field should make no difference at all. So, intuitively, it seems that this function should be applicable to any record value that has at least a y field.

Looking at the same thing from another point of view, a record with more fields is "at least as good in any context" as one with just a subset of these fields, in the sense that any value belonging to the longer record type can be used SAFELY in any context expecting the shorter record type. If the context expects something with the shorter type but we actually give it something with the longer type, nothing bad will happen (formally, the program will not get stuck).

The general principle at work here is called SUBTYPING. We say that "S is a subtype of T", informally written S <: T, if a value of type S can safely be used in any context where a value of type T is expected. The idea of subtyping applies not only to records, but to all of the type constructors in the language -- to functions, pairs, etc.

Subtyping and object-oriented languages

The principle of subtyping plays a fundamental role in many present-day programming languages -- in particular, it is closely related to the notion of SUBCLASSING in object-oriented languages.

An OBJECT (in Java, C#, etc.) can be thought of as a record, some of whose fields are functions ("methods") and some of whose fields are data values ("fields" or "instance variables"). Invoking a method m of an object o on some arguments a1..an consists of projecting out the m field of o and applying it to a1..an.

The type of an object can be given as either a CLASS or an INTERFACE. Both of these provide a description of which methods and which data fields the object offers.

Classes and interfaces are related by the SUBCLASS and SUBINTERFACE relations. An object belonging to a subclass (or subinterface) is required to provide all the methods and fields of one belonging to a superclass (or superinterface), plus possibly some more.

The fact that an object from a subclass (or subinterface) can be used in place of one from a superclass (/superinterface) provides a degree of flexibility that is is extremely handy for organizing complex libraries. For example, a graphical user interface toolkit like Java's AWT might define an abstract interface Component that collects together the common fields and methods of all objects having a graphical representation that can be displayed on the screen and that can interact with the user. Examples of such object would include the buttons, checkboxes, and scrollbars of a typical GUI. A method that relies only on this common interface can now be applied to any of these objects.

Of course, real object-oriented languages include many other features besides these. Fields can be updated. Fields and methods can be declared private. Classes also give _code_ that is used when constructing objects and implementing their methods, and the code in subclasses cooperate with code in superclasses via INHERITANCE. Classes can have static methods and fields, initializers, etc., etc.

To keep things simple, we won't deal with any of these issues, and we won't even talk any more about objects or classes. (There is a lot of discussion in Types and Programming Languages, for those that are interested.) Instead, we'll study the "essential core" of the subclass / subinterface relation in the stripped down setting of the STLC.

The subsumption rule

Our goal for this chapter is to add subtyping to the simply typed lambda-calculus with records. This involves two steps:

• 1. Defining subtyping as a binary relation between types.
• 2. Enriching the typing relation to take subtyping into account.

The second step is actually very simple. We add just a single rule to the typing relation -- the so-called RULE OF SUBSUMPTION:

                         Gamma |- t : S     S <: T
                         -------------------------                      (T_Sub)
                               Gamma |- t : T

This rule says, intuitively, that we can "forget" some of the information that we know about a term. For example, we may know that t is a record with two fields (e.g., S = {x:A->A, y:B->B} ], but choose to forget about one of the fields (T = {y:B->B}) so that we can pass t to a function that expects just a single-field record.

The subtype relation

The first step -- the definition of the relation S <: T -- is where all the action is. Let's look at each of the clauses of its definition.

To begin with, we need to formalize the basic intuition about record types that a longer record should be a subtype of a shorter one.

          for each jk in j1..jn, exists ip in i1..im with Sk <: Tp
          --------------------------------------------------------      (S_Rcd)
                     {i1:S1...im:Sm} <: {j1:T1...jn:Tn}

That is, the record on the left should have all the field labels of the one on the right (and possibly more), while the types of the common fields should be in the subtype relation. For example:

       {x:nat,y:bool} <: {x:nat}             "width subtyping"
       {x:nat} <: {}
       {x:nat,y:bool} <: {y:bool,x:nat}      "permutation"
       {x:nat,y:bool} <: {y:bool}
       {a:{x:nat}} <: {a:{}}                 "depth subtyping"

Formally, our record types are presented in a "binary" form (with constructors for nil and cons, rather than a single constructor that assembles a whole multi-field record all at once), and so we'll need to formulate subtyping in the same way. This can be accomplished by replacing the single rule above with a combination of three rules.

To state these rules, we need to begin by introducing informal notations corresponding to the constructors ty_rnil and ty_rcons. We'll write ty_rnil as {} (just like we have been), and we'll write ty_rcons k S T as k:S;T (note the semicolon).

So, for example, a record with three fields might be written j:A;k:B;l:C;{}. However, since looks a bit awkward and nonstandard, we'll introduce one final informal convention: in multi-field row types like this expression, we'll pick up the left curly brace and move it all the way to the left of the expression, changing the semicolons to commas as we go. In other words the informal expressions
     j:A;k:B;l:C;{}

and
     {j:A,k:B,l:C}

will mean the very same thing -- they will both denote this formal record type:
     ty_rcons j (ty_base A)
        (ty_rcons k (ty_base B)
           (ty_rcons l (ty_base C)
              ty_rnil))

With these notations in hand, we're ready to talk about the three subtyping rules for records.

First, any row of record types is a subtype of the empty record type:

                                 --------                          (S_RcdWidth)
                                 Tr <: {}

Second, we can apply subtyping inside the components of a compound row:

                         S1 <: T1       Sr2 <: Tr2
                        ----------------------------               (S_RcdDepth)
                          i1:S1; Sr2 <: i1:T1; Tr2

We can use S_RcdDepth and S_RcdWidth together to drop later fields of a multi-field record while keeping earlier fields, showing for example that {y:B->B, x:A->A} <: {y:B->B}. We can also use S_RcdDepth and S_RcdWidth together to show that {y:{z:B->B}, x:A->A} <: {y:{}}.

The example we originally had in mind was {x:A->A,y:B->B} <: {y:B->B}. We haven't quite achieved this yet: using just S_RcdDepth and S_RcdWidth we can only drop fields from the _end_ of a record type. To handle the original example, we also need to be able to reorder fields:

                                  i1 <> i2
                 ------------------------------------------         (S_RcdPerm)
                   i1:T1; i2:T2; Tr3 <: i2:T2; i1:T1; Tr3

For example, {x:A->A,y:B->B} <: {y:B->B,x:A->A}.

We can also include a general rule of TRANSITIVITY, which says intuitively that, if S is better than U and U is better than T, then S is better than T.

                              S <: U    U <: T
                              ----------------                        (S_Trans)
                                   S <: T

This rule allows us to paste together the proofs we've seen that {x:A->A, y:B->B} <: {y:B->B, x:A->A} (by S_RcdPerm) and that {y:B->B, x:A->A} <: {y:B->B} (by S_RcdDepth and S_RcdWidth), to yield a proof that {x:A->A, y:B->B} <: {y:B->B}.

This completes the subtyping rules for records. To finish the whole definition of subtyping, we need to consider how each of the other type constructors behaves with respect to subtyping. Since we're dealing with a very simple language with just arrows and records, we have only arrows left to deal with. A variant of the first example motivates the discussion.

Suppose we have two functions f and g with these types:

       f  :  C -> {x:A->A,y:B->B}
       g  :  (C->{y:B->B}) -> D

That is, f is a function that yields a record of type {x:A->A,y:B->B}, and g is a higher-order function that expects its (function) argument to yield a record of type {y:B->B}. Then the application g f is safe even though their types do not match up precisely, because the only thing g can do with f is to apply it to some argument (of type C); the result will actually be a two-field record, while g will be expecting only a record with a single field, but this is safe because the only thing g can then do is to project out the single field that it knows about, and this will certainly be among the two fields that are present.

This example suggests that the subtyping rule for arrow types should say that two arrow types are in the subtype relation if their results are:

                                  S2 <: T2
                              ----------------                        (S_Arrow)
                              S1->S2 <: S1->T2

We can generalize this to allow the arguments of the two arrow types to be in the subtype relation as well:

                            T1 <: S1    S2 <: T2
                            --------------------                      (S_Arrow)
                              S1->S2 <: T1->T2

Notice, here, that the argument types are subtypes "the other way round": in order to conclude that S1->S2 to be a subtype of T1->T2, it must be the case that T1 is a subtype of S1. The arrow constructor is said to be _contravariant_ in its first argument and _covariant_ in its second.

The intuition is that, if we have a function f of type S1->S2, then we know that f accepts elements of type S1; clearly, f will also accept elements of any subtype T1 of S1. The type of f also tells us that it returns elements of type S2; we can also view these results belonging to any supertype T2 of S2. That is, any function f of type S1->S2 can also be viewed as having type T1->T2.

Finally, we add one last structural rule, which (together with transitivity) ensures that the subtype relation is a preorder:

                                   ------                              (S_Refl)
                                   T <: T

We can stop here, if we like. But it is common practice to go one further step and add to the language one new type constant, called Top, together with a subtyping rule that places it above every other type in the subtype relation:

                                   --------                             (S_Top)
                                   S <: Top

The Top type is an analog of the Object type in Java and C#.

Some more examples:

• {}->{j:A} <: {k:B}->Top
• Top->{k:A->A,j:B->B} <: (C->C)->{j:B->B}

Variations

Real languages often choose not to adopt all of these subtyping rules. For example, in Java,

• A subclass may not change the argument or result types of a method of its superclass (i.e., no depth subtyping or no arrow subtyping, depending how you look at it).
• Each class has just one superclass ("single inheritance" of classes)
  • Each class member (field or method) can be assigned a single index, adding new indices "on the right" as more members are added in subclasses
  • i.e., no permutation for classes
• A class may implement multiple interfaces ("multiple inheritance" of interfaces)
  • i.e., permutation is allowed for interfaces.

Core definitions

We've already sketched the significant extensions that we'll need to make to the STLC: (1) add the subtype relation and (2) extend the typing relation with the rule of subsumption. To make everything work smoothly, we'll also implement some technical improvements to the presentation from the last chapter; in particular, we'll add "well formedness" predicates for both terms and types that verify that the record constructors are properly used. The rest of the definitions -- in particular, the syntax and operational semantics of the language -- are identical to what we saw in the last chapter. Let's first do the identical bits.

Syntax

There are just a couple of small technical differences here from the previous system. (The way we formulate it here is a little bit nicer -- the previous system should be changed to match this one.)

• The ty_rcd constructor has been eliminated from this presentation (and tm_rcd below). This is not essential (it is easy to add them back in), but it streamlines the development a little and we feel now that it's a bit cleaner.
• The constructors ty_rnil and ty_rcons have been renamed ty_rnil and ty_rcons.

Well-formedness

Next, we slightly reformulate and generalize the well-formedness condition is_tmr from the last chapter. (Again, we feel the new version is cleaner and the previous version should be rewritten to match this one.)

First, a type is a record type if it is built with either ty_nil or ty_cons.

Note that record_ty is not recursive -- it just checks the outermost constructor. The well_formed_ty predicate, on the other hand, verifies that the whole type is well formed in the sense that the tail of every record (the second argument to ty_rcons) is a record.

The predicates record_tm and well_formed_tm are similar.

Substitution

The definition of substitution remains the same as for the ordinary STLC with records.

Reduction

Likewise the definitions of the value predicate and the step relation.

Subtyping

Now we come to the interesting part. We begin by defining the subtyping relation and developing some of its important technical properties.

Definition

The definition of subtyping is essentially just what we sketched in the motivating discussion, but we need to add well-formedness side conditions to some of the rules.

Subtyping examples and exercises

The following facts are mostly easy to prove in Coq. To get full benefit from the exercises, make sure you also understand how to prove them on paper!

Exercise: 2 stars

Exercise: 1 star

Exercise: 1 star

Exercise: 2 stars

Exercise: 1 star (subtype_instances_tf_1)

Suppose we have types S, T, U, and V with S <: T and U <: V. Which of the following subtyping assertions are then true? Write TRUE or FALSE after each one.

• T->S <: T->S
• Top->U <: S->Top
• (C->C)->{x:A->A,y:B->B} <: (C->C)->{y:B->B}
• ty_rnil -> ty_rnil <: {x:A} -> Top
• T->T->U <: S->S->V
• (T->T)->U <: (S->S)->V
• ((T->S)->T)->U <: ((S->T)->S)->V
• {a:S, b:V} <: {a:T, b:U}

Exercise: 1 star (subtype_instances_tf_2)

Which of the following statements are true? Write TRUE or FALSE after each one.

• 
          forall S T,
              S <: T
           -> S->S <: T->T

• 
          forall S T,
               S <: A->A
            -> exists T,
                  S = T->T /\ T <: A

• 
          forall S T1 T1,
               S <: T1 -> T2
            -> exists S1 S2,
                  S = S1 -> S2 /\ T1 <: S1 /\ S2 <: T2

• 
          exists S,
               S <: S->S

• 
          exists S,
               S->S <: S

• 
          forall S T2 T2,
               S <: {j:T1,k:T2}
            -> exists S1 S2,
                  S = {j:S1,k:S2} /\ S1 <: T1 /\ S2 <: T2

Exercise: 1 star (subtype_concepts_tf)

Which of the following statements are true, and which are false?

• There exists a type that is a supertype of every other type.
• There exists a type that is a subtype of every other type.
• There exists a record type that is a subtype of every other record type.
• There exists a record type that is a supertype of every other record type.
• There exists an arrow type that is a subtype of every other arrow type.
• There exists an arrow type that is a supertype of every other arrow type.
• There is an infinite descending chain of distinct types in the subtype relation---that is, an infinite sequence of types S0, S1, etc., such that all the Si's are different and each S(i+1) is a subtype of Si.
• There is an infinite *ascending* chain of distinct types in the subtype relation---that is, an infinite sequence of types S0, S1, etc., such that all the Si's are different and each S(i+1) is a supertype of Si.

Exercise: 1 star (proper_subtypes)

Is the following statement true or false? Briefly explain your answer.
    forall T,
         ~(exists n, T = ty_base n)
      -> exists S,
            S <: T /\ S <> T

Exercise: 1 star (small_large_1)

• What is the *smallest* type T ("smallest" in the subtype relation) that makes the following assertion true?
        empty |- (\r:{x:T}. r.x) {x=(\z:A.z)} : A->A

• What is the *largest* type T that makes the same assertion true?

Exercise: 1 star (small_large_2)

• What is the *smallest* type T that makes the following assertion true?
      empty |- (\r:{y:B->B, x:A->A}. r) {x=(\z:A.z), y=(\z:B.z)} : T

• What is the *largest* type T that makes the same assertion true?

Exercise: 1 star (small_large_3)

• What is the *smallest* type T that makes the following assertion true?
      a:A |- (\r:{x:A, T}. (r.y) (r.x)) {y=(\z:A.z), x=a} : A

• What is the *largest* type T that makes the same assertion true?

Exercise: 1 star (small_large_4)

• What is the *smallest* row type Tr that makes the following assertion true?
      exists S,
        empty |- (\r:(x:A;Tr). (r.y) (r.x)) : S

• What is the *largest* type T that makes the same assertion true?

Exercise: 1 star (smallest_1)

What is the *smallest* type T that makes the following assertion true?
      exists S, exists t,
        empty |- (\r:{x:T}. (r.x) (r.x)) t : S

Exercise: 1 star (smallest_2)

What is the *smallest* type T that makes the following assertion true?
      empty |- (\x:Top. x) {a=(\z:A.z), b=(\z:B.z)} : T

Exercise: 1 star (count_supertypes)

How many supertypes does the type {x:A, y:C->C} have? That is, how many different types T are there such that {x:A, y:C->C} <: T? (We consider two types to be different if they are written differently, even if each is a subtype of the other. For example, {x:A,y:B} and {y:B,x:A} are different.)

Properties of subtyping

Well-formedness

Exercise: 1 star (wf_variation)

The proof of the subtype__wf lemma goes by induction on the derivation of S <: T. Suppose we change the statement of the lemma to

        Lemma subtype__wf : forall S T,
             S <: T
          -> well_formed_ty T

and again start the proof by induction on the derivation of S <: T. Will we succeed in completing it? Briefly explain.

Field lookup

Our record matching lemmas get a little more complicated in the presence of subtyping for two reasons: First, record types no longer necessarily describe the exact structure of corresponding terms. Second, reasoning by induction on has_type derivations becomes harder in general, because has_type is no longer syntax directed.

Exercise: 3 stars

Write a careful informal proof of the rcd_types_match lemma.

Inversion lemmas

Exercise: 3 stars (sub_inversion_arrow)

Typing

Again, the typing relation is exactly the same, except for (a) the rule of subsumption, T_Sub, and (b) some well-formedness side conditions.

Typing examples

Exercise: 1 star

Exercise: 2 stars

Exercise: 2 stars, optional

Properties of typing

Well-formedness

Field lookup

Progress

Exercise: 3 stars (canonical_forms_of_arrow_types)

Inversion lemmas

Context invariance

Preservation

Exercises on typing

Exercise: 2 stars, optional (variations)

Each part of this problem suggests a different way of changing the definition of the STLC with records and subtyping. (These changes are not cumulative: each part starts from the original language.) In each part, list which properties (Progress, Preservation, both, or neither) become false. If a property becomes false, give a counterexample.

• Suppose we add the following typing rule:

  
                              Gamma |- t : S1->S2
                      S1 <: S2      S2 <: S1     S2 <: T2
                      -----------------------------------              (T_Funny1)
                              Gamma |- t : T1->T2
  
  

• Suppose we add the following reduction rule:

                               ------------------                     (ST_Funny21)
                               {} ~~> (\x:Top. x)
  

• Suppose we add the following subtyping rule:

                                 --------------                        (S_Funny3)
                                 {} <: Top->Top
  

• Suppose we add the following subtyping rule:

                                 --------------                        (S_Funny4)
                                 Top->Top <: {}
  

• Suppose we add the following evaluation rule:

                               -----------------                      (ST_Funny5)
                               ({} t) ~~> (t {})
  

• Suppose we add the same evaluation rule *and* a new typing rule:

                               -----------------                      (ST_Funny5)
                               ({} t) ~~> (t {})
  
                             ----------------------                    (T_Funny6)
                             empty |- {} : Top->Top
  
  

• Suppose we *change* the arrow subtyping rule to:

  
                            S1 <: T1       S2 <: T2
                            -----------------------                    (S_Arrow')
                              -> S1->S2 <: T1->T2
  
  

The main exercise for the week:

Exercise: 4 stars (products)

Adding pairs, projections, and product types to the system we have defined is a relatively straightforward matter. Carry out this extension.

• Add constructors for pairs, first and second projections, and product types to the definitions of ty and tm. (Don't forget to add corresponding cases to ty_cases and tm_cases.)
• Extend the well-formedness relation in the obvious way.
• Extend the operational semantics with the same reduction rules as in the last chapter.
• Extend the subtyping relation with this rule:

                             S1 <: T1     S2 <: T2
                             ---------------------                     (Sub_Prod)
                               S1 * S2 <: T1 * T2
  

• Extend the typing relation with the same rules for pairs and projections as in the last chapter.
• Extend the proofs of progress, preservation, and all their supporting lemmas to deal with the new constructs. (You'll also need to add some completely new lemmas.)