An alternative to s-expressions for extensible syntax

Why do people like Lisp? I mean, it looks weird, doesn't it? Countless backronyms were designed to mock its syntax: "Lost In a Sea of Parentheses", "Lots of Irritating Superfluous Parentheses" and other jabs at the very large amounts of brackets found in typical programs. It is one of the rare languages where ")))))" is a commonplace sight and also one of the rare languages to eschew the infix arithmetic we have known and loved since grade school.

And yet, it endures. Lisp and its variants have been around for many decades and are still actively maintained and extended. Its syntax (s-expressions) has even seen a renaissance of sorts recently, with new languages such as Clojure.

What's great about Lisp isn't really the way it looks: aesthetically, it looks terrible to almost everyone. It is also not the tersest of languages. No, what's great about Lisp is that idea that "code is data". I mean, it always is, really: nearly all programming languages transform plain text into some kind of abstract syntax tree (AST). However, if you want to extend a language, if you want to adapt it to your needs or try a new paradigm, you're going to give it some kind of surface syntax, and you're going to need to manipulate code as a data structure. And that's where Lisp shines.

Some people would say that a good programming language shouldn't need to be extended, but that's wishful thinking: no one can think of all the things people might want to do, and there are no general principles that can optimize all use cases. There are many reasons to want to extend a language:

If you were to design a language to make it as extensible as possible, you would see that many factors constrain your design:

  1. Work with AST, not grammar: if extending the parser is possible, sure, the programmer can do what they want, but they'll have to learn how to write grammar rules and deal with syntax, even though what they really care about are the semantics. So it is not ideal: it would be better if they could work from the AST directly.
  2. Simplicity: if the data structures are too complex, they will have a hard time committing them to memory and they will be less motivated to write extensions. So the data structures should be as simple as possible.
  3. Homoiconicity: if the connection between surface syntax and the data structures is not clear, then they will struggle to figure out where their extension is going to fit.
  4. Genericity: finally, if extensions stand out from the rest of the language like a sore thumb, people are unlikely to use them. It's just a matter of visual coherency: the language's own structures are first class and unless extensions can be made to look just like them, they will feel second class.

Lisp does not promote the use of parser extensions: you work on the AST. ASTs are nested lists – you couldn't do simpler if you tried. The translation from source code to AST is completely obvious: (x y z) is a list with the three elements x, y and z. All of the language's constructs are defined by the first atom of a list, and so are all extensions. Nothing stands out. Lisp follows these instructions to a tee. The only cost to pay is lots of parentheses.

Plenty of people are comfortable with paying that price. Nonetheless, even more are put off by it – it is arguably the main reason why use of these languages is not more widespread. Besides subjective aesthetic preference, there are a few issues with s-expressions – the kind of issues that aficionados will gloss over because they are somewhat minor, but…

This suggests that s-expressions might not be the end-all of extensibility: there is a lot of opportunity in designing extensible syntax that "looks" better and could gain wider adoption as a result.


Of course, if improving on s-expressions was easy, it would have been done already. The issue is that it is very difficult to improve on Lisp's syntax without increasing the complexity of extending the language or crippling the kind of extensions that are possible to do. Of course, I don't think it is impossible. But it is very hard and it is very easy for language designers to fall into all sorts of traps: they know every detail of the language they made, so they are going to underestimate its complexity (me included, probably).

To expound on the difficulty, let's list things that won't fly:

Conversely, here are the language properties the language should have:

  1. Uniform behavior: all tokens should behave the same syntactically in all situations. If you have, say, {x} over here and {x} over there, they should produce the same AST at both places. If you have a, b in c, I don't care what the context is: same AST. That way you have to think about syntax a lot less.
  2. Rule ubiquity: if I was to pick some moderately sized source file at random -- big enough not to be trivial, but small enough that I can go through it in 15 minutes or so -- it should exhibit all the syntax rules I need to know. If I know the AST of some random source file, I should be able to work out the AST of every source file.
    In short, it should be virtually impossible to write any code base of significant size without using all the syntax available. This rule ensures that the syntax will be learned quickly and completely by all users of the language.

Lisp obviously satifies both criteria, but you will note that it does in the strictest way imaginable and that there is some exploitable leeway. Rule 2 has the most restrictive implications:

All we need to do now is figure out just how far we can go before these rules are bent or broken.

Infix operators

Let's start with something that is sorely missing in Lisp: infix operators. People like them, and we aim to please. We have to be parsimonious in what we add, though, so the first question we should ask ourselves is are they common enough?

Personally, I would think so. First, pretty much everyone knows what operators are, so familiarity is the norm. Sure, some may be confused about BEDMAS, but everybody knows what 2 + 2 is supposed to mean. Second, there are many common use cases outside of standard arithmetic: declaration, assignment, anonymous functions, key/value mappings, typing, concatenation, etc. We can reasonably assume that everybody who learns the language will run across infix syntax eventually.

As for what constitutes an operator and what doesn't, we can devise simple enough rules. For instance, perhaps we can have a set of "operator characters", any sequence of which defines an operator. This creates an unbounded number of operators for extension purposes.

So we have some tokens that are identifiers and some other tokens that are operators. That's a good start, but what happens when we combine them?

Iteration 1

Let's start with a few very simple rules loosely inspired from s-expressions (but not necessarily based on s-expressions).

  1. Juxtaposition using whitespace is application of a function to arguments. f x y applies f to x and y. f may be a function or a macro.
    • As a special case, f. applies f to no arguments.
  2. a <op> b <op> c applies <op> to a, b and c. It is like (<op>) a b c.
  3. Juxtaposition has higher priority than any operator.
  4. (...) groups things together. It has no presence in the syntax tree, so extra pairs of parentheses will not change the AST.
  5. Mixing different operators together is not allowed.

This is not too bad; if we were to give to the = operator the semantics of define and the semantics of lambda to the -> operator, you'd end up with code like this:

diag = (x y -> (((x * x) + (y * y)) ^ 0.5))
if (diag 3 4 == 5) (print "hello") (print "world")

Still looks a lot like s-expressions, though. Very parenthesey.

Iteration 2

One way to make the code terser would be to add some priority relationships. You'd be removing some punctuation without adding any back; but you have to be careful, because a priority graph is one more thing for users to learn. The ubiquity rule is a good guide here: the priority graph can only involve commonplace operators.

Of course, this would create a "first class" of operators: the ones in the graph. All the others would be "second class" and that is less than ideal. This being said, operators are a lot more versatile than, say, control structures. Furthermore, allowing users to define priority for custom operators is a lot easier and much less disruptive than letting them add general parsing rules.

For the default priority graph, it would be reasonable to have the usual order of operations for arithmetic because it is common knowledge. Declarations and anonymous functions can be low priority right-associative operators, because that makes sense. We just need to make sure only to define the bare minimum that lets us remove 90% of parentheses.

diag = x y -> (x * x + y * y) ^ 0.5
if (diag 3 4 == 5) (print "hello") (print "world")

Figuring out the structure is harder than for s-expressions but it is not a lot harder: once you wrap your mind around the (hopefully very simple) priority graph and understand that the context is always irrelevant, you can work out the AST of any expression or sub-expression.

Iteration 3

There is still the question of what to do with newlines: if they were equivalent to spaces, we'd have to parenthesize pretty much every line. That's not what we signed up for. But if each line was automatically wrapped in ()s, we'd have another problem: breaking up if test a b on more than one line in a natural way would be impossible unless you parenthesized the whole thing. Significant indent is an attractive solution, but we don't really want to make that a cornerstone of the syntax.

However, if test a b could be structured differently. For instance, it could be if test (a b) and then we could write this:

if test (

Unfortunately there is no difference between if test (a b) and if test ((a b)), so how exactly do we tell apart the use case where we want to execute the single statement (a b) from the one where we want to execute a, and then b?

It is at this point that we might want to enforce the syntactic distinction I mentioned earlier, which languages based on s-expressions erase: the distinction between applications and lists of things. It's a pretty ubiquitous distinction: virtually all programs contain some function applications and some lists of things or statements.

  1. (a, b, ...) defines a sequence; in normal situations, it returns the value of the last expression. The comma has lowest priority.
if test (

This is unambiguous, and since the comma now acts as a statement separator, line breaks can be normal whitespace. Alternatively, you could define line breaks to be equivalent to commas and indented blocks to be equivalent to parenthesized blocks.

diag = x y -> (x * x + y * y) ^ 0.5
if (diag 3 4 == 5) (print "hello", print "world")

We've reached a point where the syntax is definitely viable. The only real wart is nullary application (x. to apply x to no arguments). Depending on your semantics that may not be a big deal: in a pure language like Haskell, for instance, there is no such thing as nullary application.

Iteration 4

There is one way to eliminate the "wart" and make the syntax conceptually cleaner: reinterpret juxtaposition to be currying and add syntax for actual lists (that returns the list rather than the last element).

  1. (Amended) juxtaposition using whitespace is a left-associative operator: f x y applies f to x, then applies the result to y.
  1. [a, b, ...] defines a list.

Then we can make the definition of diag currying:

diag = x -> y -> (x * x + y * y) ^ 0.5
if (diag 3 4 == 5) (print "hello", print "world")

Or we can define it on a list of two arguments:

diag = [x, y] -> (x * x + y * y) ^ 0.5
if (diag[3, 4] == 5) (print "hello", print "world")

See how that works? In a similar vein, a function that takes no arguments would be called like f[].

At a glance it feels like we may have reinvented M-expressions, but we did not: a function taking a single argument would correspond to the M-expression f[x], but with our solution so far, it could be f[x], or it could also be f x. A function taking two arguments would be the M-expression f[x; y], but here we can have f[x, y] or f x y. Basically, there is no requirement for functions to take lists of arguments. This is why unlike m-expressions, this syntax can't be directly translated to s-expressions. For instance, what's the s-expression for f[x, y]? (f x y)? But then what's the s-expression for f(x[y])? (f x y) again?

That's also why I did not bother changing if: I mean, sure, we could define it like if[test, body, ...], but that would look terrible, and we don't have to do it. We can just think of it as a kind of currying macro where if test returns a macro that processes a body. Why not? (Okay, there is a good reason why not; I'll explain later).


Ideally, operators should be translated in terms of juxtaposition, since they are function or macro calls. a <op> b could either be (<op>) a b or (<op>)[a, b]. The former is handy for things like partial application, whereas the latter is more consistent with the idea that operators may take a variable number number of arguments. It depends on the semantics you are shooting for.

Either way we can write a generic draft for our syntax:

  1. Classification: given a set of "operator characters", tokens made from characters of that set are operators and others are identifiers or literals.
  2. Juxtaposition: left-associative operator. f x applies the function f to the value x. It produces the AST node Apply[f, x].
  3. Operators: a <op> b is sugar for (<op>) a b, or (<op>)[a, b].
  4. Lists: [a, b, ...] creates the AST node List[a, b, ...].
  5. Sequencing: a, b, ... creates the AST node Seq[a, b, ...].
  6. Grouping: (...) ensures that the enclosed expression corresponds to a single node in the AST. However, it does not show up in the AST.
  7. Priority: from higher to lower priority, we have juxt -> op -> comma -> brackets. For priority relations between operators, a priority graph is provided (or not).

I dub these expressions o-expressions (operator expressions).

With only three internal node types, the AST is rather simple. We could actually make it even simpler by removing Seq nodes and using List nodes instead (I think Seq is worth it, though: Seq[expr] being semantically equivalent to expr in all situations makes it a better candidate for the body of a function or control structure, whereas using List would make it awkward to return lists).

Is this AST more complex than s-expressions? I'm not sure. Apply[f, List[x, y]] is the equivalent of the s-expression (f x y), but s-expressions are rigged in such a way that every List node is always the second argument of some Apply node. If that's the case, then you can just reduce both to a single datatype, but it's a bit disingenuous to claim that this makes things simpler: the point of having two types of nodes is that you can decouple them.

For instance, decoupling means that you don't need an apply function any more. Apply[f, Apply[g, List[x]]] has no equivalent s-expression because erasing node type means that you cannot hope to distinguish it from Apply[f, List[g, x]]. To get the same semantics you must write (apply f (g x)). That's not very elegant.

Solving technical issues

There is still one outstanding issue: AST manipulation.

while (x < 0) (x += 1)

That looks nice enough and it will work. Unfortunately, juxtaposition is left-associative, which means we get the following AST:

Apply[Apply[while, (x < 0)], (x += 1)]

Surely you can see the problem: the "important token", while, is not accessible. You have to dig for it. The idea of currying macros might not be entirely devoid of sense, but as far as AST manipulation goes, this is a big no-no.

Unfortunately, while[test, x < 0, x += 1] looks terrible. So what do we do?

One option would be to use an operator as syntactic sugar. For instance:

while (x < 0): (x += 1)  ==>  while[x < 0, x += 1]

Every bit of sugar we add is a disadvantage versus s-expressions, so this is not thrilling, but nothing's perfect.

Aggregative operators

Here's another idea, though.

Imagine that instead of figuring out operator priority between random operators, we simply "aggregated" them together, concatenating all their names:

x <op1> y <op2> z    ==>    (<op1>_<op2>)[x, y, z]

Not only is this conceptually simple, it opens up a massive range of ternary, quaternary and what-have-you operators. Suppose that we define a set of operator tokens that contain identifiers, so that they don't look like confusing garbage characters. For instance, perhaps we could interpret @word as an operator. If we do this we can get interesting mileage out of existing rules:

@while (x < 0): (x += 1)        ==> (@while_:)[x < 0, x += 1]

The potential for extensibility is massive.

@if (x < 0) @then "-" @else "+" ==> (@if_@then_@else)[x < 0, "-", "+"]
@for x <- xs: print[x]          ==> (@for_<-_:)[x, xs, print[x]]

The method has several advantages:

It might not be a good idea to aggregate all operators, since this would lead to loads of spurious definitions for all sorts of common combinations. For instance, I probably wouldn't actually aggregate @for and <- in the previous example. Once these kinks are worked out, though, I think the idea has a lot of potential.


I haven't finalized a syntax in this essay but I think I've expounded the aspects and properties that it must have in order to be truly extensible. To summarize:

As I mentioned, it's not really possible to do better than s-expressions on any of these points. Only data formats like XML or JSON could be said to have similar homoiconic properties, but they are completely unusable to define programs. Much like Lisp requires a sacrifice in the form of parentheses, its alternatives require a sacrifice in how directly the structure of source code can be visualized.

O-expressions attempt to reduce the amount of punctuation by leveraging operators and introducing a distinction between application nodes and list nodes. This adds what I consider to be a small amount of complexity. Which tradeoff is "better" is a subjective matter, but I believe that most people would prefer working with o-expressions.

There are a few advantages of o-expressions that I think are difficult to argue:

I have written an implementation of o-expressions for Racket. That implementation translates o-expressions into s-expressions in such a way to keep maximal compatibility. The resulting syntax is a bit more complex than pure o-expressions would be if they were the basis of a new language, but that's always to be expected when shoehorning a system into another. You can read about it here.

Olivier Breuleux January 16, 2014