Scheme (and Racket) Macros

Scheme, Racket included (and much extended), has a solution that is better than defmacro: it has define-syntax and syntax-rules. First of all, define-syntax is used to create the “magical connection” between user code and some macro transformation code that does some rewriting. This definition:

makes foo be a special syntax that, when used in the head of an expression, will lead to transforming the expression itself, where the result of this transformation is what gets used instead of the original expression. The “...something...” in this code fragment should be a transformation function — one that consumes the expression that is to be transformed, and returns the new expression to run.

Next, syntax-rules is used to create such a transformation in an easy way. The idea is that what we thought to be an informal specification of such rewrites, for example:

and

can actually be formalized by automatically creating a syntax transformation function from these rule specifications. (Note that this example has round parentheses so we don’t fall into the illusion that square brackets are different: the resulting transformation would be the same.) The main point is to view the left hand side as a pattern that can match some forms of syntax, and the right hand side as producing an output that can use some matched patterns.

syntax-rules is used with such rewrite specifications, and it produces the corresponding transformation function. For example, this:

evaluates to a function that is somewhat similar to:

but match is a little closer, since it uses similar input patterns:

Such transformations are used in a define-syntax expression to tie the transformer back into the compiler by hooking it on a specific keyword. You can now appreciate how all this work when you see how easy it is to define macros that are very tedious with define-macro. For example, the above bind:

and let* with its two rules:

These transformations are so convenient to follow, that Scheme specifications (and reference manuals) describe forms by specifying their definition. For example, the Scheme report, specifies let* as a “derived form”, and explains its semantics via this transformation.

The input patterns in these rules are similar to match patterns, and the output patterns assemble an s-expression using the matched parts in the input. For example:

does the thing you expect it to do — matches a parenthesized form with two sub-forms, and produces a form with the two sub-forms swapped. The rules for “...” on the left side are similar to match, as we have seen many times, and on the right side it is used to splice a matched sequence into the resulting expression and it is required to use the ... for sequence-matched pattern variables. For example, here is a list of some patterns, and a description of how they match an input when used on the left side of a transformation rule and how they produce an output expression when they appear on the right side:

• (x ...)

  LHS: matches a parenthesized sequence of zero or more expressions, and the x pattern variable is bound to this whole sequence; match analogy: (match ? [(list x ...) ?])

  RHS: when x is bound to a sequence, this will produce a parenthesized expression containing this sequence; match analogy: (match ? [(list x ...) x])

• (x1 x2 ...)

  LHS: matches a parenthesized sequence of one or more expressions, the first is bound to x1 and the rest of the sequence is bound to x2;

  match analogy: (match ? [(list x1 x2 ...) ?])

  RHS: produces a parenthesized expression that contains the expression bound to x1 first, then all of the expressions in the sequence that x2 is bound to;

  match analogy: (match ? [(list x1 x2 ...) (cons x1 x2)])

• ((x y) ...)

  LHS: matches a parenthesized sequence of 2-form parenthesized sequences, binding x to all the first forms of these, and y to all the seconds of these (so they will both have the same number of items);

  match analogy: (match ? [(list (list x y) ...) ?])

  RHS: produces a list of forms where each one is made of consecutive forms in the x sequence and consecutive forms in the y sequence (both sequences should have the same number of elements);

  match analogy:

  (match ? [(list (list x y) ...)
            (map (lambda (x y) (list x y)) x y)])

Some examples of transformations that would be very tedious to write code manually for:

• ((x y) ...) --> ((y x) ...)

  Matches a sequence of 2-item sequences, produces a similar sequence with all of the nested 2-item sequences flipped.

• ((x y) ...) --> ((x ...) (y ...))

  Matches a similar sequence, and produces a sequence of two sequences, one of all the first items, and one of the second ones.

• ((x y ...) ...) --> ((y ... x) ...)

  Similar to the first example, but the nested sequences can have 1 or more items in them, and the nested sequences in the result have the first element moved to the end. Note how the ... are nested: the rule is that for each pattern variable you count how many ...s apply to it, and that tells you what it holds — and you have to use the same ... nestedness for it in the output template.

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This is solving the problems of easy code — no need for list, cons etc, not even for quasiquotes and tedious syntax massaging. But there were other problems. First, there was a problem of bad scope, one that was previously solved with a gensym:

Translating this to define-syntax and syntax-rules we get something simpler:

Even simpler, Racket has a macro called define-syntax-rule that expands to a define-syntax combined with a syntax-rules — using it, we can write:

This looks like like a function — but you must remember that it is a transformation rule specification which is a very different beast, as we’ll see.

The main thing here is that Racket takes care of making bindings follow the lexical scope rules:

works fine. In fact, it fully respects the scoping rules: there is no confusion between bindings that the macro introduces and bindings that are introduced where the macro is used. (Think about different colors for bindings introduced by the macro and other bindings.) It’s fine with many cases that are much harder to cope with otherwise (eg, cases where there is no gensym magic solution):

You can even use both:

and use DrRacket’s macro debugger to see how the various bindings get colored differently.

define-macro advocates will claim that it is difficult to make a macro that intentionally plants an identifier. Think about a loop macro that has an abort that can be used inside its body. Or an if-it form that is like if, but makes it possible to use the condition’s value in the “then” branch as an it binding. It is possible with all Scheme macro systems to “break hygiene” in such ways, and we will later see how to do this in Racket. However, Racket also provides a better way to deal with such problems (think about it being always “bound to a syntax error”, but locally rebound in an if-it form).

Scheme macros are said to be hygienic — a term used to specify that they respect lexical scope. (All of this can get much more important in the presence of a module system, since you can write a module that provides transformations rules, not just values and functions.) There are several implementations of hygienic macro systems across Scheme implementations.

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The way that Scheme implementations achieve hygiene in a macro system is by making it deal with more than just raw S-expressions. Roughly speaking, it deals with syntax objects that are sort of a wrapper structure around S-expression, carrying additional information. The important part of this information when it gets to dealing with hygiene is the “lexical scope” — which can roughly be described as having identifiers be represented as symbols plus a “color” which represents the scope. This way such systems can properly avoid confusing identifiers with the same name that come from different scopes.

There was also the problem of making debugging difficult, because a macro can introduce errors that are “coming out of nowhere”. In the implementation that we work with, this is solved by adding yet more information to these syntax objects — in addition to the underlying S-expression and the lexical scope, they also contain source location information. This allows Racket (and DrRacket) to locate the source of a specific syntax error, so locating the offending code is easy. DrRacket’s macro debugger heavily relies on this information to provide a very useful tool — since writing macros can easily become a hard job.

Finally, there was the problem of writing bad macros. For example, it is easy to forget that you’re dealing with a macro definition and write:

just because you want to inline the addition — but in this case you end up duplicating the input expression which can have a disastrous effect. For example:

expands to a lot of code to compile.

Another example is:

the problem here is that (* foo 2) will be used as an identifier to be bound by the let expression — which can lead to a confusing syntax error.

Racket provides many tools to help macro programmers — in addition to a user-interface tool like the macro debugger there are also programmer-level tools where you can reject an input if it doesn’t contain an identifier at a certain place etc. Still, writing macros is much harder than writing functions — some of these problems are inherent to the problem that macros solve; for example, you may want a twice macro that replicates an expression. By specifying a transformation to the core language, a macro writer has full control over which expressions get evaluated and how, which identifiers are binding instances, and how is the scope of the given expression is shaped.

Meta Macros

One of the nice results of syntax-rules dealing with the subtle points of identifiers and scope is that things works fine even when we “go up a level”. For example, the short define-syntax-rule form that we’ve seen is itself a defined as a simple macro:

In fact, this is very similar to something that we have already seen: the rewrite form that we have used in Schlac is implemented in just this way. The only difference is that rewrite requires an actual => token to separate the input pattern from the output template. If we just use it in a syntax rule:

it won’t work. Racket treats the above => just like any identifier, which in this case acts as a pattern variable which matches anything. The solution to this is to list the => as a keyword which is expected to appear in the macro use as-is — and that’s what the mysterious () of syntax-rules is used for: any identifier listed there is taken to be such a keyword. This makes the following version

do what we want and throw a syntax error unless rewrite is used with an actual => in the proper place.

Lazy Constructions in an Eager Language

PLAI §37 (has some examples)

This is not really lazy evaluation, but it gets close, and provides the core useful property of easy-to-use infinite lists.

Using it:

Actually, all Scheme implementations come with a generalized tool for (local) laziness: a delay form that delays computation of its body expression, and a force function that forces such promises. Here is a naive implementation of this:

Proper definitions of delay/force cache the result — and practical ones can get pretty complex, for example, in order to allow tail calls via promises.

Recursive Macros

Syntax transformations can be recursive. For example, we have seen how let* can be implemented by a transformation that uses two rules, one of which expands to another use of let*:

When Racket expands a let* expression, the result may contain a new let* which needs extending as well. An important implication of this is that recursive macros are fine, as long as the recursive case is using a smaller expression. This is just like any form of recursion (or loop), where you need to be looping over a well-founded set of values — where each iteration uses a new value that is closer to some base case.

For example, consider the following macro:

It seems like this is a good implementation of a while loop — after all, if you were to implement it as a function using thunks, you’d write very similar code:

But if you look at the nested while form in the transformation rule, you’ll see that it is exactly the same as the input form. This means that this macro can never be completely expanded — it specifies infinite code! In practice, this makes the (Racket) compiler loop forever, consuming more and more memory. This is unlike, for example, the recursive let* rule which uses one less binding-value pair than specified as its input.

The reason that the function version of while is fine is that it iterates using the same code, and the condition thunk will depend on some state that converges to a base case (usually the body thunk will perform some side-effects that makes the loop converge). But in the macro case there is no evaluation happening, if the transformed syntax contains the same input pattern, we end up having a macro that expands infinitely.

The correct solution for a while macro is therefore to use plain recursion using a local recursive function:

A popular way to deal with macros like this that revolve around a specific control flow is to separate them into a function that uses thunks, and a macro that does nothing except wrap input expressions as thunks. In this case, we get this solution:

Another example: a simple loop

Here is an implementation of a macro that does a simple arithmetic loop:

(Note that this is not complete code: it suffers from the usual problem of multiple evaluations of the n expression. We’ll deal with it soon.)

This macro combines both control flow and lexical scope. Control flow is specified by the loop (done, as usual in Racket, as a tail-recursive function) — for example, it determines how code is iterated, and it also determines what the for form will evaluate to (it evaluates to whatever when evaluates to, the void value in this case). Scope is also specified here, by translating the code to a function — this code makes x have a scope that covers the body so this is valid:

but it also makes the boundary expression n be in this scope, making this:

valid. In addition, while evaluating the condition on each iteration might be desirable, in most cases it’s not — consider this example:

This is easily solved by using a let to make the expression evaluate just once:

which makes the previous use result in a “reference to undefined identifier: i” error.

Furthermore, the fact that we have a hygienic macro system means that it is perfectly fine to use nested for expressions:

The transformation is, therefore, completely specifying the semantics of this new form.

Extending this syntax is easy using multiple transformation rules — for example, say that we want to extend it to have a step optional keyword. The standard idiom is to have the step-less pattern translated into one that uses step 1:

Usually, you should remember that syntax-rules tries the patterns one by one until a match is found, but in this case there is no problems because the keywords make the choice unambiguous:

We can even extend it to do a different kind of iteration, for example, iterate over list:

Yet Another: List Comprehension

At this point it’s clear that macros are a powerful language feature that makes it relatively easy to implement new features, making it a language that is easy to use as a tool for quick experimentation with new language features. As an example of a practical feature rather than a toy, let’s see how we can implement Python’s list comprehenions. These are expressions that conveniently combine map, filter, and nested uses of both.

First, a simple implementation that uses only the map feature:

It is a good exercise to see how everything that we’ve seen above plays a role here. For example, how we get the ID to be bound in EXPR.

Next, add a condition expression with an if keyword, and implemented using a filter:

Again, go over it and see how the binding structure makes the identifier available in both expressions. Note that since we’re just playing around we’re not paying too much attention to performance etc. (For example, if we cared, we could have implemented the if-less case by not using filter at all, or we could implement a filter that accepts #t as a predicate and in that case just returns the list, or even implementing it as a macro that identifies a (lambda (_) #t) pattern and expands to just the list (a bad idea in general).)

The last step: Python’s comprehension accepts multiple for-ins for nested loops, possibly with if filters at each level:

A collection of examples that I found in the Python docs and elsewhere, demonstrating all of these:

Problems of syntax-rules Macros

As we’ve seen, using syntax-rules solves many of the problems of macros, but it comes with a high price tag: the macros are “just” rewrite rules. As rewrite rules they’re pretty sophisticated, but it still loses a huge advantage of what we had with define-macro — the macro code is no longer Racket code but a simple language of rewrite rules.

There are two big problems with this which we will look into now. (DrRacket’s macro stepper tool can be very useful in clarifying these examples.) The first problem is that in some cases we want to perform computations at the macro level — for example, consider a repeat macro that needs to expand like this:

With a syntax-rules macro we can match over specific integers, but we just cannot do this with any integer. Note that this specific case can be done better via a function — better by not replicating the expression:

or even better, assuming the above for is already implemented:

But still, we want to have the ability to do such computation. A similar, and perhaps better example, is better error reporting. For example, the above for implementation blindly expands its input, so:

we get a bad error message in terms of lambda, which is breaking abstraction (it comes from the expansion of for, which is an implementation detail), and worse — it is an error about something that the user didn’t write.

Yet another aspect of this problem is that sometimes we need to get creative solutions where it would be very simple to write the corresponding Racket code. For example, consider the problem of writing a rev-app macro — (rev-app F E …) should evaluate to a function similar to (F E …), except that we want the evaluation to go from right to left instead of the usual left-to-right that Racket does. This code is obviously very broken:

because it generates a malformed let form — there is no way for the macro expander to somehow know that the reverse should happen at the transformation level. In this case, we can actually solve this using a helper macro to do the reversing:

There are still problems with this — it complains about x ... because there is a single x there rather than a sequence of them; and even if it did somehow work, we also need the xs in that last line in the original order rather than the reversed one. So the solution is complicated by collecting new xs while reversing — and since we need them in both orders, we’re going to collect both orders:

So, this worked, but in this case the simplicity of the syntax-rules rewrite language worked against us, and made a very inconvenient solution. This could have been much easier if we could just write a “meta-level” reverse, and a use of map to generate the names.

… And all of that was just the first problem. The second one is even harder: syntax-rules is designed to avoid all name captures, but what if we want to break hygiene? There are some cases where you want a macro that “injects” a user-visible identifier into its result. The most common (and therefore the classic) example of this is an anaphoric if macro, that binds it to the result of the test (which can be any value, not just a boolean):

which we want to turn to:

The obvious definition of `if-it’ doesn’t work:

The reason it doesn’t work should be obvious now — it is designed to avoid the it that the macro introduced from interfering with the it that the user code uses.

Next, we’ll see Racket’s “low level” macro system, which can later be used to solve these problems.

Racket’s “Low-Level” Macros

As we’ve previously seen, syntax-rules creates transformation functions — but there are other more direct ways to write these functions. These involve writing a function directly rather than creating one with syntax-rules — and because this is a more low-level approach than using syntax-rules to generate a transformer, it is called a “low level macro system”. All Scheme implementations have such low-level systems, and these systems vary from one to the other. They all involve some particular type that is used as “syntax” — this type is always related to S-expressions, but it cannot be the simple define-macro tool that we’ve seen earlier if we want to avoid the problems of capturing identifiers.

Historical note: For a very long time the Scheme standard had avoided a concrete specification of this low-level system, leaving syntax-rules as the only way to write portable Scheme code. This had lead some people to explore more thoroughly the things that can be done with just syntax-rules rewrites, even beyond the examples we’ve seen. As it turns out, there’s a lot that can be done with it — in fact, it is possible to write rewrite rules that implement a lambda calculus, making it possible to write things that look close to “real code”. This is, however, awkward to get used to, and redundant with a macro system that can use the full language for arbitrary computations. It has also became less popular recently, since R6RS dictates something that is known as a “syntax-case macro system” (not really a good name, since syntax-case is just a tool in this system).

Racket uses an extended version of this syntax-case system, which is what we will discuss now. In the Racket macro system, “syntax” is a new type, not just S-expressions as is the case with define-macro. The way to think about this type is as a wrapper around S-expressions, where the S-expression is the “raw” symbolic form of the syntax, and a bunch of “stuff” is added. Two important bits of this “stuff” are the source location information for the syntax, and its lexical scope. The source location is what you’d expect: the file name for the syntax (if it was read from a file), its position in the file, and its line and column numbers; this information is mostly useful for reporting errors. The lexical scope information is used in a somewhat peculiar way: there is no direct way to access it, since usually you don’t need to do so — instead, for the rare cases where you do need to manipulate it, you copy the lexical scope of an existing syntax to create another. This allows the macro interface to be usable without specification of a specific representation for the scope.

The main piece of functionality in this system is syntax-case (which has lead to its common name) — a form that is used to deconstruct the input via pattern-matching similar to syntax-rules. In fact, the syntax of syntax-case looks very similar to the syntax of syntax-rules — there are zero or more parenthesized keywords, and then clauses that begin with the same kind of patterns to match the syntax against. The first obvious difference is that the syntax to be matched is given explicitly:

A macro is written as a plain function, usually used as the value in a define-syntax form (but it could also be used in plain helper functions). For example, here’s how the orelse macro is written using this:

Racket’s define-syntax can also use the same syntactic sugar for functions as define:

The second significant change from syntax-rules is that the right-hand-side expressions in the branches are not patterns. Instead, they’re plain Racket expressions. In this example (as in most uses of syntax-case) the result of the syntax-case form is going to be the result of the macro, and therefore it should return a piece of syntax. So far, the only piece of syntax that we see in this code is just the input stx — and returning that will get the macro expander in an infinite loop (because we’re essentially making a transformer for orelse expressions that expands the syntax to itself).

To return a new piece of syntax, we need a way to write new syntax values. The common way to do this is using a new special form: syntax. This form is similar to quote — except that instead of an S-expression, it results in a syntax. For example, in this code:

the first printout happens during macro expansion, and the second is part of the generated code. Like quote, syntax has a convenient abbreviation — “#'”:

Now the question is how we can get the actual macro working. The thing is that syntax is not completely quoting its contents as a syntax — there could be some meta identifiers that are bound as “pattern variables” in the syntax-case pattern that was matched for the current clause — in this case, we have x and y as such pattern variables. (Actually, orelse is a pattern variable too, but this doesn’t matter for our purpose.) Using these inside a syntax will have them replaced by the syntax that they matched against. The complete orelse definition is therefore very easy:

The same treatment of ... holds here too — in the matching pattern they specify 0 or more occurrences of the preceding pattern, and in the output template they mean that the matching sequence is “spliced” in. Note that syntax-rules is now easy to define as a macro that expands to a function that uses syntax-case to do the actual rewrite work: