15 Checking Program Invariants Statically: Types

As programs grow larger or more subtle, developers need tools to help them describe and validate statements about program invariants. Invariants, as the name suggests, are statements about program elements that are expected to always hold of those elements. For example, when we write x : number in our typed language, we mean that x will always hold a number, and that all parts of the program that depend on x can rely on this statement being enforced. As we will see, types are just one point in a spectrum of invariants we might wish to state, and static type checking—itself a diverse family of techniques—is also a point in a spectrum of methods we can use to enforce the invariants.

15.1 Types as Static Disciplines

In this chapter, we will focus especially on static type checking: that is, checking (declared) types before the program even executes. We have already experienced a form of this in our programs by virtue of using a typed programming language. We will explore some of the design space of types and their trade-offs. Finally, though static typing is an especially powerful and important form of invariant enforcement, we will also examine some other techniques that we have available.

Now consider the following Racket program:

(define f n

(+ n 3))

This too fails before program execution begins, with a parse error. Though we think of parsing as being somehow distinct from type-checking—usually because the type-checker assumes it has a parsed program to begin with—it can be useful to think of parsing as being simply the very simplest kind of type-checking: determining (typically) whether the program obeys a context-free syntax. Type-checking then asks whether it obeys a context-sensitive (or richer) syntax. In short, type-checking is a generalization of parsing, in that both are concerned with syntactic methods for enforcing disciplines on programs.

15.2 A Classical View of Types

We will begin by introducing a traditional core language of types. Later, we will explore both extensions [REF] and variations [REF].

15.2.1 A Simple Type Checker

Before we can define a type checker, we have to fix two things: the syntax of our typed core language and, hand-in-hand with that, the syntax of types themselves.

To begin with, we’ll return to our language with functions-as-values [REF] but before we added mutation and other complications (some of which we’ll return to later). To this language we have to add type annotations. Conventionally, we don’t impose type annotations on constants or on primitive operations such as addition; instead, we impose them on the boundaries of functions or methods. Over the course of this study we will explore why this is a good locus for annotations.

Even numeric types may not be straightforward: What information does a number type need to record? In most languages, there are actually many numeric types, and indeed there may not even be a single one that represents “numbers”. However, we have ignored these gradations between numbers [REF], so it’s sufficient for us to have just one. Having decided that, do we record additional information about which number? We could in principle, but we would soon run into decidability problems.

A natural starting signature for the type-checker would be that it is a procedure that consumes an expression and returns a boolean value indicating whether or not the expression type-checked. Because we know expressions contain identifiers, it soon becomes clear that we will want a type environment, which maps names to types, analogous to the value environment we have seen so far.

Thus, we might begin our program as follows:

As the abbreviated set of cases above suggests, this approach will not work out. We’ll soon see why.

Let’s begin with the easy case: numbers. Does a number type-check? Well, on its own, of course it does; it may be that the surrounding context is not expecting a number, but that error would be signaled elsewhere. Thus:

Now let’s handle identifiers. Is an identifier well-typed? Again, on its own it would appear to be, provided it is actually a bound identifier; it may not be what the context desires, but hopefully that too would be handled elsewhere. Thus we might write

This should make you a little uncomfortable: lookup already signals an error if an identifier isn’t bound, so there’s no need to repeat it (indeed, we will never get to this error invocation). But let’s push on.

Now we tackle applications. We should type-check both the function part, to make sure it’s a function, then ensure that the actual argument’s type is consistent with what the function declares to be the type of its formal argument. For instance, the function could be a number and the application could itself be a function, or vice versa, and in either case we want to prevent such mis-applications.

How does the code look?

The recursive call to tc can only tell us whether the function expression type-checks or not. If it does, how do we know what type it has? If we have an immediate function, we could reach into its syntax and pull out the argument (and return) types. But if we have a complex expression, we need some procedure that will calculate the resulting type of that expression. Of course, such a procedure could only provide a type if the expression is well-typed; otherwise it would not be able to provide a coherent answer. In other words, our type “calculator” has type “checking” as a special case! We should therefore strengthen the inductive invariant on tc: that it not only tells us whether an expression is typed but also what its type is. Indeed, by giving any type at all it confirms that the expression types, and otherwise it signals an error.

Let’s now define this richer notion of a type “checker”.

Now let’s fill in the pieces. Numbers are easy: they have the numeric type.

Similarly, identifiers have whatever type the environment says they do (and if they aren’t bound, this signals an error).

Observe, so far, the similarity to and difference from interpreting: in the identifier case we did essentially the same thing (except we returned a type rather than an actual value), whereas in the numeric case we returned the abstract “number” rather than indicate which specific number it was.

Let’s now examine addition. We must make sure both sub-expressions have numeric type; only if they do will the overall expression evaluate to a number itself.

We’ve usually glossed over multiplication after considering addition, but now it will be instructive to handle it explicitly:

That’s right: practically nothing! (The multC instead of plusC in the type-case, and the slightly different error message, notwithstanding.) That’s because, from the perspective of type-checking (in this type language), there is no difference between addition and multiplication, or indeed between any two functions that consume two numbers and return one.

Observe another difference between interpreting and type-checking. Both care that the arguments be numbers. The interpreter then returns a precise sum or product, but the type-checker is indifferent to the differences between them: therefore the expression that computes what it returns ((numT)) is a constant, and the same constant in both cases.

Finally, the two hard cases: application and funcions. We’ve already discussed what application must do: compute the value of the function and argument expressions; ensure the function expression has function type; and check that the argument expression is of compatible type. If all this holds up, then the type of the overall application is whatever type the function body would return (because the value that eventually returns at run-time is the result of evaluating the function’s body).

That leaves function definitions. The function has a formal parameter, which is presumably used in the body; unless this is bound in the environment, the body most probably will not type-check properly. Thus we have to extend the type environment with the formal name bound to its type, and in that extended environment type-check the body. Whatever value this computes must be the same as the declared type of the body. If that is so, then the function itself has a function type from the type of the argument to the type of the body.

Observe another curious difference between the interpreter and type-checker. In the interpreter, application was responsible for evaluating the argument expression, extending the environment, and evaluating the body. Here, the application case does check the argument expression, but leaves the environment alone, and simply returns the type of the body without traversing it. Instead, the body is actually traversed by the checker when checking a function definition, so this is the point at which the environment actually extends.

15.2.2 Type-Checking Conditionals

Suppose we extend the above language with conditionals. Even the humble if introduces several design decisions. We’ll discuss two here, and return to one of them later [REF].

1. What should be the type of the test expression? In some languages it must evaluate to a boolean value, in which case we have to enrich the type language to include booleans (which would probably be a good idea anyway). In other languages it can be any value, and some values are considered “truthy” while others “falsy”.

2. What should be the relationship between the then- and else-branches? In some languages they must be of the same type, so that there is a single, unambiguous type for the overall expression (which is that one type). In other languages the two branches can have distinct types, which greatly changes the design of the type-language and -checker, but also of the nature of the programming language itself.

15.2.3 Recursion in Code

Now that we’ve obtained a basic programming language, let’s add recursion to it. We saw earlier [REF] that this could be done easily through desugaring. It’ll prove to be a more complex story here.

15.2.3.1 A First Attempt at Typing Recursion

Now that we have a typed language, and one that forces us to annotate all functions, let’s annotate it. For simplicity, from now on we’ll assume we’re writing programs in a typed surface syntax, and that desugaring takes care of constructing core language terms.

Observe, first, that we have two identical terms being applied to each other. Historically, the overall term is called Ω (capital omega in Greek) and each of the identical sub-terms is called ω (lower-case omega in Greek). It is not a given that identical terms must have precisely the same type, because it depends on what invariants we want to assert of the context of use. In this case, however, observe that x binds to ω, so the second ω goes into both the first and second positions. As a result, typing one effectively types both.

Therefore, let’s try to type ω; let’s call this type γ. It’s clearly a function type, and the function takes one argument, so it must be of the form φ -> ψ. Now what is that argument? It’s ω itself. That is, the type of the value going into φ is itself γ. Thus, the type of ω is γ, which is φ -> ψ, which expands into (φ -> ψ) -> ψ, which further expands to ((φ -> ψ) -> ψ) -> ψ, and so on. In other words, this type cannot be written as any finite string!

15.2.3.2 Program Termination

We observed that the obvious typing of Ω, which entails typing γ, seems to run into serious problems. From that, however, we jumped to the conclusion that this type cannot be written as any finite string, for which we’ve given only an intuition, not a proof. In fact, something even stranger is true: in the type system we’ve defined so far, we cannot type Ω at all!

This is a strong statement, but we can actually say something much stronger. The typed language we have so far has a property called strong normalization: every expression that has a type will terminate computation after a finite number of steps. In other words, this special (and peculiar) infinite loop program isn’t the only one we can’t type; we can’t type any infinite loop (or even potential infinite loop). A rough intuition that might help is that any type—which must be a finite string—can have only a finite number of ->’s in it, and each application discharges one, so we can perform only a finite number of applications.

If our language permitted only straight-line programs, this would be unsurprising. However, we have conditionals and even functions being passed around as values, and with those we can encode any datatype we want. Yet, we still get this guarantee! That makes this a somewhat astonishing result.

This result also says something deeper. It shows that, contrary to what you may believe—that a type system only prevents a few buggy programs from running—a type system can change the semantics of a language. Whereas previously we could write an infinite loop in just one to two lines, now we cannot write one at all. It also shows that the type system can establish invariants not just about a particular program, but about the language itself. If we want to absolutely ensure that a program will terminate, we simply need to write it in this language and pass the type checker, and the guarantee is ours!

What possible use is a language in which all programs terminate? For general-purpose programming, none, of course. But in many specialized domains, it’s a tremendously useful guarantee to have. For instance, suppose you are implementing a complex scheduling algorithm; you would like to know that your scheduler is guaranteed to terminate so that the tasks being scheduled will actually run. There are many other domains, too, where we would benefit from such a guarantee: a packet-filter in a router; a real-time event processor; a device initializer; a configuration file; the callbacks in single-threaded JavaScript; and even a compiler or linker. In each case, we have an almost unstated expectation that these programs will always terminate. And now we have a language that can offer this guarantee—something it is impossible to test for, no less!These are not hypothetical examples. In the Standard ML language, the language for linking modules uses essentially this typed language for writing module linking specifications. This means developers can write quite sophisticated abstractions—they have functions-as-values, after all!—while still being guaranteed that linking will always terminate, producing a program.

15.2.3.3 Typing Recursion

How do we type such an expression? Clearly, we must have n bound in the body of the function as we type it (but not of course, in the use of the function); this much we know from typing functions. But what about Σ? Obviously it must be bound in the type environment when checking the use ((Σ 10)), and its type must be num -> num. But it must also be bound, to the same type, when checking the body of the function. (Observe, too, that the type returned by the body must match its declared return type.)

Now we can see how to break the shackles of the finiteness of the type. It is certainly true that we can write only a finite number of ->’s in types in the program source. However, this rule for typing recursion duplicates the -> in the body that refers to itself, thereby ensuring that there is an inexhaustible supply of applications. It’s our infinite quiver of arrows.

The code to implement this rule would be as follows. Assuming f is bound to the function’s name, aT is the function’s argument type and rT is its return type, b is the function’s body, and u is the function’s use:

15.2.4 Recursion in Data

We have seen how to type recursive programs, but this doesn’t yet enable us to create recursive data. We already have one kind of recursive datum—the function type—but this is built-in. We haven’t yet seen how developers can create their own recursive datatypes.

15.2.4.1 Recursive Datatype Definitions

This concludes our initial presentation of recursive types, but it has a fatal problem. We have not actually explained where this new type, BTnum, comes from. That is because we have had to pretend it is baked into our type-checker. However, it is simply impractical to keep changing our type-checker for each new recursive type definition—it would be like modifying our interpreter each time the program contains a recursive function! Instead, we need to find a way to make such definitions intrinsic to the type language. We will return to this problem later [REF].

This style of data definition is sometimes also known as a sum of products. “Product” refers to the way fields combine in one variant: for instance, the legal values of a BTnd are the cross-product of legal values in each of the fields, supplied to the BTnd constructor. The “sum” is the aggregate of all these variants: any given BTnum value is just one of these. (Think of “product” as being “and”, and “sum” as being “or”.)

15.2.4.2 Introduced Types

15.2.4.3 Pattern-Matching and Desugaring

Except, that’s not quite true. Somehow, the macro that generates the code above in terms of cond needs to know that the three positional selectors for a BTnd are BTnd-n, BTnd-l, and BTnd-r, respectively. This information is explicit in the type definition but only implicitly present in the use of the pattern-matcher (that, indeed, being the point). Thus, somehow this information must be communicated from definition to use. Thus the macro expander needs something akin to the type environment to accomplish its task.

Observe, furthermore, that expressions such as e1 and e2 cannot be type-checked—indeed, cannot even be reliable identified as expressions—until macro expansion expands the use of type-case. Thus, expansion depends on the type environment, while type-checking depends on the result of expansion. In other words, the two are symbiotic and need to happen, not quite in “parallel”, but rather in lock-step. Thus, building desugaring for a typed language, where the syntactic sugar makes assumptions about types, is a little more intricate than doing so for an untyped language.

15.2.5 Types, Time, and Space

It is evident that types already bestow a performance benefit in safe languages. That is because the checks that would have been performed at run-time—e.g., + checking that both its arguments are indeed numbers—are now performed statically. In a typed language, an annotation like : number already answers the question of whether or not something is of a particular a type; there is nothing to ask at run-time. As a result, these type-level predicates can (and need to) disappear entirely, and with them any need to use them in programs.

This is at some cost to the developer, who must convince the static type system that their program does not induce type errors; due to the limitations of decidability, even programs that might have run without error might run afoul of the type system. Nevertheless, for programs that meet this requirement, types provide a notable execution time saving.

Now let’s discuss space. Until now, the language run-time system has needed to store information attached to every value indicating what its type is. This is how it can implement type-level predicates such as number?, which may be used both by developers and by primitives. If those predicates disappear, so does the space needed to hold information to implement them. Thus, type-tags are no longer necessary.They would, however, still be needed by the garbage collector, though other representations such as BIBOP can greatly reduce their space impact.

The net result, however, is that the run-time representation must still store enough information to accurately answer these questions. However, previously it needed to use enough bits to record every possible type (and variant). Now, because the types have been statically segregated, for a type with no variants (e.g., there is only one kind of string), there is no need to store any variant information at all; that means the run-time system can use all available bits to store actual dynamic values.

In contrast, when variants are present, the run-time system must sacrifice bits to distinguish between the variants, but the number of variants within a type is obviously far smaller than the number of variants and types across all types. In the BTnum example above, there are only two variants, so the run-time system needs to use only one bit to record which variant of BTnum a value represents.

Observe, in particular, that the type system’s segregation prevents confusion. If there are two different datatypes that each have two variants, in the untyped world all these four variants require distinct representations. In contrast, in the typed world these representations can overlap across types, because the static type system will ensure one type’s variants are never confused for that the another. Thus, types have a genuine space (saving representation) and time (eliminating run-time checks) performance benefit for programs.

15.2.6 Types and Mutation

To avoid this morass, traditional type checkers adopt a simple policy: types must be invariant across mutation. That is, a mutation operation—whether variable mutation or structure mutation—cannot change the type of the mutant. Thus, the above examples would not type in our type language so far. How much flexibility this gives the programmer is, however, a function of the type language. For instance, if we were to admit a more flexible type that stands for “number or string”, then the examples above would type, but x would always have this, less precise, type, and all uses of x would have to contend with its reduced specificity, an issue we will return to later [REF].

In short, mutation is easy to account for in a traditional type system because its rule is simply that, while the value can change in ways below the level of specificity of the type system, the type cannot change. In the case of an operation like set! (or our core language’s setC), this means the type of the assigned value must match that of the variable. In the case of structure mutation, such as boxes, it means the assigned value must match that the box’s contained type.

15.2.7 The Central Theorem: Type Soundness

We have seen earlier [REF] that certain type languages can offer very strong theorems about their programs: for instance, that all programs in the language terminate. In general, of course, we cannot obtain such a guarantee (indeed, we added general recursion precisely to let ourselves write unbounded loops). However, a meaningful type system—indeed, anything to which we wish to bestow the noble title of a type systemWe have repeatedly used the term “type system”. A type system is usually a combination of three components: a language of types, a set of type rules, and an algorithm that applies these rules to programs. By largely presenting our type rules embedded in a function, we have blurred the distinction between the second and third of these, but can still be thought of as intellectually distinct.—ought to provide some kind of meaningful guarantee that all typed programs enjoy. This is the payoff for the programmer: by typing this program, she can be certain that certain bad things will certainly not happen. Short of this, we have just a bug-finder; while it may be useful, it is not a sufficient basis for building any higher-level tools (e.g., for obtaining security or privacy or robustness guarantees).

What theorem might we want of a type system? Remember that the type checker runs over the static program, before execution. In doing so, it is essentially making a prediction about the program’s behavior: for instance, when it states that a particular complex term has type num, it is effectively predicting that when run, that term will produce a numeric value. How do we know this prediction is sound, i.e., that the type checker never lies? Every type system should be accompanied by a theorem that proves this.

The central result we wish to have for a given type-system is called soundness. It says this. Suppose we are given an expression (or program) e. We type-check it and conclude that its type is t. When we run e, let us say we obtain the value v. Then v will also have type t.

The standard way of proving this theorem is to prove it in two parts, known as progress and preservation. Progress says that if a term passes the type-checker, it will be able to make a step of evaluation (unless it is already a value); preservation says that the result of this step will have the same type as the original. If we interleave these steps (first progress, then preservation; repeat), we can conclude that the final answer will indeed have the same type as the original, so the type system is indeed sound.

For instance, consider this expression: (+ 5 (* 2 3)). It has the type num. In a sound type system, progress offers a proof that, because this term types, and is not already a value, it can take a step of execution—which it clearly can. After one step the program reduces to (+ 5 6). Sure enough, as preservation proves, this has the same type as the original: num. Progress again says this can take a step, producing 11. Preservation again shows that this has the same type as the previous (intermediate) expressions: num. Now progress finds that we are at an answer, so there are no steps left to be taken, and our answer is of the same type as that given for the original expression.

The latter caveat looks like a cop-out. In fact, it is easy to forget that it is really a statement about what cannot happen at run-time: any exception not in this set will provably not be raised. Of course, in languages designed with static types in the first place, it is not clear (except by loose analogy) what these exceptions might be, because there would be no need to define them. But when we retrofit a type system onto an existing programming language—especially languages with only dynamic enforcement, such as Racket or Python—then there is already a well-defined set of exceptions, and the type-checker is explicitly stating that some set of those exceptions (such as “non-function found in application position” or “method not found”) will simply never occur. This is therefore the payoff that the programmer receives in return for accepting the type system’s syntactic restrictions.

15.3 Extensions to the Core

Now that we have a basic typed language, let’s explore how we can extend it to obtain a more useful programming language.

15.3.1 Explicit Parametric Polymorphism

Actually, none of these is quite the same. But the first and third are very alike, because the first is in Java and the third in our typed language, whereas the second, in C++, is different. All clear? No? Good, read on!

15.3.1.1 Parameterized Types

Obviously, it is impossible to load all these functions into our standard library: there’s an infinite number of these! We’d rather have a way to obtain each of these functions on demand. Our naming convention offers a hint: it is as if map takes two parameters, which are types. Given the pair of types as arguments, we can then obtain a map that is customized to that particular type. This kind of parameterization over types is called parametric polymorphism.Not to be confused with the “polymorphism” of objects, which we will discuss below [REF].

15.3.1.2 Making Parameters Explicit

Observe that once we start parameterizing, more code than we expect ends up being parameterized. For instance, consider the type of the humble cons. Its type really is parametric over the type of values in the list (even though it doesn’t actually depend on those values!—more on that in a bit [REF]) so every use of cons must be instantiated at the appropriate type. For that matter, even empty must be instantiated to create an empty list of the correct type! Of course, Java and C++ programmers are familiar with this pain.

15.3.1.3 Rank-1 Polymorphism

Instead, we will limit ourselves to one particularly useful and tractable point in this space, which is the type system of Standard ML, of the typed language of this book, of earlier versions of Haskell, roughly that of Java and C# with generics, and roughly that obtained using templates in C++. This language defines what is called predicative, rank-1, or prenex polymorphism. It answers the above questions thus: nothing, no, and yes. Let’s explore this below.

In rank-1 polymorphism, the type variables can only be substituted with monotypes. (Furthermore, these can only be concrete types, because there would be nothing left to substitute any remaining type variables.) As a result, we obtain a clear separation between the type variable-parameters and regular parameters. We don’t need to provide a “type annotation” for the type variables because we know precisely what kind of thing they can be. This produces a relatively clean language that still offers considerable expressive power.Impredicative languages erase the distinction between monotypes and polytypes, so a type variable can be instantiated with another polymorphic type.

Observe that because type variables can only be replaced with monotypes, they are all independent of each other. As a result, all type parameters can be brought to the front of the parameter list. This is what enables us to write types in the form ∀ tv, ... : t where the tv are type variables and t is a monotype (that might refer to those variables). This justifies not only the syntax but also the name “prenex”. It will prove to also be useful in the implementation.

15.3.1.4 Interpreting Rank-1 Polymorphism as Desugaring

However, this approach has two important limitations.

1. Let’s try to define a recursive polymorphic function, such as filter. Earlier we have said that we ought to instantiate every single polymorphic value (such as even cons and empty) with types, but to keep our code concise we’ll rely on the fact that the underlying typed language already does this, and focus just on type parameters for filter. Here’s the code:

   (define-poly (filter t)

   (lambda ([f : (t -> boolean)] [l : (listof t)]) : (listof t)

   (cond

   [(empty? l) empty]

   [(cons? l) (if (f (first l))

   (cons (first l)

   ((filter t) f (rest l)))

   ((filter t) f (rest l)))])))

   Observe that at the recursive uses of filter, we must instantiate it with the appropriate type.
   This is a perfectly good definition. There’s just one problem. When we try to use it—e.g.,

   (define filter_num (filter number))

   DrRacket does not terminate. Specifically, macro expansion does not terminate, because it is repeatedly trying to make new copies of the code of filter. If, in contrast, we write the function as follows, expansion terminates—

   (define-poly (filter2 t)

   (letrec ([fltr

   (lambda ([f : (t -> boolean)] [l : (listof t)]) : (listof t)

   (cond

   [(empty? l) empty]

   [(cons? l) (if (f (first l))

   (cons (first l) (fltr f (rest l)))

   (fltr f (rest l)))]))])

   fltr))

   but this needlessly pushes pain onto the user. Indeed, some template expanders will cache previous values of expansion and avoid re-generating code when given the same parameters. (Racket cannot do this because, in general, the body of a macro can depend on mutable variables and values and even perform input-output, so Racket cannot guarantee that a given input will always generate the same output.)

2. Consider two instantiations of the identity function. We cannot compare id_num and id_str because they are of different types, but even if they are of the same type, they are not eq?:

   (test (eq? (id number) (id number)) #f)

   This is because each use of id creates a new copy of the body. Now even if the optimization we mentioned above were applied, so for the same type there is only one code body, there would still be different code bodies for different typesIndeed, C++ templates are notorious for creating code bloat; this is one of the reasons.—but even this is unnecessary! There’s absolutely nothing in the body of id, for instance, that actually depends on the type of the argument. Indeed, the entire infinite family of id functions can share just one implementation. The simple desugaring strategy fails to provide this.

In other words, the desugaring based strategy, which is essentially an implementation by substitution, has largely the same problems we saw earlier with regards to substitution as an implementation of parameter instantiation. However, in other cases substitution also gives us a ground truth for what we expect as the program’s behavior. The same will be true with polymorphism, as we will soon see [REF].

Observe that one virtue to the desugaring strategy is that it does not require our type checker to “know” about polymorphism. Rather, the core type language can continue to be monomorphic, and all the (rank-1) polymorphism is handled entirely through expansion. This offers a cheap strategy for adding polymorphism to a language, though—as C++ shows—it also introduces significant overheads.

Finally, though we have only focused on functions, the preceding discussions applies equally well to data structures.

15.3.1.5 Alternate Implementations

There are other implementation strategies that don’t suffer from these problems. We won’t go into them here, but the essence of at least some of them is the “caching” approach we sketched above. Because we can be certain that, for a given set of type parameters, we will always get the same typed body, we never need to instantiate a polymorphic function at the same type twice. This avoids the infinite loop. If we type-check the instantiated body once, we can avoid checking at other instantiations of the same type (because the body will not have changed). Furthermore, we do not need to retain the instantiated sources: once we have checked the expanded program, we can dispose of the expanded terms and retain just one copy at run-time. This avoids all the problems discussed in the pure desugaring strategy shown above, while retaining the benefits.

Actually, we are being a little too glib. One of the benefits of static types is that they enable us to pick more precise run-time representations. For instance, a static type can tell us whether we have a 32-bit or 64-bit number, or for that matter a 32-bit value or a 1-bit value (effectively, a boolean). A compiler can then generate specialized code for each representation, taking advantage of how the bits are laid out (for example, 32 booleans can be packed into a single 32-bit word). Thus, after type-checking at each used type, the polymorphic instantiator may keep track of all the special types at which a function or data structure was used, and provide this information to the compiler for code-generation. This will then result in several copies of the function, none of which are eq? with each other—but for good reason and, because their operations are truly different, rightly so.

15.3.1.6 Relational Parametricity

There’s one last detail we must address regarding polymorphism.

We earlier said that a function like cons doesn’t depend on the specific values of its arguments. This is also true of map, filter, and so on. When map and filter want to operate on individual elements, they take as a parameter another function which in turn is responsible for making decisions about how to treat the elements; map and filter themselves simply obey their parameter functions.

One way to “test” whether this is true is to substitute some different values in the argument list, and a correspondingly different parameter function. That is, imagine we have a relation between two sets of values; we convert the list elements according to the relation, and the parameter function as well. The question is, will the output from map and filter also be predictable by the relation? If, for some input, this was not true of the output of map, then it must be that map inspected the actual values and did something with that information. But in fact this won’t happen for map, or indeed most of the standard polymorphic functions.

Functions that obey this relational rule are called relationally parametricRead Wadler’s Theorems for Free! and Reynolds’s Types, Abstraction and Parametric Polymorphism.. This is another very powerful property that types give us, because they tell us there is a strong limit on the kinds of operations such polymorphic functions can perform: essentially, that they can drop, duplicate, and rearrange elements, but not directly inspect and make decisions on them.

At first this sounds very impressive (and it is!), but on inspection you might realize this doesn’t square with your experience. In Java, for instance, a polymorphic method can still use instanceof to check which particular kind of value it obtained at run-time, and change its behavior accordingly. Such a method would indeed not be relationally parametric!On the Web, you will often find this property described as the inability of a function to inspect the argument—which is not quite right. Indeed, relational parametricity can equally be viewed as a statement of the weakness of the language: that it permits only a very limited set of operations. (You could still inspect the type—but not act upon what you learned, which makes the inspection pointless. Therefore, a run-time system that wants to simulate relational parametricity would have to remove operations like instanceof as well as various proxies to it: for instance, adding 1 to a value and catching exceptions would reveal whether the value is a number.) Nevertheless, it is a very elegant and surprising result, and shows the power of program reasoning possible with rich type systems.

15.3.2 Type Inference

First, let’s understand what type inference is doing. Some people mistakenly think of languages with inference as having no type declarations, with inference taking their place. This is confused at multiple levels. For one thing, even in languages with inference, programmers are free (and for documentation purposes, often encouraged—as you have been)Sometimes, inference is also undecidable and programmers have no choice but to declare some of the types. Finally, writing explicit annotations can greatly reduce indecipherable error messages. to declare types. Furthemore, in the absence of such declarations, it is not quite clear what inference actually means.

Instead, it is better to think of the underlying language as being fully, explicitly typed—just like the polymorphic language we have just studied [REF]. We will simply say that we are free to leave the type annotations after the :’s blank, and assume some programming environment feature will fill them in for us. (And if we can go that far, we can drop the :’s and extra embellishments as well, and let them all be inserted automatically. Thus, inference becomes simply a user convenience for alleviating the burden of writing type annotations, but the language underneath is explicitly typed.

How do we even think about what inference does? Suppose we have an expression (or program) e, written in an explicitly typed language: i.e., e has type annotations everywhere they are required. Now suppose we erase all annotations in e, and use a procedure infer to deduce them back.

We could demand many things of it. One might be that it produces precisely those annotations that e originally had. This is problematic for many reasons, not least that e might not even type-check, in which case how could infer possibly know what they were (and hence should be)? This might strike you as a pedantic trifle: after all, if e didn’t type-check, how can erasing its annotations and filling them back in make it do so? Since neither program type-checks, who cares?

With these preliminaries out of the way, we are now ready to delve into the mechanics of type inference. The most important thing to note is that our simple, recursive-descent type-checking algorithm [REF] will no longer work. That was possible because we already had annotations on all function boundaries, so we could descend into function bodies carrying information about those annotations in the type environment. Sans these annotations, it is not clear how to descend.

In fact, it is not clear that there is any particular direction that makes more sense than another. In a definition like mapper above, each fragment of code influences the other. For instance, applying empty?, cons?, first, and rest to l all point to its being a list. But a list of what? We can’t tell from any of those operations. However, the fact that we apply f to each (or indeed, any) first element means the list members must be of a type that can be passed to f. Similarly, we know the output must be a list because of cons and empty. But what are its elements? They must be the return type of f. Finally, note something very subtle: when the argument list is empty, we return empty, not l (which we know is bound to empty at that point). Using the former leaves the type of the return free to be any kind of list at all (constrained only by what f returns); using the latter would force it to be the same type as the argument list.

All this information is in the function. But how do we extract it systematically and in an algorithm that terminates and enjoys the property we have stated above? We do this in two steps. First we generate constraints, based on program terms, on what the types must be. Then we solve constraints to identify inconsistencies and join together constraints spread across the function body. Each step is relatively simple, but the combination creates magic.

15.3.2.1 Constraint Generation

Our goal, ultimately, is to find a type to fill into every type annotation position. It will prove to be just as well to find a type for every expression. A moment’s thought will show that this is likely necessary anyway: for instance, how can we determine the type to put on a function without knowing the type of its body? It is also sufficient, in that if every expression has had its type calculated, this will include the ones that need annotations.

First, we must generate constraints to (later) solve. Constraint generation walks the program source, emitting appropriate constraints on each expression, and returns this set of constraints. It works by recursive descent mainly for simplicity; it really computes a set of constraints, so the order of traversal and generation really does not matter in principle—so we may as well pick recursive descent, which is easy—though for simplicity we will use a list to represent this set.

(define-type Constraints

[eqCon (lhs : Term) (rhs : Term)])

(define-type Term

[tExp (e : ExprC)]

[tVar (s : symbol)]

[tNum]

[tArrow (dom : Term) (rng : Term)])

Now we can define the process of generating constraints:

When the expression is a number, all we can say is that we expect the type of the expression to be numeric:

This might sound trivial, but what we don’t know is what other expectations are being made of this expression by those containing it. Thus, there is the possibility that some outer expression will contradict the assertion that this expression’s type must be numeric, leading to a type error.

For an identifier, we simply say that the type of the expression is whatever we expect to be the type of that identifier:

If the context limits its type, then this expression’s type will automatically be limited, and must then be consistent with what its context expects of it.

Addition gives us our first look at a contextual constraint. For an addition expression, we must first make sure we generate (and return) constraints in the two sub-expressions, which might be complex. That done, what do we expect? That each of the sub-expressions be of numeric type. (If the form of one of the sub-expressions demands a type that is not numeric, this will lead to a type error.) Finally, we assert that the entire expression’s type is itself numeric.append3 is just a three-argument version of append.

The case for multC is identical other than the variant name.

Now we get to the other two interesting cases, function declaration and application. In both cases, we must remember to generate and return constraints of the sub-expressions.

In a function definition, the type of the function is a function (“arrow”) type, whose argument type is that of the formal parameter, and whose return type is that of the body:

Finally, we have applications. We cannot directly state a constraint on the type of the application. Rather, we can say that the function in the application position must consume arguments of the actual parameter expression’s type, and return types of the application expression’s type:

And that’s it! We have finished generating constraints; now we just have to solve them.

15.3.2.2 Constraint Solving Using Unification

The process used to solve constraints is known as unification. A unifier is given a set of equations. Each equation maps a variable to a term, whose datatype is above. Note one subtle point: we actually have two kinds of variables. Both tvar and tExp are “variables”, the former evidently so but the latter equally so because we need to solve for the types of these expressions. (An alternate formulation would introduce fresh type variables for each expression, but we would still need a way to identify which ones correspond to which expression, which eq? on the expressions already does automatically. Also, this would generate far larger constraint sets, making visual inspection daunting.)

For our purposes, the goal of unification is generate a substitution, or mapping from variables to terms that do not contain any variables. This should sound familiar: we have a set of simultaneous equations in which each variable is used linearly; such equations are solved using Gaussian elimination. In that context, we know that we can end up with systems that are both under- and over-constrained. The same thing can happen here, as we will soon see.

The unification algorithm works iteratively over the set of constraints. Because each constraint equation has two terms and each term can be one of four kinds, there are essentially sixteen cases to consider. Fortunately, we can cover all sixteen with fewer actual code cases.

The algorithm begins with the set of all constraints, and the empty substitution. Each constraint is considered once and removed from the set, so in principle the termination argument should be utterly simple, but it will prove to be only slightly more tricky in reality. As constraints are disposed, the substitution set tends to grow. When all constraints have been disposed, unification returns the final substitution set.

For a given constraint, the unifier examines the left-hand-side of the equation. If it is a variable, it is now ripe for elimination. The unifier adds the variable’s right-hand-side to the substitution and, to truly eliminate it, replaces all occurrences of the variable in the substitution with the this right-hand-side. In practice this needs to be implemented efficiently; for instance, using a mutational representation of these variables can avoid having to search-and-replace all occurrences. However, in a setting where we might need to backtrack (as we will, in the presence of unification [REF]), the mutational implementation has its own disadvantages.

The subtle error is this. We said that the unifier eliminates the variable by replacing all instances of it in the substitution. However, that assumes that the right-hand-side does not contain any instances of the same variable. Otherwise we have a circular definition, and it becomes impossible to perform this particular substitution. For this reason, unifiers include a occurs check: a check for whether the same variable occurs on both sides and, if it does, decline to unify.

Do you remember ω?

Let us now consider the implementation of unification. It is traditional to denote the substitution by the Greek letter Θ.

(define-type-alias Subst (listof Substitution))

(define-type Substitution

[sub [var : Term] [is : Term]])

(define (unify [cs : (listof Constraints)]) : Subst

(unify/Θ cs empty))

Let’s get the easy parts out of the way:

Now we’re ready for the heart of unification. We will depend on a function, extend+replace, with this signature: (Term Term Subst -> Subst). We expect this to perform the occurs test and, if it fails (i.e., there is no circularity), extends the substituion and replaces all existing instances of the first term with the second in the substitution. Similarly, we will assume the existence of lookup: (Term subst -> (optionof Term))

If the left-hand of a constraint equation is a variable, we first look it up in the substitution. If it is present, we replace the current constraint with a new one; otherwise, we extend the substitution:

The same logic applies when it is an expression designator:

Finally, we left with function types. Here the argument is almost exactly the same as for numeric types.

Note that we do not always shrink the size of the constraint set, so a simple argument does not suffice for proving termination. Instead, we must make an argument based on the size of the constraint set, and on the size of the substitution (including the number of variables in it).

The algorithm above is very general in that it works for all sorts of type terms, not only numbers and functions. We have used numbers as a stand-in for all form of base types; functions, similarly, stand for all constructed types, such as listof and vectorof.

With this, we are done. Unification produces a substitution. We can now traverse the substitution and find the types of all the expressions in the program, then insert the type annotations accordingly. A theorem, which we will not prove here, dictates that the success of the above process implies that the program would have typed-checked, so we need not explicitly run the type-checker over this program.

Observe, however, that the nature of a type error has now changed dramatically. Previously, we had a recursive-descent algorithm that walked a expressions using a type environment. The bindings in the type environment were programmer-declared types, and could hence be taken as (intended) authoritative specifications of types. As a result, any mismatch was blamed on the expressions, and reporting type errors was simple (and easy to understand). Here, however, a type error is a failure to notify. The unification failure is based on events that occur at the confluence of two smart algorithms—constraint generation and unification—and hence are not necessarily comprehensible to the programmer. In particular, the equational nature of these constraints means that the location reported for the error, and the location of the “true” error, could be quite far apart. As a result, producing better error messages remains an active research area.In practice the algorithm will maintain metadata on which program source terms were involved and probably on the history of unification, to be able to trace errors back to the source program.

Finally, remember that the constraints may not precisely dictate the type of all variables. If the system of equations is over-constrained, then we get clashes, resulting in type errors. If instead the system is under-constrained, that means we don’t have enough information to make definitive statements about all expressions. For instance, in the expression (lambda (x) x) we do not have enough constraints to indicate what the type of x, and hence of the entire expression, must be. This is not an error; it simply means that x is free to be any type at all. In other words, its type is “the type of x -> the type of x” with no other constraints. The types of these underconstrained identifiers are presented as type variables, so the above expression’s type might be reported as (’a -> ’a).

The unification algorithm actually has a wonderful property: it automatically computes the most general types for an expression, also known as principal types. That is, any actual type the expression can have can be obtained by instantiating the inferred type variables with actual types. This is a remarkable result: in another example of computers beating humans, it says that no human can generate a more general type than the above algorithm can!

15.3.2.3 Let-Polymorphism

The reason for this is because the types we have inferred through unification are not actually polymorphic. This is important to remember: just because you type variables, you haven’t seen polymorphism! The type variables could be unified at the next use, at which point you end up with a mere monomorphic function. Rather, true polymorphism only obtains when you have true instantiation of type variables.

In languages with true polymorphism, then, constraint generation and unification are not enough. Instead, languages like ML, Haskell, and even our typed programming language, implement something colloquially called let-polymorphism. In this strategy, when a term with type variables is bound in a lexical context, the type is automatically promoted to be a quantified one. At each use, the term is effectively automatically instantiated.

There are many implementation strategies that will accomplish this. The most naive (and unsatisfying) is to merely copy the code of the bound identifier; thus, each use of id above gets its own copy of (lambda (x) x), so each gets its own type variables. The first might get the type (’a -> ’a), the second (’b -> ’b), the third (’c -> ’c), and so on. None of these type variables clash, so we get the effect of polymorphism. Obviously, this not only increases program size, it also does not work in the presence of recursion. However, it gives us insight into a better solution: instead of copying the code, why not just copy the type? Thus at each use, we create a renamed copy of the inferred type: id’s (’a -> ’a) becomes (’b -> ’b) at the first use, and so on, thus achieving the same effect as copying code but without its burdens. Because all these strategies effectively mimic copying code, however, they only work within a lexical context.

15.3.3 Union Types

Either way, the point of a union type is to represent a disjunction, or “or”. A value’s type is one of the types in the union. A value usually belongs to only one of the types in the union, though this is a function of precisely how the union types are defined, whether there are rules for normalizing them, and so on.

15.3.3.1 Structures as Types

15.3.3.2 Untagged Unions

15.3.3.3 Discriminating Untagged Unions

15.3.3.4 Retrofitting Types

It is unsurprising that Typed Racket uses union types. They are especially useful when retrofitting types onto existing programming languages whose programs were not defined with an ML-like type discipline in mind, such as in scripting languages. A common principle of such retrofitted types is to statically catch as many dynamic exceptions as possible. Of course, the checker must ultimately reject some programs,Unless it implements an interesting idea called soft typing, which rejects no programs but provides information about points where the program would not have been typeable. and if it rejects too many programs that would have run without an error, developers are unlikely to adopt it. Because these programs were written without type-checking in mind, the type checker may therefore need to go to heroic lengths to accept what are considered reasonable idioms in the language.

15.3.3.5 Design Choices

15.3.4 Nominal Versus Structural Systems

In our initial type-checker, two types were considered equivalent if they had the same structure. In fact, we offered no mechanism for naming types at all, so it is not clear what alternative we had.

There are two perfectly reasonable interpretations. One is to say that v was declared to be of type NB1, which is a different name than NB2, and hence should be considered a different type, so the above application should result in an error. Such a system is called nominal, because the name of a type is paramount for determining type equality.

In contrast, another interpretation is that because the structure of NB1 and NB2 are identical, there is no way for a developer to write a program that behaves differently on values of these two types, so these two types should be considered identical.If you want to get especially careful, you would note that there is a difference between being considered the same and actually being the same. We won’t go into this issue here, but consider the implication for a compiler writer choosing representations of values, especially in a language that allows run-time inspection of the static types of values. Such a type system is called structural, and would successfully type the above expression. (Typed Racket follows a structural discipline, again to reduce the burden of importing existing untyped code, which—in Racket—is usually written with a structural interpretation in mind. In fact, Typed Racket not only types (f v), it prints the result as having type NB1, despite the return type annotation on f!)

The difference between nominal and structural typing is most commonly contentious in object-oriented languages, and we will return to this issue briefly later [REF]. However, the point of this section is to illustrate that these questions are not intrinsically about “objects”. Any language that permits types to be named—as all must, for programmer sanity—must contend with this question: is naming merely a convenience, or are the choices of names intended to be meaningful? Choosing the former answer leads to structural typing, while choosing the latter leads down the nominal path.

15.3.5 Intersection Types

Since we’ve just explored union types, you must naturally wonder whether there are also intersection types. Indeed there are.

If a union type means that a value (of that type) belongs to one of the types in the union, an intersection type clearly means the value belongs to all the types in the intersection: a conjunction, or “and”. This might seem strange: how can a value belong to more than one type?

Observe that with this type, the return type on all invocations is (number U string). Thus, on every return we must distinguish beween numeric and string returns, or else we will get a type error. Thus, even though we know that if given two numeric arguments we will get a numeric result, this information is lost to the type system.

More subtly, this type permits each argument’s type to be chosen independently of the other. Thus, according to this type, the invocation (+ 3 "x") is perfectly valid (and produces a value of type (number U string)). But of course the addition operation we have specified is not defined for these inputs at all!

15.3.6 Recursive Types

Observe that even with this informal understanding of μ, we can now provide a type to ω, and hence to Ω.

15.3.7 Subtyping