Most real languages include impure features ("computational effects")...

• mutable pointer structures
• non-local control constructs (exceptions, continuations, etc.)
• process synchronization and communication
• etc.

Goal for this chapter: formalize pointers.

Definitions

In most real-world programming languages, the mechanisms of name binding and storage allocation are intentionally confused: every name refers to a mutable piece of storage. Conceptually, it's cleaner to separate the two:

• keep the mechanisms for name binding (abstraction, let) the same;
• introduce new, explicit operations for allocating, changing, and looking up the contents of references (pointers).

Syntax

In this chapter, we study adding mutable references to the simply-typed lambda calculus with natural numbers.

The basic operations on references are allocation, dereferencing, and assignment.

• To allocate a reference, we use the ref operator, providing an initial value for the new cell. For example, ref 5 creates a new cell containing the value 5, and reduces to a reference to that cell.
• To read the current value of this cell, we use the dereferencing operator !; for example, !(ref 5) reduces to 5.
• To change the value stored in a cell, we use the assignment operator. If r is a reference, r := 7 will store the value 7 in the cell referenced by r.

Types

If T is a type, then Ref T is the type of references to cells holding values of type T.
      T ::= Nat
          | Unit
          | T → T
          | Ref T

Terms

Besides variables, abstractions, applications, natural-number-related terms, and unit, we need four more sorts of terms in order to handle mutable references:

      t ::= ...              Terms
          | ref t              allocation
          | !t                 dereference
          | t := t             assignment
          | l                  location

Typing (Preview)

Informally, the typing rules for allocation, dereferencing, and assignment will look like this:

Gamma ⊢ t1 : T1	(T_Ref)
Gamma ⊢ ref t1 : Ref T1

Gamma ⊢ t1 : Ref T1	(T_Deref)
Gamma ⊢ !t1 : T1

Gamma ⊢ t1 : Ref T2
Gamma ⊢ t2 : T2	(T_Assign)
Gamma ⊢ t1 := t2 : Unit

The rule for locations will require a bit more machinery, and this will motivate some changes to the other rules; we'll come back to this later.

Values and Substitution

Besides abstractions and numbers, we have two new types of values: the unit value, and locations.

Extending substitution to handle the new syntax of terms is straightforward.

Pragmatics

Side Effects and Sequencing

We can write

       r:=succ(!r); !r

as an abbreviation for

       (\x:Unit. !r) (r := succ(!r)).

References and Aliasing

It is important to bear in mind the difference between the reference that is bound to some variable r and the cell in the store that is pointed to by this reference. If we make a copy of r, for example by binding its value to another variable s, what gets copied is only the reference, not the contents of the cell itself. For example, after reducing

      let r = ref 5 in
      let s = r in
      s := 82;
      (!r)+1

the cell referenced by r will contain the value 82, while the result of the whole expression will be 83. The references r and s are said to be aliases for the same cell. The possibility of aliasing can make programs with references quite tricky to reason about. For example, the expression

      r := 5; r := !s

assigns 5 to r and then immediately overwrites it with s's current value; this has exactly the same effect as the single assignment

      r := !s

unless we happen to do it in a context where r and s are aliases for the same cell!

Shared State

Of course, aliasing is also a large part of what makes references useful. In particular, it allows us to set up "implicit communication channels" -- shared state -- between different parts of a program. For example, suppose we define a reference cell and two functions that manipulate its contents:

      let c = ref 0 in
      let incc = \_:Unit. (c := succ (!c); !c) in
      let decc = \_:Unit. (c := pred (!c); !c) in
      ...

The Unit-abstractions ("thunks") are used to prevent reduction until later.

Objects

We can go a step further and write a function that creates c, incc, and decc, packages incc and decc together into a record, and returns this record:

      newcounter =
          \_:Unit.
             let c = ref 0 in
             let incc = \_:Unit. (c := succ (!c); !c) in
             let decc = \_:Unit. (c := pred (!c); !c) in
             {i=incc, d=decc}

Now, each time we call newcounter, we get a new record of functions that share access to the same storage cell c. The caller of newcounter can't get at this storage cell directly, but can affect it indirectly by calling the two functions. In other words, we've created a simple form of object.

      let c1 = newcounter unit in
      let c2 = newcounter unit in
      // Note that we've allocated two separate storage cells now!
      let r1 = c1.i unit in
      let r2 = c2.i unit in
      r2  // yields 1, not 2!

References to Compound Types

A reference cell need not contain just a number: the primitives we've defined above allow us to create references to values of any type, including functions. For example, we can use references to functions to give an (inefficient) implementation of arrays of numbers, as follows. Write NatArray for the type Ref (Nat→Nat). Recall the equal function from the MoreStlc chapter:

      equal =
        fix
          (\eq:Nat->Nat->Bool.
             \m:Nat. \n:Nat.
               if m=0 then iszero n
               else if n=0 then false
               else eq (pred m) (pred n))

To build a new array, we allocate a reference cell and fill it with a function that, when given an index, always returns 0.

      newarray = \_:Unit. ref (\n:Nat.0)

To look up an element of an array, we simply apply the function to the desired index.

      lookup = \a:NatArray. \n:Nat. (!a) n

The interesting part of the encoding is the update function. It takes an array, an index, and a new value to be stored at that index, and does its job by creating (and storing in the reference) a new function that, when it is asked for the value at this very index, returns the new value that was given to update, while on all other indices it passes the lookup to the function that was previously stored in the reference.

      update = \a:NatArray. \m:Nat. \v:Nat.
                   let oldf = !a in
                   a := (\n:Nat. if equal m n then v else oldf n);

References to values containing other references can also be very useful, allowing us to define data structures such as mutable lists and trees.

Null References

One more difference between our references and C-style mutable variables: null pointers

• in C, a pointer variable can contain either a valid pointer into the heap or the special value NULL
• source of many errors and much tricky reasoning
  • (any pointer may potentially be "not there")
  • but occasionally useful
• easy to implement here using references plus options (which can be built out of disjoint sum types)

              Option T       =  Unit + T
              NullableRef T  =  Option (Ref T)
  

Garbage Collection

A last issue that we should mention before we move on with formalizing references is storage de-allocation. We have not provided any primitives for freeing reference cells when they are no longer needed. Instead, like many modern languages (including ML and Java) we rely on the run-time system to perform garbage collection, automatically identifying and reusing cells that can no longer be reached by the program. This is not just a question of taste in language design: it is extremely difficult to achieve type safety in the presence of an explicit deallocation operation. One reason for this is the familiar dangling reference problem: we allocate a cell holding a number, save a reference to it in some data structure, use it for a while, then deallocate it and allocate a new cell holding a boolean, possibly reusing the same storage. Now we can have two names for the same storage cell -- one with type Ref Nat and the other with type Ref Bool.

Exercise: 2 stars, standard (type_safety_violation)

Show how this can lead to a violation of type safety.

Operational Semantics

Locations

A reference names a location in the store (also known as the heap or just the memory). What is the store?

• Concretely: An array of 8-bit bytes, indexed by 32-bit integers.
• More abstractly: a list (or array) of values
• Even more abstractly: a partial function from locations to values.

We'll choose the middle way here: A store is a list of values, and a location is a natural number index into this list.

Stores

A store is just a list of values. (This more concrete representation will be more convenient for proofs than the functional representation we used in Imp.)

We use store_lookup n st to retrieve the value of the reference cell at location n in the store st. Note that we must give a default value to nth in case we try looking up an index which is too large. (In fact, we will never actually do this, but proving that we don't will require a bit of work.)

To update the store, we use the replace function, which replaces the contents of a cell at a particular index.

Reduction

First, we augment existing reduction rules with stores:

value v2	(ST_AppAbs)
(\x:T2.t1) v2 / st --> [x:=v2]t1 / st

t1 / st --> t1' / st'	(ST_App1)
t1 t2 / st --> t1' t2 / st'

value v1     t2 / st --> t2' / st'	(ST_App2)
v1 t2 / st --> v1 t2' / st'

Now we can give the rules for the new constructs:

	(ST_RefValue)
ref v / st --> loc |st| / st,v

t1 / st --> t1' / st'	(ST_Ref)
ref t1 / st --> ref t1' / st'

l < |st|	(ST_DerefLoc)
!(loc l) / st --> lookup l st / st

t1 / st --> t1' / st'	(ST_Deref)
!t1 / st --> !t1' / st'

l < |st|	(ST_Assign)
loc l := v / st --> unit / replace l v st

t1 / st --> t1' / st'	(ST_Assign1)
t1 := t2 / st --> t1' := t2 / st'

t2 / st --> t2' / st'	(ST_Assign2)
v1 := t2 / st --> v1 := t2' / st'

One slightly ugly point should be noted here: In the ST_RefValue rule, we extend the state by writing st ++ v::nil rather than the more natural st ++ [v]. The reason for this is that the notation we've defined for substitution uses square brackets, which clash with the standard library's notation for lists.

Typing

The contexts assigning types to free variables are exactly the same as for the STLC: partial maps from identifiers to types.

Store typings

Having extended our syntax and reduction rules to accommodate references, our last job is to write down typing rules for the new constructs (and, of course, to check that these rules are sound!). Naturalurally, the key question is, "What is the type of a location?" First of all, notice that this question doesn't arise when typechecking terms that programmers actually write. Concrete location constants arise only in terms that are the intermediate results of reduction; they are not in the language that programmers write. So we only need to determine the type of a location when we're in the middle of a reduction sequence, e.g., trying to apply the progress or preservation lemmas. Thus, even though we normally think of typing as a static program property, it makes sense for the typing of locations to depend on the dynamic progress of the program too. As a first try, note that when we reduce a term containing concrete locations, the type of the result depends on the contents of the store that we start with. For example, if we reduce the term !(loc 1) in the store [unit, unit], the result is unit; if we reduce the same term in the store [unit, \x:Unit.x], the result is \x:Unit.x. With respect to the former store, the location 1 has type Unit, and with respect to the latter it has type Unit→Unit. This observation leads us immediately to a first attempt at a typing rule for locations:

Gamma ⊢ lookup l st : T1
Gamma ⊢ loc l : Ref T1

That is, to find the type of a location l, we look up the current contents of l in the store and calculate the type T₁ of the contents. The type of the location is then Ref T₁. Having begun in this way, we need to go a little further to reach a consistent state. In effect, by making the type of a term depend on the store, we have changed the typing relation from a three-place relation (between contexts, terms, and types) to a four-place relation (between contexts, stores, terms, and types). Since the store is, intuitively, part of the context in which we calculate the type of a term, let's write this four-place relation with the store to the left of the turnstile: Gamma; st ⊢ t : T. Our rule for typing references now has the form

Gamma; st ⊢ lookup l st : T1
Gamma; st ⊢ loc l : Ref T1

and all the rest of the typing rules in the system are extended similarly with stores. (The other rules do not need to do anything interesting with their stores -- just pass them from premise to conclusion.) However, this rule will not quite do. For one thing, typechecking is rather inefficient, since calculating the type of a location l involves calculating the type of the current contents v of l. If l appears many times in a term t, we will re-calculate the type of v many times in the course of constructing a typing derivation for t. Worse, if v itself contains locations, then we will have to recalculate their types each time they appear. Worse yet, the proposed typing rule for locations may not allow us to derive anything at all, if the store contains a cycle. For example, there is no finite typing derivation for the location 0 with respect to this store:

   [\x:Nat. (!(loc 1)) x, \x:Nat. (!(loc 0)) x]

Exercise: 2 stars, standard (cyclic_store)

Can you find a term whose reduction will create this particular cyclic store? ☐ These problems arise from the fact that our proposed typing rule for locations requires us to recalculate the type of a location every time we mention it in a term. But this, intuitively, should not be necessary. After all, when a location is first created, we know the type of the initial value that we are storing into it. Suppose we are willing to enforce the invariant that the type of the value contained in a given location never changes; that is, although we may later store other values into this location, those other values will always have the same type as the initial one. In other words, we always have in mind a single, definite type for every location in the store, which is fixed when the location is allocated. Then these intended types can be collected together as a store typing -- a finite function mapping locations to types. As with the other type systems we've seen, this conservative typing restriction on allowed updates means that we will rule out as ill-typed some programs that could reduce perfectly well without getting stuck. Just as we did for stores, we will represent a store type simply as a list of types: the type at index i records the type of the values that we expect to be stored in cell i.

The store_Tlookup function retrieves the type at a particular index.

Suppose we are given a store typing ST describing the store st in which some term t will be reduced. Then we can use ST to calculate the type of the result of t without ever looking directly at st. For example, if ST is [Unit, Unit→Unit], then we can immediately infer that !(loc 1) has type Unit→Unit. More generally, the typing rule for locations can be reformulated in terms of store typings like this:

l < |ST|
Gamma; ST ⊢ loc l : Ref (lookup l ST)

That is, as long as l is a valid location, we can compute the type of l just by looking it up in ST. Typing is again a four-place relation, but it is parameterized on a store typing rather than a concrete store. The rest of the typing rules are analogously augmented with store typings.

The Typing Relation

l < |ST|	(T_Loc)
Gamma; ST ⊢ loc l : Ref (lookup l ST)

Gamma; ST ⊢ t1 : T1	(T_Ref)
Gamma; ST ⊢ ref t1 : Ref T1

Gamma; ST ⊢ t1 : Ref T1	(T_Deref)
Gamma; ST ⊢ !t1 : T1

Gamma; ST ⊢ t1 : Ref T2
Gamma; ST ⊢ t2 : T2	(T_Assign)
Gamma; ST ⊢ t1 := t2 : Unit

Of course, these typing rules will accurately predict the results of reduction only if the concrete store used during reduction actually conforms to the store typing that we assume for purposes of typechecking. This proviso exactly parallels the situation with free variables in the basic STLC: the substitution lemma promises that, if Gamma ⊢ t : T, then we can replace the free variables in t with values of the types listed in Gamma to obtain a closed term of type T, which, by the type preservation theorem will reduce to a final result of type T if it yields any result at all. We will see below how to formalize an analogous intuition for stores and store typings. However, for purposes of typechecking the terms that programmers actually write, we do not need to do anything tricky to guess what store typing we should use. Concrete locations arise only in terms that are the intermediate results of reduction; they are not in the language that programmers write. Thus, we can simply typecheck the programmer's terms with respect to the empty store typing. As reduction proceeds and new locations are created, we will always be able to see how to extend the store typing by looking at the type of the initial values being placed in newly allocated cells; this intuition is formalized in the statement of the type preservation theorem below.

Properties

Standard theorems...

• Progress -- pretty much same as always
• Preservation -- needs to be stated more carefully!

Well-Typed Stores

Evaulation and typing relations take more parameters now, so at a minumum we have to add these to the statement of preservation...

Obviously wrong: no relation between assumed store typing and provided store! We need a way of saying "this store satisfies the assumptions of that store typing"...

Informally, we will write ST ⊢ st for store_well_typed ST st. We can now state something closer to the desired preservation property:

This works... for all but one of the reduction rules!

Extending Store Typings

Intuition: Since the store can grow during reduction, we need to let the store typing grow too...

We'll need a few technical lemmas about extended contexts. First, looking up a type in an extended store typing yields the same result as in the original:

Next, if ST' extends ST, the length of ST' is at least that of ST.

Finally, ST ++ T extends ST, and extends is reflexive.

Preservation, Finally

We can now give the final, correct statement of the type preservation property:

Note that this gives us just what we need to "turn the crank" when applying the theorem to multi-step reduction sequences.

Substitution Lemma

To prove preservation, we need to re-develop the rest of the machinery that we saw for the pure STLC (plus a couple of new things about store typings and extension)...

Assignment Preserves Store Typing

Next, we must show that replacing the contents of a cell in the store with a new value of appropriate type does not change the overall type of the store. (This is needed for the ST_Assign rule.)

Weakening for Stores

Finally, we need a lemma on store typings, stating that, if a store typing is extended with a new location, the extended one still allows us to assign the same types to the same terms as the original. (The lemma is called store_weakening because it resembles the "weakening" lemmas found in proof theory, which show that adding a new assumption to some logical theory does not decrease the set of provable theorems.)

We can use the store_weakening lemma to prove that if a store is well typed with respect to a store typing, then the store extended with a new term t will still be well typed with respect to the store typing extended with t's type.

Preservation!

Now that we've got everything set up right, the proof of preservation is actually quite straightforward. Begin with one technical lemma:

And here, at last, is the preservation theorem and proof:

Exercise: 3 stars, standard (preservation_informal)

Write a careful informal proof of the preservation theorem, concentrating on the T_App, T_Deref, T_Assign, and T_Ref cases. (* FILL IN HERE *)
☐

Progress

As we've said, progress for this system is pretty easy to prove; the proof is very similar to the proof of progress for the STLC, with a few new cases for the new syntactic constructs.

References and Nontermination

An important fact about the STLC (proved in chapter Norm) is that it is is normalizing -- that is, every well-typed term can be reduced to a value in a finite number of steps. What about STLC + references? Surprisingly, adding references causes us to lose the normalization property: there exist well-typed terms in the STLC + references which can continue to reduce forever, without ever reaching a normal form! How can we construct such a term? The main idea is to make a function which calls itself. We first make a function which calls another function stored in a reference cell; the trick is that we then smuggle in a reference to itself!

   (\r:Ref (Unit -> Unit).
        r := (\x:Unit.(!r) unit); (!r) unit)
   (ref (\x:Unit.unit))

First, ref (\x:Unit.unit) creates a reference to a cell of type Unit → Unit. We then pass this reference as the argument to a function which binds it to the name r, and assigns to it the function \x:Unit.(!r) unit -- that is, the function which ignores its argument and calls the function stored in r on the argument unit; but of course, that function is itself! To start the divergent loop, we execute the function stored in the cell by evaluating (!r) unit. Here is the divergent term in Coq:

This term is well typed:

To show formally that the term diverges, we first define the step_closure of the single-step reduction relation, written -->+. This is just like the reflexive step closure of single-step reduction (which we're been writing -->*), except that it is not reflexive: t -->+ t' means that t can reach t' by one or more steps of reduction.

Now, we can show that the expression loop reduces to the expression !(loc 0) unit and the size-one store [r:=(loc 0)]loop_fun. As a convenience, we introduce a slight variant of the normalize tactic, called reduce, which tries solving the goal with multi_refl at each step, instead of waiting until the goal can't be reduced any more. Of course, the whole point is that loop doesn't normalize, so the old normalize tactic would just go into an infinite loop reducing it forever!

Next, we use reduce to show that loop steps to !(loc 0) unit, starting from the empty store.

Finally, we show that the latter expression reduces in two steps to itself!

Exercise: 4 stars, standard (factorial_ref)

Use the above ideas to implement a factorial function in STLC with references. (There is no need to prove formally that it really behaves like the factorial. Just uncomment the example below to make sure it gives the correct result when applied to the argument 4.)

If your definition is correct, you should be able to just uncomment the example below; the proof should be fully automatic using the reduce tactic.

Additional Exercises

Exercise: 5 stars, standard, optional (garabage_collector)

Challenge problem: modify our formalization to include an account of garbage collection, and prove that it satisfies whatever nice properties you can think to prove about it. ☐