On this page:
32.1 Separating Meaning from Notation
32.2 Mutation and Closures
32.3 Mutable Structures
32.3.1 Extending the Language Representation
32.3.2 The Interpretation of Boxes
32.3.3 Can the Environment Help?
32.3.4 Welcome to the Store
32.3.5 Interpreting Boxes
32.3.6 Implementing Mutation:   Subtleties and Variations
32.4 Variables
32.4.1 The Syntax of Variable Assignment
32.4.2 Interpreting Variables
32.4.3 Reference Parameter Passing
32.5 The Design of Stateful Language Operations
32.6 Typing State
32.6.1 Mutation and Polymorphism
32.6.2 Typing the Initial Value

32 Mutation: Structures and Variables

    32.1 Separating Meaning from Notation

    32.2 Mutation and Closures

    32.3 Mutable Structures

      32.3.1 Extending the Language Representation

      32.3.2 The Interpretation of Boxes

      32.3.3 Can the Environment Help?

      32.3.4 Welcome to the Store

      32.3.5 Interpreting Boxes

      32.3.6 Implementing Mutation: Subtleties and Variations

    32.4 Variables

      32.4.1 The Syntax of Variable Assignment

      32.4.2 Interpreting Variables

      32.4.3 Reference Parameter Passing

    32.5 The Design of Stateful Language Operations

    32.6 Typing State

      32.6.1 Mutation and Polymorphism

      32.6.2 Typing the Initial Value

32.1 Separating Meaning from Notation

Which of these is the same?
  • f = 3

  • o.f = 3

  • f = 3

Assuming all three are in Java, the first and third could behave exactly like each other or exactly like the second: it all depends on whether f is a local identifier (such as a parameter) or a field of the object (i.e., the code is really this.f = 3).

In either case, we are asking the evaluator to permanently change the value bound to f. This has important implications for other observers. Until now, for a given set of inputs, a computation always returned the same value. Now, the answer depends on when it was invoked: above, it depends on whether it was invoked before or after the value of f was changed. The introduction of time has profound effects on predicting the behavior of programs.

However, there are really two quite different notions of change buried in the uniform syntax above. Changing the value of a field (o.f = 3 or this.f = 3) is extremely different from changing that of an identifier (f = 3 where f is bound as a parameter or a local inside the method, not by the object). We will explore these in turn. We’ll tackle fields below, and return to identifiers in Variables.

To study both these features, we will as usual write interpreters. However, to make sure we expose their essence, we will write these interpreters without the use of state. That is, we will do something quite remarkable: write mutation-free interpreters that faithfully mimic languages with mutation. The key to this will be a special pattern of passing information around in a computation.

32.2 Mutation and Closures

Before we proceed, make sure you’ve understood boxes (A Canonical Mutable Structure), and especially the interaction between mutation and closures (Interaction of Mutation with Closures: Counters).

The interaction with closures (and their first-cousin, objects [Objects: Interpretation and Types]) is particularly subtle. Consider the following situation:We first heard this example described by Corky Cartwright. you are writing a GUI program to implement a calculator. The GUI element corresponding to each calculator button needs a callback—essentially a closure—that, when invoked, will report that that particular button was pressed. Therefore, it is tempting to initialize this as follows:

for(var i = 0; i < 10; i++) {

  button[i] = function() { return i; }


This pseudo-code translates with minimal change to any number of languages, with and without static types, ranging from Python to Swift. (Assume button has been initialized appropriately.)

Notice that the closures created above all have i in their environment. Now let us try to inspect the behavior of these closures:

for(var i = 0; i < 10; i++) {



That is, we extract the ith closure from button and apply it to no arguments. This evaluates the return i statement.

We might have liked this to produce the sequence of values 0, 1, 2, and so on through to 9. In fact, however, it produces ten outputs, all the same: 10.

Do Now!

Do you see why? How would you fix this?

The problem is that i in the for loop is allocated only once. Therefore, all the closures share the same i. Because that value had to become 10 for the for loop to terminate, all the closures report the value 10.

With traditional for loops, there is no obvious way out of this problem. This seemingly confusing behavior often confounds programmers new to languages that make extensive use of closures. Because they cannot change the behavior of for loops, many languages have introduced new versions of for (or new keywords inside for) to address this problem. The solution is always to allocate a fresh i on each iteration, so that each closure is over a different variable; the looping construct copies the previous value of i as the initial value of the new one before applying the updater (in this case, i++) and then performing the comparison and loop body.

Observe that when programming functionally, the desired behavior comes for free. For instance:

funs = for map(i from range(0, 10)): lam(): i end end check: map(lam(c): c() end, funs) is range(0, 10) end

It is easier to see why the definition of funs works by writing the use of map more explicitly:

funs = map( lam(i): lam(): i end end, range(0, 10))

Thus, we can see that on each iteration the outer lam(i): ... is applied, allocating a new i, which is the one closed over by the inner lam(): ....

32.3 Mutable Structures

Equipped with these examples, let us now return to adding mutation to the language in the form of mutable structures (which are also a good basis for objects with mutable members). Besides mutable structures themselves, note that we must sometimes perform mutation in groups (e.g., removing money from one bank account and depositing it in another). Therefore, it is useful to be able to sequence a group of mutable operations. We will call this begin: it evaluates its sub-terms terms in order and returns the value of the last one.


Why does it matter whether begin evaluates its sub-terms in some particular, specified order?

Does it matter what this order is?


Define begin by desugaring into let (and hence into anonymous functions).

This is an excellent illustration of the non-canonical nature of desugaring. We’ve chosen to add to the core a construct that is certainly not necessary. If our goal was to shrink the size of the interpreter—perhaps at some cost to the size of the input program—we would not make this choice. But our goal here is to understand key ideas in languages, so we choose the combination of features that will be most instructive. Even though it is possible to eliminate begin as syntactic sugar, it will prove extremely useful for understanding how mutation works. Therefore, we will add a simple, two-term version of sequencing (seqC) to the core. In turn, because our core language is becoming unwieldy, we will drop Boolean values and conditional values except where their presence makes things more interesting: and it does not here.

32.3.1 Extending the Language Representation

First, let’s extend our core language datatype:

data ExprC: | numC(n :: Number) | plusC(l :: ExprC, r :: ExprC) | multC(l :: ExprC, r :: ExprC) | idC(s :: String) | appC(f :: ExprC, a :: ExprC) | fdC(arg :: String, body :: ExprC) | boxC(v :: ExprC) | unboxC(b :: ExprC) | setboxC(b :: ExprC, v :: ExprC) | seqC(b1 :: ExprC, b2 :: ExprC) end

Observe that in a setboxC expression, both the box position and its new value are expressions. The latter is unsurprising, but the former might be. It means we can write programs such as this in corresponding Pyret:
fun make-box-list():
  b0 = box(0)
  b1 = box(1)
  l = [list: b0, b1]
  index(l, 0)!{v : 1}
  index(l, 1)!{v : 2}
  l = make-box-list()
  index(l, 0)!v is 1
  index(l, 1)!v is 2
This evaluates to a list of boxes, the first containing 1 and the second 2. Observe that in each of the mutation statements, we are using a complex expression—e.g., index(l, 0)rather than a literal or an identifier to obtain the box before mutating it (!{v : 1}). the first argument to the This is precisely analogous to languages like Java, where one can (taking some type liberties) write

public static void main (String[] args) {

    Box<Integer> b0 = new Box<Integer>(0);

    Box<Integer> b1 = new Box<Integer>(1);


    ArrayList<Box<Integer>> l = new ArrayList<Box<Integer>>();







Notice that l.get(0) is a compound expression being used to find the appropriate box, and evaluates to the box object on which set is invoked.

For convenience, we will assume that we have implemented desugaring to provide us with (a) let and (b) if necessary, more than two terms in a sequence (which can be desugared into nested sequences). We will also sometimes write expressions in the original Pyret syntax, both for brevity (because the core language terms can grow quite large and unwieldy) and so that you can run these same terms in Pyret and observe what answers they produce. As this implies, we are taking the behavior in Pyret—which mutable structures are similar in behavior to those of just about every mainstream language with mutable objects and structures—as the reference behavior.

32.3.2 The Interpretation of Boxes

First, because we’ve introduced a new kind of value, the box, we have to update the set of values:
<mut-str-value/1> ::=

  data Value:

    | numV(n :: Number)

    | closV(f :: ExprC%(is-fdC), e :: Env)

    | boxV(v :: Value)


Now let’s begin by reproducing our current interpreter:
<mut-str-interp/1> ::=

  fun interp(e :: ExprC, nv :: Env) -> Value:

    cases (ExprC) e:

      | numC(n) => numV(n)

      | plusC(l, r) => plus-v(interp(l, nv), interp(r, nv))

      | multC(l, r) => mult-v(interp(l, nv), interp(r, nv))

      | idC(s) => lookup(s, nv)

      | fdC(_, _) => closV(e, nv)

      | appC(f, a) =>

        clos = interp(f, nv)

        arg-val = interp(a, nv)


          xtnd-env(bind(clos.f.arg, arg-val), clos.e))






(You’ll soon see why the setboxC case is missing.)

Two of these cases should be easy. When we’re given a box expression, we simply evaluate the content and return it wrapped in a boxV:
<mut-str-interp/1-boxC> ::=

  | boxC(v) => boxV(interp(v, nv))

Similarly, extracting a value from a box is easy:
<mut-str-interp/1-unboxC> ::=

  | unboxC(b) => interp(b, nv).v

By now, you should be constructing a healthy set of test cases to make sure these behave as you’d expect.

Of course, we haven’t done any hard work yet. All the interesting behavior is, presumably, hidden in the treatment of setboxC. It may therefore surprise you that we’re going to look at seqC first instead (and you’ll see why we included it in the core).

Let’s take the most natural implementation of a sequence of two instructions:
<mut-str-interp/1-seqC> ::=

  | seqC(b1, b2) =>

    b1-value = interp(b1, nv)

    b2-value = interp(b2, nv)


That is, we evaluate the first term, then the second, and return the result of the second.

You should immediately spot something troubling. We bound the result of evaluating the first term, but didn’t subsequently do anything with it. That’s okay: presumably the first term contained a mutation expression of some sort, and its value is uninteresting. Thus, an equivalent implementation might be this:
<mut-str-interp/1-seqC/2> ::=

  | seqC(b1, b2) =>

    interp(b1, nv)

    interp(b2, nv)

Not only is this slightly dissatisfying in that it just uses Pyret’s sequential behavior, it can’t possibly be right! This can only work only if the result of the mutation is being stored somewhere. But because our interpreter only computes values and does not perform any mutation itself—because that would be cheating—any mutations in interp(b1, nv) are completely lost. This is obviously not what we want. (And therefore, we’re not going to even attempt to define what to do in the setbox case.)

32.3.3 Can the Environment Help?

Here is an input example that can help:

(let ([b (box 0)])

  (begin (begin (set-box! b (+ 1 (unbox b)))

                (set-box! b (+ 1 (unbox b))))

         (unbox b)))

In Racket, this evaluates to 2.


Represent this expression in ExprC.

Let’s consider the evaluation of the inner sequence. In both cases, the expression (the representation of (set-box! ...)) is exactly identical. Yet something is changing underneath, because this causes the value of the box to go from 0 to 2! We can “see” this even more clearly if instead we evaluate

(let ([b (box 0)])

  (+ (begin (set-box! b (+ 1 (unbox b)))

            (unbox b))

     (begin (set-box! b (+ 1 (unbox b)))

            (unbox b))))

which evaluates to 3. Here, the two calls to interp in the rule for addition are evaluating exactly the same textual expression in both cases. Yet somehow the effects from the left branch of the addition are being felt in the right branch.Spukhafte Fernwirkung.

If the interpreter is being given precisely the same expression, how can it possibly avoid producing precisely the same answer? The most obvious way is if the interpreter’s other parameter, the environment, were somehow different. As of now the exact same environment is sent to both both branches of the sequence and both arms of the addition, so our interpreter—which produces the same output every time on a given input—cannot possibly produce the answers we want.

Here is what we know so far:
  1. We must somehow make sure the interpreter is fed different arguments on calls that are expected to potentially produce different results.

  2. We must return from the interpreter some record of the mutations made when evaluating its argument expression.

Because the expression is what it is, the first point suggests that we might try to use the environment to reflect the differences between invocations. In turn, the second point suggests that each invocation of the interpreter should also return the environment, so it can be passed to the next invocation. So the interpreter should presumably be modified to return both the value and an updated environment. That is, the interpreter consumes an expression and environment; it evaluates in that environment, updating it as it proceeds; when the expression is done evaluating, the interpreter returns the answer (as it did before), along with an updated environment, which in turn is sent to the next invocation of the interpreter. And the treatment of setboxC would somehow impact the environment to reflect the mutation.

Before we dive into the implementation, however, we should consider the consequences of such a change. The environment already serves an important purpose: it holds deferred substitutions. In that respect, it already has a precise semantics—given by substitution—and we must be careful to not alter that. One consequence of its tie to substitution is that it is also the repository of lexical scope information. If we were to allow the extended environment escape from one branch of addition and be used in the other, for instance, consider the impact on the equivalent of the following program:

(+ (let ([b (box 0)])



It should be evident that this program has an error: b in the right branch of the addition is unbound (the scope of the b in the left branch ends with the closing of the letif this is not evident, desugar the above expression to use functions). But the extended environment at the end of interpreting the let clearly has b bound in it.


Work out the above problem in detail and make sure you understand it.

You could try various other related proposals, but they are likely to all have similar failings. For instance, you may decide that, because the problem has to do with additional bindings in the environment, you will instead remove all added bindings in the returned environment. Sounds attractive? Did you remember we have closures?


Consider the representation of the following program:

(let ([a (box 1)])

  (let ([f (fun x (+ x (unbox a)))])


      (set-box! a 2)

      (f 10))))

What problems does this example cause?

Rather, we should note that while the constraints described above are all valid, the solution we proposed is not the only one. Observe that neither condition actually requires the environment to be the responsible agent. Indeed, it is quite evident that the environment cannot be the principal agent. We need something else.

32.3.4 Welcome to the Store

The preceding discussion tells us that we need two repositories to accompany the expression, not one. One of them, the environment, continues to be responsible for maintaining lexical scope. But the environment cannot directly map identifiers to their value, because the value might change. Instead, something else needs to be responsible for maintaining the dynamic state of mutated boxes. This latter data structure is called the store.

Like the environment, the store is a partial map. Its domain could be any abstract set of names, but it is natural to think of these as numbers, meant to stand for memory locations. This is because the store in the semantics maps directly onto (abstracted) physical memory in the machine, which is traditionally addressed by numbers. Thus the environment maps names to locations, and the store maps locations to values:

data Binding: | bind(name :: String, location :: Number) end type Env = List<Binding> mt-env = empty xtnd-env = link data Storage: | cell(location :: Number, value :: Value) end type Store = List<Storage> mt-sto = empty xtnd-sto = link

We’ll also equip ourselves with a function to look up values in the store, just as we already have one for the environment (which now returns locations instead):

fun lookup(s :: String, nv :: Env) -> Number: ... fun fetch(n :: Number, st :: Store) -> Value: ...


Fill in the bodies of lookup and fetch.

With this, we can refine our notion of values to the correct one:
<mut-str-value> ::=

  data Value:

    | numV(n :: Number)

    | closV(f :: ExprC, e :: Env)

    | boxV(l :: Number)


32.3.5 Interpreting Boxes

Now we have something that the interpreter can return, updated, reflecting mutations during the evaluation of the expression, without having to change the environment in any way. Because a function can return only one value, we will use an ad hoc object with two fields: v for the value (which will (effectively) be the same as the one the interpreter originally returned), and st for the (potentially) updated store. We will use the following helper function to assemble the result:

fun ret(v :: Value, st :: Store): {v : v, st : st} end


Why do we say “effectively” and “potentially” above?

Hint for “effectively”: look at closures.

Thus our interpreter becomes:
<mut-str-interp> ::=

  fun interp(e :: ExprC, nv :: Env, st :: Store):

    cases (ExprC) e:












The easiest one to dispatch is numbers. Remember that we have to return the store reflecting all mutations that happened while evaluating the given expression. Because a number is a constant, no mutations could have happened, so the returned store is the same as the one passed in:
<mut-str-interp/numC> ::=

  | numC(n) => ret(numV(n), st)

A similar argument applies to closure creation; observe that we are speaking of the creation, not use, of closures:
<mut-str-interp/fdC> ::=

  | fdC(_, _) => ret(closV(e, nv), st)

Identifiers are almost as straightforward, though if you are simplistic, you’ll get a type error that will alert you that to obtain a value, you must now look up both in the environment and in the store:
<mut-str-interp/idC> ::=

  | idC(s) => ret(fetch(lookup(s, nv), st), st)

Notice how lookup and fetch compose to produce the same result that lookup alone produced before.

Now things get interesting.

Let’s take sequencing. Clearly, we need to interpret the two terms.
<mut-str-interp/seqC/alt> ::=

  | seqC(b1, b2) =>

    interp(b1, nv, st)

    interp(b2, nv, st)

Oh, but wait. The whole point was to evaluate the second term in the store returned by the first oneotherwise there would have been no point to all these changes. Therefore, instead we must evaluate the first term, capture the resulting store, and use it to evaluate the second. (Evaluating the first term also yields its value, but sequencing ignores this value and assumes the first term was run purely for its potential mutations.) Thus:
<mut-str-interp/seqC> ::=

  | seqC(b1, b2) =>

    b1-value = interp(b1, nv, st)

    interp(b2, nv, b1-value.st)

This says to interpret the first term: interp(b1, nv, st); name the resulting value, which contains v and st fields, b1-value; and evaluate the second term in the store from the first: interp(b2, nv, b1-value.st). The result will be the value and store returned by the second term, which is what we expect. The fact that the first term’s effect is only on the store can be read from the code because we never use b1-value.v.

Do Now!

Spend a moment contemplating the code above. You’ll soon need to adjust your eyes to read this pattern fluently.

Now let’s move on to the binary arithmetic primitives. These are similar to sequencing in that they have two sub-terms, but in this case we really do care about the value from each branch. As usual, we’ll look at only plusC since multC is virtually identical.
<mut-str-interp/plusC> ::=

  | plusC(l, r) =>

    lv = interp(l, nv, st)

    rv = interp(r, nv, lv.st)

    ret(plus-v(lv.v, rv.v), rv.st)

Observe that we’ve repeated the pattern because we have two sub-expressions to evaluate whose values we want to use. Thus the first store (lv.st) is used to interpret the second expression, and the overall result returns that of the second (rv.st).

Here’s an important distinction. When we evaluate a term, we usually use the same environment for all its sub-terms in accordance with the scoping rules of the language. The environment thus flows in a recursive-descent pattern. In contrast, the store is threaded: rather than using the same store in all branches, we take the store from one branch and pass it on to the next, and take the result and send it back out. This pattern is called store-passing style.

Now the penny drops. We see that store-passing style is our secret ingredient: it enables the environment to preserve lexical scope while still giving a binding structure that can reflect changes. Our intution told us that the environment had to somehow participate in obtaining different results for the same syntactic expression, and we can now see how it does: not directly, by itself changing, but indirectly, by referring to the store, which updates. Now we only need to see how the store itself “changes”.

Let’s begin with boxing. To store a value in a box, we have to first allocate a new place in the store where its value will reside. The value corresponding to a box will then remember this location, for use in box mutation. To obtain a fresh value each time, we will use the stateful counter example we have seen earlier [Interaction of Mutation with Closures: Counters]:

new-loc = mk-counter()

Given this, we can now define the interpretation of box creation:
<mut-str-interp/boxC> ::=

  | boxC(v) =>

    val = interp(v, nv, st)

    loc = new-loc()


      xtnd-sto(cell(loc, val.v), st))

Do Now!

Observe that we have relied above on new-loc, which is itself implemented in terms of boxes! This is outright cheating. How would you modify the interpreter so that we no longer need mutation for this little bit of state?

To eliminate new-loc, the simplest option would be to add another parameter to and return value from the interpreter, representing the largest address used so far. Every operation that allocates in the store would return an incremented address, while all others would return it unchanged. In other words, this is precisely another application of the store-passing pattern. Writing the interpreter this way would make it extremely unwieldy and might obscure the more important use of store-passing for the store itself, which is why we have not done so. However, it is important to make sure that we can: that’s what tells us that we are not reliant on state to add state to the language.

Now that boxes are recording the location in memory, getting the value corresponding to them is easy.
<mut-str-interp/unboxC> ::=

  | unboxC(b) =>

    val = interp(b, nv, st)

    ret(fetch(val.v.l, val.st), val.st)

It’s the same pattern we saw before, where we have to use fetch to obtain the actual value residing at that location. Note that we are relying on Racket to halt with an error if the underlying value isn’t actually a boxV; otherwise it would be dangerous to not check, since this would be tantamount to dereferencing arbitrary memory [REF memory safety].

Let’s now see how to update the value held in a box. First we have to evaluate the box expression to obtain a box, and the value expression to obtain the new value to store in it. The box’s value is going to be a boxV holding a location.

In principle we want to “change”, or override, the value at that location in the store. We can do this in two ways.
  1. One is to traverse the store, find the old binding for that location, and replace it with the new one, copying all the other store bindings unchanged.

  2. The other, lazier, option is to simply extend the store with a new binding for that location, which works provided we always obtain the most recent binding for a location (which is how lookup works in the environment, so fetch can do the same in the store).Observe that this latter option forces us to commit to lists rather than to sets.

The code below is written to be independent of these options:
<mut-str-interp/setboxC> ::=

  | setboxC(b, v) =>

    b-val = interp(b, nv, st)

    v-val = interp(v, nv, b-val.st)


      xtnd-sto(cell(b-val.v.l, v-val.v), v-val.st))

If we’ve implemented xtnd-sto as link above, we’ve actually taken the lazier (and slightly riskier, because of its dependence on the implementation of fetch) option.


Implement the other version of store alteration, whereby we update an existing binding and thereby avoid multiple bindings for a location in the store.


When we look for a location to override the value stored at it, can the location fail to be present? If so, write a program that demonstrates this. If not, explain what invariant of the interpreter prevents this from happening.

Alright, we’re now done with everything other than application! Most of application should already be familiar: evaluate the function position, evaluate the argument position, interpret the closure body in an extension of the closure’s environment...but how do stores interact with this?
<mut-str-interp/appC> ::=

  | appC(f, a) =>

    clos = interp(f, nv, st)

    clos-v :: Value = clos.v

    clos-st :: Store = clos.st

    arg-val = interp(a, nv, clos-st)


Let’s start by thinking about extending the closure environment. The name we’re extending it with is obviously the name of the function’s formal parameter. But what location do we bind it to? To avoid any confusion with already-used locations (a confusion we will explicitly introduce later!—Reference Parameter Passing), let’s just allocate a new location. This location is used in the environment, and the value of the argument resides at this location in the store:
<mut-str-interp/appC/core> ::=

  new-loc = new-loc()


    xtnd-env(bind(clos-v.f.arg, new-loc), clos-v.e),

    xtnd-sto(cell(new-loc, arg-val.v), arg-val.st))

Because we have not said the function parameter is mutable, there is no real need to have implemented procedure calls this way. We could instead have followed the same strategy as before. Indeed, observe that the mutability of this location will never be used: only setboxC changes what’s in an existing store location (the xtnd-sto above is technically a store initialization), and then only when they are referred to by boxVs, but no box is being allocated above.You could call this the useless app store. However, we have chosen to implement application this way for uniformity, and to reduce the number of cases we’d have to handle.


It’s a useful exercise to try to limit the use of store locations only to boxes. How many changes would you need to make?

32.3.6 Implementing Mutation: Subtleties and Variations

Even though we’ve finished the implementation, there are still many subtleties and insights to discuss.

  1. Implicit in our implementation is a subtle and important decision: the order of evaluation. For instance, why did we not implement addition thus?

    <mut-str-interp/plusC/alt> ::=

      | plusC(l, r) =>

        rv = interp(r, nv, st)

        lv = interp(l, nv, rv.st)

        ret(plus-v(lv.v, rv.v), lv.st)

    It would have been perfectly consistent to do so. Similarly, embodied in the pattern of store-passing is the decision to evaluate the function position before the argument. Observe that:

    1. Previously, we delegated such decisions to the underlying language implementation. Now, store-passing has forced us to sequentialize the computation, and hence make this decision ourselves (whether we realized it or not).

    2. Even more importantly, this decision is now a semantic one. Before there were mutations, one branch of an addition, for instance, could not affect the value produced by the other branch.The only effect they could have was halting with an error or failing to terminate—which, to be sure, are certainly observable effects, but at a much more gross level. A program would not terminate with two different answers depending on the order of evaluation. Because each branch can have mutations that impact the value of the other, we must choose some order so that programmers can predict what their program is going to do! Being forced to write a store-passing interpreter has made this clear.

  2. Observe that in the application rule, we are passing along the dynamic store, i.e., the one resulting from evaluating both function and argument. This is precisely the opposite of what we said to do with the environment. This distinction is critical. The store is, in effect, “dynamically scoped”, in that it reflects the history of the computation, not its lexical shape. Because we are already using the term “scope” to refer to the bindings of identifiers, however, it would be confusing to say “dynamically scoped” to refer to the store. Instead, we simply say that it is persistent.

    Languages sometimes dangerously conflate these two. In C, for instance, values bound to local identifiers are allocated (by default) on the stack. However, the stack matches the environment, and hence disappears upon completion of the call. If the call, however, returned references to any of these values, these references are now pointing to unused or even overridden memory: a genuine source of serious errors in C programs. The problem is that programmers want the values themselves to persist; but the storage for those values has been conflated with that for identifiers, who come and go with lexical scope.

  3. We have already discussed how there are two strategies for overriding the store: to simply extend it (and rely on fetch to extract the newest one) or to “search-and-replace”. The latter strategy has the virtue of not holding on to useless store bindings that will can never be obtained again.

    However, this does not cover all the wasted memory. Over time, we cease to be able to access some boxes entirely: e.g., if they are bound to only one identifier, and that identifier is no longer in scope. These locations are called garbage. Thinking more conceptually, garbage locations are those whose elimination does not have any impact on the value produced by a program. There are many strategies for automatically identifying and reclaiming garbage locations, usually called garbage collection [REF].

  4. It’s very important to evaluate every expression position and thread the store that results from it. Consider, for instance, this alternate implementation of unboxC (compare with <mut-str-interp/unboxC>):
    <mut-str-interp/unboxC/alt-1> ::=

      | unboxC(b) =>

        val = interp(b, nv, st)

        ret(fetch(val.v.l, st), val.st)

    Did you notice? We fetched the location from st, not val.st. But st reflects mutations up to but before the evaluation of the unboxC expression, not any within it. Could there possibly be any? Mais oui!

    (let ([b (box 0)])

      (unbox (begin (set-box! b 1)


    With the incorrect code above, this would evaluate to 0 rather than 1.

  5. Here’s another, similar, error (again compare with <mut-str-interp/unboxC>):
    <mut-str-interp/unboxC/alt-2> ::=

      | unboxC(b) =>

        val = interp(b, nv, st)

        ret(fetch(val.v.l, val.st), st)

    How do we break this? In the end we’re returning the old store, the one before any mutations in the unboxC happened. Thus, we just need the outside context to depend on one of them.

    (let ([b (box 0)])

      (+ (unbox (begin (set-box! b 1)


         (unbox b)))

    This should evaluate to 2, but because the store being returned is one where b’s location is bound to the representation of 0, the result is 1.

    If we combined both bugs above—i.e., using st twice in the last line instead of s-a twice—this expression would evaluate to 0 rather than 2.


    Go through the interpreter; replace every reference to an updated store with a reference to one before update; make sure your test cases catch all the introduced errors!

  6. Observe that these uses of “old” stores enable us to perform a kind of time travel: because mutation introduces a notion of time, these enable us to go back in time to when the mutation had not yet occurred. This sounds both interesting and perverse; does it have any use?

    It does! Imagine that instead of directly mutating the store, we introduce the idea of a journal of intended updates to the store. The journal flows in a threaded manner just like the real store itself. Some instruction creates a new journal; after that, all lookups first check the journal, and only if the journal cannot find a binding for a location is it looked for in the actual store. There are two other new instructions: one to discard the journal (i.e., travel back in time), and the other to commit it (i.e., all of its edits get applied to the real store).

    This is the essence of software transactional memory. Each thread maintains its own journal. Thus, one thread does not see the edits made by the other before committing (because each thread sees only its own journal and the global store, but not the journals of other threads). At the same time, each thread gets its own consistent view of the world (it sees edits it made, because these are recorded in the journal). If the transaction ends successfully, all threads atomically see the updated global store. If the transaction aborts, the discarded journal takes with it all changes and the state of the thread reverts (modulo global changes committed by other threads).

    Software transactional memory offers one of the most sensible approaches to tackling the difficulties of multi-threaded programming, if we insist on programming with shared mutable state. Because most computers have only one global store, however, maintaining the journals can be expensive, and much effort goes into optimizing them. As an alternative, some hardware architectures have begun to provide direct support for transactional memory by making the creation, maintenance, and commitment of journals as efficient as using the global store, removing one important barrier to the adoption of this idea.


    Augment the language with the journal features of software transactional memory.


An alternate implementation strategy is to have the environment map names to boxed Values. We don’t do it here because it: (a) would be cheating, (b) wouldn’t tell us how to implement the same feature in a language without boxes, (c) doesn’t necessarily carry over to other mutation operations, and (d) most of all, doesn’t really give us insight into what is happening here.

It is nevertheless useful to understand, not least because you may find it a useful strategy to adopt when implementing your own language. Therefore, alter the implementation to obey this strategy. Do you still need store-passing style? Why or why not?

32.4 Variables

Now that we’ve got structure mutation worked out, let’s consider the other case: variable mutation. We have already discussed [From Identifiers to Variables] our choice of terminology, and seen examplse of their use in Pyret. In particular, Whereas other languages overload the mutation syntax, as we have seen [Separating Meaning from Notation], in Pyret they are kept distinct: ! mutates fields of objects while := mutates variables. This forces Pyret programmers to confront the distinction we introduced at the beginning of Separating Meaning from Notation. We will, of course, sidestep these syntactic issues in our core language by using different constructs for boxes and for variables.

32.4.1 The Syntax of Variable Assignment

The first thing to note about variable mutation is that, although it too has two sub-terms like box mutation (setboxC), its syntax is fundamentally different. To understand why, let’s return to our Java fragment:

x = 3;

In this setting, we cannot write an arbitrary expression in place of x: we must literally write the name of the identifier itself. That is because, if it were an expression position, then we could evaluate it, yielding a value: for instance, if x were previously bound to 1, this would be tantamout to writing the following statement:

1 = 3;

But this is, of course, nonsensical! We can’t assign a new value to 1, and indeed 1 is pretty much the definition of immutable. What we instead want is to find where x is in the store, and change the value held over there.

Here’s another way to see this. Suppose, in Java, the local variable o is already bound to some String object:

o = new String("an initial string");

Say the program now executes

o = new String("a new string");

Is it trying to change the content of the original string ("an initial string")? Certainly not: the second assignment intends to leave that original string alone; it only wants to change the value that o is referring to, so that subsequent references evaluate to this new string ("a new string") object instead.

32.4.2 Interpreting Variables

We’ll start by reflecting this in our syntax:

data ExprC: | numC(n :: Number) | plusC(l :: ExprC, r :: ExprC) | multC(l :: ExprC, r :: ExprC) | varC(s :: String) | appC(f :: ExprC, a :: ExprC) | fdC(arg :: String, body :: ExprC) | setC(v :: String, b :: ExprC) | seqC(b1 :: ExprC, b2 :: ExprC) end

Observe that we’ve jettisoned the box operations, but kept sequencing because it’s handy around mutation. Importantly, we’ve now added the setC case, and its first sub-term is not an expression but the literal name of a variable. We’ve also renamed idC to varC.

Because we’ve gotten rid of boxes, we can also get rid of the special box values. When the only kind of mutation you have is variables, you don’t need new kinds of values.

data Value: | numV (n :: Number) | closV (f :: ExprC, e :: List<Binding>) end

As you might imagine, to support variables we need the same store-passing style that we’ve seen before [Interpreting Boxes], and for the same reasons. What differs is in precisely how we use it. Because sequencing is interpreted in just the same way (observe that the verb for it does not depend on boxes versus variables), that leaves us just the variable mutation case to handle.

First, we might as well evaluate the value expression and obtain the updated store:
<mut-var-interp/setC> ::=

  | setC(v, b) =>

    new-val = interp(b, nv, st)


What now? Remember we just said that we don’t want to fully evaluate the variable, because that would just give the value it is bound to. Instead, we want to know which memory location it corresponds to, and update what is stored at that memory location; this latter part is just the same thing we did when mutating boxes:
<mut-var-interp/setC/core> ::=

  var-loc = lookup(v, nv)


    xtnd-sto(cell(var-loc, new-val.v), new-val.st))

The very interesting new pattern we have here is this. When we added boxes, in the idC case (<mut-str-interp/idC>), we looked up an identifier in the environment (just as above), and then immediately fetched the value at that location from the store; the composition yielded a value, just as it used to before we added stores. Now, however, we have a new pattern: looking up an identifier in the environment without subsequently fetching its value from the store, i.e., we have “half” of a variable’s evaluation. The result of invoking just lookup is traditionally called an l-value, for “left-hand-side (of an assignment) value”. This is a fancy way of saying “memory address”, and stands in contast to the actual values that the store yields: observe that it does not directly correspond to anything in the type Value.

And we’re done! We did all the hard work when we implemented store-passing style (and also in that application allocated new locations for variables).

32.4.3 Reference Parameter Passing

Let’s return to the parenthetical statement above: that every application allocates a fresh location in the store for the parameter.

Do Now!

Why does this matter? Consider the following Pyret program:

fun f(x): x := 3 end var y = 5 f(y)

After this runs, what do we expect to be the value of y?

In the example above, y evaluates to 5, not 3. That is because the value of the formal parameter x is held at a different location than that of the actual parameter y, so the mutation affects the location of x, leaving y unscathed.

Now suppose, instead, that application behaved as follows. When the actual parameter is a variable, and hence has a location in memory, instead of allocating a new location for the value, it simply passes along the existing one for the variable. Now the formal parameter is referring to the same store location as the actual: i.e., they are variable aliases. Thus any mutation on the formal will leak back out into the calling context; the above program would evaluate to 3 rather than 5. These is called a call-by-reference parameter-passing strategy.Instead, our interpreter implements call-by-value, and this is the same strategy followed by languages like Java. This causes confusion because when the value is itself mutable, changes made to the value in the callee are observed by the caller. However, that is simply an artifact of mutable values, not of the calling strategy. Please avoid this confusion!

For some years, this power was considered a good idea. It was useful because programmers could write abstractions such as swap, which swaps the value of two variables in the caller. However, the disadvantages greatly outweigh the advantages:

At the very least, then, if the language is going to permit by-reference parameters, it should let the caller determine whether to pass the reference—i.e., let the callee share the memory address of the caller’s variable—or not. However, even this option is not quite as attractive as it may sound, because now the callee faces a symmetric problem, not knowing whether its parameters are aliased or not. In traditional, sequential programs this is less of a concern, but if the procedure is reentrant, the callee faces precisely the same predicaments.

At some point, therefore, we should consider whether any of this fuss is worthwhile. Instead, callers who want the callee to perform a mutation could simply send a boxed value to the callee. The box signals that the caller accepts—indeed, invites—the callee to perform a mutation, and the caller can extract the value when it’s done. This does obviate the ability to write a simple swapper, but that’s a small price to pay for genuine software engineering concerns.

32.5 The Design of Stateful Language Operations

Though most programming languages include one or both kinds of state we have studied, their admission should not be regarded as a trivial or foregone matter. On the one hand, state brings some vital benefits:

On the other hand, state imposes real costs on programmers as well as on programs that process programs (such as compilers). One is “aliasing”, which we discuss later [REF]. Another is “referential transparency”, which too we hope to return to [REF]. Finally, we have described above how state provides a form of modularity. However, this same description could be viewed as that of a back-channel of communication that the intermediaries did not know and could not monitor. In some (especially security and distributed system) settings, such back-channels can lead to collusion, and can hence be extremely dangerous and undesirable.

Because there is no optimal answer, it is probably wise to include mutation operators but to carefully delinate them. In Standard ML, for instance, there is no variable mutation, because it is considered unnecessary. Instead, the language has the equivalent of boxes (called refs). One can easily simulate variables using boxes, so no expressive power is lost, though it does create more potential for aliasing than variables alone would have ([REF aliasing]) if the boxes are not used carefully.

In return, however, developers obtain expressive types: every data structure is considered immutable unless it contains a ref, and the presence of a ref is a warning to both developers and programs (such as compilers) that the underlying value may keep changing.This same argument applies to Pyret, where the absence of a ref declaration means that a field is immutable, and the absence of a var declaration means an identifier is immutable, i.e., not a variable. Thus, for instance, suppose b is a box and v is bound to the unboxing of b. A developer should be aware that replacing all instances of the unboxing b with references to v is not safe, because the former always fetches the current value in the box while the latter holds the value only at the instant when v was computed, and may now be inconsistent. The declaration that a field is mutable provides this information to both the developer and to programming tools (such as compilers); in particular, the absence of such a declaration permits caching of values, thereby trading computation time for space.

32.6 Typing State

Adding stateful operations to a type-checker is easy: the only safe thing to do is make sure the type of the new value is exactly the same as that of the old one. If that is true, the behavior of the program will be indistinguishable to the type system before and after the mutation. That is, it is safest to follow invariance [The Principle of Substitutability].

32.6.1 Mutation and Polymorphism

Later, once we encounter subtyping [Subtyping], we will find it is no longer trivial to type mutation. Already, however, mutation is complex in the presence of polymorphism. Let us first give polymorphic types to boxes, where Box is a type constructor for boxes:

box<T> :: T -> Box(T) unbox<T> :: Box(T) -> T set-box<T> :: Box(T), T -> Box(T)

(assuming that the mutation operation, set-box, returns the box as its result).


Implement the above three functions.

Now let us consider a simple example that uses these:

let f = box(lam(x): x end): set-box(f, lam(x): x + 5 end) unbox(f)(true) end

Initially, the type of f is Box((A -> A)) where (A -> A) represents the type of the polymorphic identity function. When performing inference, a copy of this type certainly unifies with Number -> Number, the type of lam(x): x + 5 end. Another copy is used at the application site; the argument type unifies with that of true, giving the whole expression the type Boolean. However, when the actual function is applied, it attempts to add true to 5, resulting in a run-time error. If the compiler had assumed the type-system was sound and had not compiled in checks, this program could even result in a segmentation fault.

There are many ways to try to understand this problem, which is beyond the scope of this study. The simplest way is that polymorphism and mutation do not work together well. The essence of polymorphism is to imagine a quantified type is instantiated at each use; however, the essence of mutation is to silently transmit values from one part of the program to the other. Thus, the values being unified at two different sites are only guaranteed to be compatible with the let-bound identifier—not with each other. This is normally not a problem because the two do not communicate directly, except where mutation is involved. Therefore, a simple solution in this case is to prevent polymorphic generalization for mutable let-bound identifiers.

32.6.2 Typing the Initial Value

There is one last issue we should address where mutation is concerned: the typing of cycle creation, and in particular dealing with the problem of Premature Evaluation. We have discussed several approaches to handling the initial value; each of these has consequences for typing:
  1. Using a fixed initial value of a standard type means the value subsequently mutated into place may not be type-compatible, thereby failing invariance.

  2. Using a different initial value of the type that will eventually be put into the mutable has the problem that prematurely observing it is even more deadly, because it may not be distinguishable from the eventual value.

  3. Using a new value just for this case works provided there is one of each type. Otherwise, again, we violate invariance. But having one of each type is a problem in itself, because now the run-time system has to check for all of them.

  4. Syntactically restricting recursion to functions is the safest, because the initial value is never seen. As a result, there is no need to provide any meaningful type for it.

In short, this is a place where we have to confront unsurmountable trade-offs. The first option sacrifices typability; the second option sacrifices program reliability (because the dummy values are of the right type, and may hence be inadvertently used without noticing they are wrong); the third sacrifices run-time simplicity; and the fourth sacrifices programmer flexibility.