JMCT

1a: Basic Notions and Notations (Expressions)

Jan 5, 2017

Preface

Welcome to part 2 of “The Burge School of Functional Programming”. This post is where we really pinpoint what an expression is. If I’m going to convince you that functional programing is about expressions, we should be really clear what we mean by that!

I want to emphasize that the goal here is not to parrot or rephrase the contents of the book, but instead to highlight the lessons that help us think like functional programmers. As such, don’t expect to come out of this having learned a language. There are plenty of great resources for learning functional languages; we aren’t aiming for that here. If you have a language you know you want to learn, and you don’t know where to start, just reach out to me on Twitter and I’ll send you some recommendations.

Throughout this series, I will use the notation that Burge used. However, I aim to point out when those notations differ from modern conventions. If I’ve missed a spot, or a notation is confusing, please let me know and I’ll clarify as quickly as possible.

I’m going to quote directly from the book a lot early on in this post, as I feel there’s a lot of insight to be found at the beginning of Burge’s first chapter. In future posts we’ll be able to leverage the vocabulary we get from Burge early on and we won’t need to quote him directly so much.

Similarly, this is likely to be the longest post in the series, so please bear with me!

Introduction

Burge hits the ground running with this chapter, making it clear that the style of programming this book advocates is a bit different from what the reader may be used to. On the very first page he states:

All the linguistic devices introduced [in this book] are based upon two methods of constructing expressions from smaller expressions […] Thus the extra notation that is added to this basis adds no new structural features

He names the extra notation additions. These days we would call it syntactic sugar, but the idea is identical: New forms of expression that can be translated to a few core constructs. Let that sink in: all of the language constructs that Burge describes in his book can be rewritten with just the two following constructs:

  1. “An operator/operand construction that denotes function application”
  2. “An expression format which denotes a function”

Burge does remind the reader that there will be some ‘constants’ (we’d call them primitives) required for certain tasks. This sets up one of the first great insights of the text, stating that the constructs

create a practical and powerful programming system, which is more like a family of programming languages than a single language, because the features introduced are concerned more with combining functions to produce new ones than with the nature of the primitive functions that are being combined.

Here comes the kicker:

A programming language for a particular range of applications can be obtained by adding an appropriate set of primitives to this basic structure.

These days we call a “langguage for a particular range of applications” a Domain Specific Language (DSL). DSLs are a very powerful tool in the functional programmer’s toolbox. So in just under a page Burge has set the stage for understanding functional programming through just a few core constructs and its usefulness in creating DSLs but avoids using any jargon. So Burge’s lesson so far: understand two core constructs, then pick primitives according to your domain.

We haven’t gotten much further in the decades since.

He’s not done yet, there’s one more insight waiting for us, right before he starts introducing the core constructs he reminds us that the focus is on expressions and not mechanisms. He argues that expressions have a great property:

the value, or meaning, of an expression depends in a simple way only on the values of its subexpressions and on no other properties of them.

So again, without leaning on fancy language, Burge lays it all out in front of us. He states that this property allows you solve large and complex problems by breaking them down into small, simple, and independent problems. And

it is possible to make the structure of the program match the structure of the problem being solved.

The Language

Operator/Operand Expressions

Let’s start with the obvious: Functional Programming deals with functions. This makes it critical to specify what we mean when we say function. Burge does this by talking about the relationship a function has to its arguments. More specifically, he talks about types, starting with function types.

But remember, this is a book from decades ago, so we’re not bringing in any heavy machinery here. Note that Burge never talks about type-checking or even implementing a type-system. Even if your compiler doesn’t do type checking, you can still benefit from thinking about types.

Functions and types

I’ve been going on about expressions over mechanisms for a while and I haven’t even shown you an expression yet. So let’s take a look at one: \(f(x)\). This is how Burge typesets the application of the function \(f\) to the value \(x\). This is the first expression we’ve seen, so let’s unpack it. \(f(x)\) is a function application, which has two parts; an operator (in this case \(f\)), and an operand (in this case \(x\)). This makes function application a type of compound expression.

In the introduction I claimed that all of the language constructs we will introduce can be understood with just two concepts; function application is the first one! The basic idea of looking at application as a compound expression is that in order to make sense of the expression \(f(x)\) you’re going to have to make sense of \(f\) and of \(x\). That may seem obvious, but the insight Burge is trying to get across to you is that when you program in this expression-based style you make sense of each in isolation. That’s a very powerful idea!

Okay, so a function is a value that you can apply to other values (a.k.a. the inputs). However, not all values make sense as inputs; for example, the function \(square\) that takes a number and multiplies it by itself, does not make much sense if you give it the letter ‘a’ as an input. Let’s be a bit more concrete and say functions have a type, usually written as

\[A \longrightarrow B\]

Here the \((\rightarrow)\) is what indicates this is a function. The \(A\) is the types of values it accepts, and \(B\) is the type of values it returns. Burge calls these the domain and range, respectively.1 So if you have a function \(f\) of type \(A \rightarrow B\), and you apply it to a value \(x\) which has type \(A\), the result is of type \(B\). Burge typesets function application as both \(f(x)\) or \(f\ x\).

Earlier I mentioned the function \(square\). One possible type for \(square\) would be2

\[square \in (\text{integer} \rightarrow \text{integer})\]

Other common examples:

\[ sin \in (\text{real} \rightarrow \text{real}) \\ log \in (\text{positive} \rightarrow \text{real}) \\ negate \in (\text{positive} \rightarrow \text{negative}) \]

So in general anything of the form \(g \in (A \rightarrow B)\) is an assertion that \(g\) is a function that takes arguments of type \(A\) and returns values of type \(B\). This means if we have a value \(x\) of type \(A\), we can be assured that \(g(x) \in B\).

Okay, this is all well and good, but every function we’ve looked at takes only a single argument, what about things like \(+\)? Here are some examples:

\[ + \in (\text{real} \times \text{real} \rightarrow \text{real})\\ min \in (\text{real} \times \text{real} \rightarrow \text{real})\\ equal \in (\text{real} \times \text{real} \rightarrow \text{truth value}) \]

This is just saying that these functions take two arguments, both reals, and return a single value. This generalizes in the obvious way to functions with an arbitrary number of arguments. So a function that takes \(N\) arguments would have a type like

\[A_{1} \times A_{2} \times \dots \times A_{N} \rightarrow B\]

Where \(A_{i}\) is the appropriate type of the \(i^{th}\) argument.

Now, the traditional way to apply functions that take multiple arguments is by extending the syntax we already have, giving us \(min(x,y)\) or \(+(x,y)\). Many languages have a predetermined set of special functions that can be applied differently, the arithmetic operations are usually such an exception, so you could write \(x + y\) instead of \(+(x,y)\). Burge is no different here and his language allows for certain functions to be applied in this special manner.

It’s worth pointing out that even if you have a special syntax for the application of certain functions, the operator/operand relationship is unchanged. It could be argued that allowing any such special syntax obscures this relationship and therefore obscures the meaning of the expression3. The important bit is that regardless of the syntax it is crucial that you be able to identify the operator and the operand of a function application.

Quick aside about types

This is pretty much the extent of what we’ll say about types, possibly for the whole series. Though types are used to describe things (as we’ll see in the next section), Burge never defines a type system, or any form of static enforcement of types. So why mention types at all? Because it’s important to think about types when you’re doing functional programming, particularly to distinguish between things that you can apply (functions), what they expect as argument values, etc.; and things you can’t apply (the number \(5\), for instance).

Many of the experts of dynamic languages I’ve interacted with will be the first to tell you: thinking about types is important when writing programs. But we don’t have to get fancy with our type system in order to have any benefit from the concept of types.

Meaning of expressions

We now know what makes up a function application, but we still don’t know the meaning of anything. I claimed earlier that you can determine the meaning of a function application by finding the meaning of its operator and operand. But eventually you’ll reach an expression that is not a function application, Burge calls these simple expressions. In this section we will explain the meaning of one type of simple expression: constants (variables are the other type of simple expression, which we’ll get to a bit later).

In any programming system there is a set of constants (what we would call pprimitives). The meaning of these constants is given by the system and its implementation. The most obvious constants are things like numbers: \(4\), \(-128\), etc.4 or arithmetic operations: \(+\), \(-\), \(\times\), etc.

So if your language has the constant \(+\) which takes two numbers and adds them together, and it has numeric constants we can now find the meaning of expressions like \(+(4,5)\), or \(square(\times(2,3))\).

We do this with the following procedure:

  1. If the expression is simple, what is the meaning of the constant?
  2. If the expression is composite, what is the meaning of its operator, and what is the meaning of its operand?

Try applying this procedure to \(+(4,5)\), or \(square(\times(2,3))\). Do yourself a favor and force yourself to actually go through the steps. In this instance it is not very hard, but it is the practice of finding the meaning of expressions via the meaning of their constituent parts that is important.

For the rest of the post we’ll allow ourselves to use the arithmetic operators as infix, i.e. \(4 + 5\).

Functions are values too

Burge uses two conventions when writing function application. The first is what we’ve seen up to this point: \(f(x)\) or \(f(x,y)\). This is very common and is likely to be what most languages you’re familiar with use. The second convention Burge uses is more sparse: \(f\ x\) or \(f\ x\ y\), respectively. Burge explains that these aren’t really different syntaxes for function application, but really represent functions of different types!

It’s clearer with two argument functions. Take the functions \(addT\) and \(addC\)5, both of which add two numbers together. These are their types:

\[ addT \in (\text{real} \times \text{real} \rightarrow \text{real})\\ addC \in (\text{real} \rightarrow (\text{real} \rightarrow \text{real})) \]

The types tell us that \(addT\) takes a pair of real numbers and returns a pair, while \(addC\) takes a single real number and returns a function of type \((\text{real} \rightarrow \text{real})\). Therefore, when we write \(addC\ x\ y\), we’re really writing \((addC\ x)\ y\), i.e. we’re applying the function \((addC\ x)\) to \(y\). We can omit the brackets because function application associates the left. Similarly the function type \((\rightarrow)\) associates to the right, so \((A \rightarrow (B \rightarrow C))\) is the same as \((A \rightarrow B \rightarrow C)\).

To really hit the lesson home, convince yourself that the following are all equivalent for a function, \(f\), that takes three arguments

\[ ((f\ x)\ y) \ z\\ (f\ x\ y) \ z\\ f\ x\ y \ (z) \]

and understand why \(f\ x\ (y\ z)\) is not equivalent to the three above6. Here’s a hint: think of what the operator/operand relationships are for every function application.

For extra credit, determine the operator/operand (function/argument) parts for each version, including \(f\ x\ (y\ z)\).

This is not just some clever rationalization of multi-argument functions, this is a direct consequence of functions themselves being values. Because they are values just like anything else we can also pass functions to other functions, take the following example:

\[ twice\ f\ x = f\ (f\ x) \]

\(twice\) is a function that takes two arguments, a function and some other value, and then applies the function ‘twice’ to the value. So if we had a function \(add\text{-}one\) which adds 1 to a number and called \(twice\ add\text{-}one\ 5\) we would get 7.

Treating functions as ‘first-class’ values has become quite pervasive these days, and for good reason! First-class functions allow you separate the form of a computation from the task being accomplished. This is one of the more powerful ideas from functional programming, and as such, Burge will explore it in depth later on.

Variables and Lambdas

Up to now we’ve used variables without discussing them, because many of us will have an intuition for what variables mean from other languages or from algebra in school. In this section we’ll be more precise about what a variable means.

Take the following mathematical equation:

\[ f\ x = (5 \times x) + 2 \]

This is a function over the variable \(x\). When you plug in different values for \(x\) you get different results.

There are two very important properties about variables like the ones in the equation above. The first is that it doesn’t matter what name we give \(x\): \(f\ y = 5y +2\) is the exact same equation with the exact same meaning. The second is that in mathematics we don’t actually change the values of a variable: Once we’ve plugged in a value for \(x\) that value remains the same.

To better formalise how variables work we’ll use a notation developed by Alonzo Church. The following defines the same function using lambda (\(\lambda\)) notation:

\[ \lambda x.(5 \times x) + 2 \]

We haven’t given the lambda expression a name, which is why in some languages they call lambdas ‘anonymous functions’. There’s nothing stopping us from giving the above a name though

\[ f = \lambda x.(5 \times x) + 2 \]

defines the same function again, this time giving it the name \(f\). Here is the crucial point: \(f\ x = (5 \times x) + 2\) and \(f = \lambda x.(5 \times x) + 2\) are the same function.

Taking lambdas apart

In the same way that function application had two parts, the operator and the operand, lambda expressions also have two parts: the bound variable and the body. Everything between the lambda (\(\lambda\)) and the period (\(.\)) is the bound variable and everything after the period is the body.

As functions you can apply values to lambda expressions, using them as the operator part of a function application. When you apply a value to a lambda expression you substitute that value wherever the bound variable appears in the body and get rid of the bound variable part of the lambda expression, leaving only the body. So \((\lambda x. x + x)\ 5\) becomes \(5 + 5\).

We can happily pass lambda expressions to other functions as well (since they are values themselves). Remembering the definition of \(twice\) from earlier, the expression

\[ twice\ (\lambda x. x + 1)\ 5 \]

is equal to \(7\).

What about multi-argument functions? Well we already learned that multi-argument functions are just single argument functions that return functions, so let’s apply that idea here and define \(add\) with lambdas:

\[ add = \lambda x.\lambda y. x + y \]

This actually makes what’s happening when we partially apply a function more clear. If we only pass \(add\) one argument what do we get? \((\lambda x.\lambda y. x + y)\ 1\) becomes \(\lambda y. 1 + y\), or the function that adds \(1\) to its argument.

Expressions

So we’ve spent many inches going on about function application, lambda expressions, and simple expressions (constants and variables). What do we know now? Well we know all the forms an expression can take. To paraphrase Burge:

An expression is

  • either simple and is a identifier
  • or a lambda expression
    • and has a bound variable which is a variable
    • and a body which is an expression
  • or it is a function application
    • and has an operator and an operand, both of which are expressions

In the next post we’ll show you the language Burge describes in full, but it’s important to emphasize: Every language construct we will show you can be translated to expressions consisting only of what’s described above. A whole family of languages arises from the three possibilities above. As promised, we have two ways of combining expressions: lambda expressions and function application. The only other piece is that we need a way to refer to things, which is where the identifiers (variables or constants) come in..

Conclusion

If you’ve made it this far, thanks for your patience! The rest of the series is going to be a pretty crazy ride, and it all comes back to what we’ve learned here. The core of Burge’s book, and Functional Programming more generally, is in what is possible with expressions.

Buckle up.

Epilogue

Josh Triplett, Kelley Robinson, and Michael Banks all gave constructive feedback on drafts of this post. Thank them if any of this made any sense.

As promised, we’ve got another Prog Rock hit from the early 70’s, this one’s about a show “you’ve got to see”, a.k.a. how all of the language constructs we are used to can be made up of just expressions ;)

-JMCT


  1. Note that in modern programming language texts and papers range would almost universally be called the codomain. The reasoning is simple: The range of a function is the set of values it actually returns, whereas the codomain is the set of possible return values. So when reasoning about types we’re actually reasoning about the codomain.

  2. Here the symbol \(\in\) can be pronounced as ‘has the type’. The use of this symbol, which is borrowing from Set Theory, has fallen out of favor since the 80’s. The reason for this is that types aren’t really sets, and so using that notation implied something that wasn’t always true. These days you’ll almost universally see the symbol ‘\(:\)’ used (e.g. \(\ f : A \rightarrow B\)), unless you’re reading something written with Haskell in mind, where it would be \(f :: A \rightarrow B\).

  3. The LISP, Racket, and Clojure communities might make this argument, for example.

  4. The constants are defined by the system in use and aren’t necessarily equivalent to their pure mathematical counterparts. For example adding two numbers in many languages introduces the possibility of overflow, whereas the ‘true’ \(+\) has no such possibility. Or how some languages automatically convert all numeric values to floating point, etc.

  5. For those that are curious about the naming, \(addT\) is the adding a tuple, and \(addC\) is the ‘curried’ addition.

  6. The difference is that in \(f\ x\ (y\ z)\) only two arguments have been passed to \(f\): \(x\) and \((y\ z)\). The expression \((y\ z)\) is itself a function application with \(y\) as the operator and \(z\) as the operand.