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---
title: "How Many Values Does a Boolean Have?"
date: 2020-08-20T18:37:50-07:00
draft: ["Java", "Haskell"]
---
A friend of mine recently had an interview for a software
engineering position. They later recounted to me the content
of the techical questions that they had been asked. Some had
been pretty standard:
* __"What's the difference between concurrency
and parallelism?"__ -- a reasonable question given that Go was
the company's language of choice.
* __"What's the difference between a method and a function?"__ --
a little more strange, in my opinion, since the difference
is of little _practical_ use.
But then, they recounted a rather interesting question:
> How many values does a bool have?
Innocous at first, isn't it? Probably a bit simpler, in fact,
than the questions about methods and functions, concurrency
and parallelism. It's plausible that a programmer
has not done much concurrent or parallel programming in their
life, or that they came from a language in which functions
were rare and methods were ubiquitous. It's not plausible,
on the other hand, that a candidate applying to a software
engineering position has not encountered booleans.
If you're genuinely unsure about the answer to the question,
I think there's no reason for me to mess with you. The
simple answer to the question -- as far as I know -- is that a boolean
has two values. They are `true` and `false` in Java, or `True` and `False`
in Haskell, and `1` and `0` in C. A boolean value is either true or false.
So, what's there to think about? There are a few things, _ackshually_.
Let's explore them, starting from the theoretical perspective.
### What's a Type, Anyway?
Boolean, or `bool`, is a type. Broadly speaking, a type
is a property of _something_ that defines what the _something_
means and what you can do with it. That _something_ can be
several things; for our purposes, it can either be an
_expression_ in a programming language (in the form of `fact(n)`)
or a value in that same programming langauge (like `5`).
Dealing with values is rather simple. Most languages have finite numbers,
usually with \\(2^{32}\\) values, which have type `int`,
`i32`, or something in a similar vein. Most languages also have
strings, of which there are as many as you have memory to contain,
and which have the type `string`, `String`, or occasianlly
the more confusing `char*`. Most languages also have booleans,
as we discussed above.
The deal with expressions is a more interesting. Presumably
expressions evaluate to values, and the type of an expression
is then the type of values it can yield. Consider the following
snippet in C++:
```C
int square(int x) {
return x * x;
}
```
Here, the expression `x` is known to have type `int` from
the type signature provided by the user. Multiplication
of integers yields an integer, and so the type of `x*x` is also
of type `int`. Since `square(x)` returns `x*x`, it is also
of type `int`. So far, so good.
Okay, how about this:
```C++
int meaningOfLife() {
return meaningOfLife();
}
```
No, wait, doesn't say "stack overflow" just yet. That's no fun.
And anyway, this is technically a tail call, so maybe our
C++ compiler can avoid growing the stack And indeed,
flicking on the `-O2` flag in this [compiler explorer example](https://godbolt.org/z/9cv4nY),
we can see that no stack growth is necessary: it's just
an infinite loop. But `meaningOfLife` will never return a value. One could say
this computation _diverges_.
Well, if it diverges, just throw the expression out of the window! That's
no `int`! We only want _real_ `int`s!
And here, we can do that. But what about the following:
```C++
inf_int collatz(inf_int x) {
if(x == 1) return 1;
if(x % 2 == 0) return collatz(x/2);
return collatz(x * 3 + 1);
}
```
Notice that I've used the fictitious type
`inf_int` to represent integers that can hold
arbitrarily large integer values, not just the 32-bit ones.
That is important for this example, and I'll explain why shortly.
The code in the example is a simulation of the process described
in the [Collatz conjecture](https://en.wikipedia.org/wiki/Collatz_conjecture).
Given an input number `x`, if the number is even, it's divided in half,
and the process continues with the halved number. If, on the other
hand, the number is odd, it's multiplied by 3, 1 is added to it,
and the process continues with _that_ number. The only way for the
process to terminate is for the computation to reach the value 1.
Why does this matter? Because as of right now, __nobody knows__
whether or not the process terminates for all possible input numbers.
We have a strong hunch that it does; we've checked a __lot__
of numbers and found that the process terminates for them.
This is why 32-bit integers are not truly sufficient for this example;
we know empirically that the function will terminate for them.
But why does _this_ matter? Well, it matters because we don't know
whether or not this function will diverge, and thus, we can't
'throw it out of the window' like we wanted to with `meaningOfLife`!
In general, it's _impossible to tell_ whether or not a program will
terminate; that is the [halting prorblem](https://en.wikipedia.org/wiki/Halting_problem).
So, what do we do?
It turns out to be convenient -- formally -- to treat the result of a diverging computation
as its own value. This value is usually called 'bottom', and written as \\(\\bot\\).
Since in most programming languages, you can write a nonterminating expression or
function of any type, this 'bottom' is included in _all_ types. So in fact, the
set of possible values for `unsigned int`: \\(\\bot, 0, 1, 2, ...\\) and so on.
As you may have by now guessed, the same is true for a boolean: we have \\(\\bot\\), `true`, and `false`.
### Haskell and Bottom
You may be thinking:
> Now he's done it; he's gone off the deep end with all that programming language
theory. Tell me, Daniel, where the heck have you ever encountered \\(\\bot\\) in
code? This question was for a software engineering interview, after all!
You're right; I haven't _specifically_ seen the symbol \\(\\bot\\) in my time
programming. But I have frequently used an equivalent notation for the same idea:
`undefined`. In fact, here's a possible definition of `undefined` in Haskell:
```
undefined = undefined
```
Just like `meaningOfLife`, this is a divergent computation! What's more is that
the type of this computation is, in Haskell, `a`. More explicitly -- and retreating
to more mathematical notation -- we can write this type as: \\(\\forall \\alpha . \\alpha\\).
That is, for any type \\(\\alpha\\), `undefined` has that type! This means
`undefined` can take on _any_ type, and so, we can write:
```Haskell
myTrue :: Bool
myTrue = True
myFalse :: Bool
myFalse = False
myBool :: Bool
myBool = undefined
```
In Haskell, this is quite useful. For instance, if one's in the middle
of writing a complicated function, and wants to check their work so far,
they can put 'undefined' for the part of the function they haven't written.
They can then compile their program; the typechecker will find any mistakes
they've made so far, but, since the type of `undefined` can be _anything_,
that part of the program will be accepted without second thought.
The language `Idris` extends this practice with the idea of typed holes,
where you can leave fragments of your program unwritten, and ask the
compiler what kind of _thing_ you need to write to fill that hole.
### Java and `null`
Now you may be thinking:
> This whole deal with Haskell's `undefined` is beside the point; it doesn't
really count as a value, since it's just a nonterminating
expression. What you're doing is a kind of academic autofellatio.
Alright, I can accept this criticism. Perhaps just calling a nonterminating
function a value _is_ far-fetched (even though denotational semantics
_do_ extend types with \\(\\bot\\)). But denotational semantics is not
the only place where types are implcitily extend with an extra value;
let's look at Java.
In Java, we have `null`. At the
core language level, any function or method that accepts a class can also take `null`;
if `null` is not to that function or method's liking, it has to
explicitly check for it using `if(x == null)`.
Java's booleans are not, at first glance, classes. Unlike classes, which you have
to allocate using `new`, you can just throw around `true` and `false` as you see
fit. Also unlike classes, you can't assign `null` to a boolean value.
The trouble is, the _generics_ part of Java, which allows you to write
polymorphic functions, can't handle 'primitives' like `bool`. If you want to have an `ArrayList`
of something, that something _must_ be a class.
But what if you really _do_ want an `ArrayList` of booleans? Java solves this problem by introducing
'boxed' booleans: they're primitives wrapped in a class, called `Boolean`. This class
can then be used for generics.
But see, this is where `null` has snuck in again. By allowing `Boolean` to be a class
(thereby granting it access to generics), we've also given it the ability to be null.
This example is made especially compelling because Java supports something
they call [autoboxing](https://docs.oracle.com/javase/tutorial/java/data/autoboxing.html):
you can directly assign a primitive to a variable of the corresponding boxed type.
Consider the example:
```Java
Boolean myTrue = true;
Boolean myFalse = false;
Boolean myBool = null;
```