blog-static/content/blog/boolean_values.md

title	date	draft
How Many Values Does a Boolean Have?	2020-08-20T18:37:50-07:00	Java Haskell

A friend of mine recently had an interview for a software engineering position. They later recounted to me the content of the technical 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?

Innocuous 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 candidate 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 language (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 occasionally 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++:

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:

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, 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 ints!

And here, we can do that. But what about the following:

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. 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 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:

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 implicitly 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: you can directly assign a primitive to a variable of the corresponding boxed type. Consider the example:

Boolean myTrue = true;
Boolean myFalse = false;
Boolean myBool = null;

Beautiful, isn't it? And, unlike Haskell, where you can't really check if your Bool is undefined (because you can't tell whether a non-terminating computation is as such), you can very easily check if your Boolean is true, false, or null:

assert myTrue != myFalse;
assert myFalse != myBool;
assert myTrue != myBool;

We're okay to use != here, instead of equals, because it so happens each boxed instance of a boolean value refers to the same Boolean object. In fact, this means that a Boolean variable can have exactly 3 values!

C and Integers

Oh the luxury of having a type representing booleans in your language! It's almost overly indulgent compared to the spartan minimalism of C. In C, boolean conditions are represented as numbers. You can perhaps get away with throwing around char or short int, but even then, these types allow far more values than two!

unsigned char test = 255;
while(test) test -= 1;

This loop will run 255 times, thereby demonstrating that C has at least 255 values that can be used to represent the boolean true. There are other languages with the notion of 'truthy' and 'falsey' values. However, some of them differ from C in that they also extend this notion to apply to equality. In JavaScript:

console.assert(true == 1)
console.assert(false == 0)

Then, there are still exactly two distinct values modulo ==. No such luck in C, though! We have 256 values that fit in unsigned char, all of which are also distinct modulo ==. Our boolean variable can contain all of these values. And there is no respite to be found with enums, either. We could try define:

enum bool { TRUE, FALSE };

But unfortunately, all this does is define bool to be a numeric type that can hold at least 2 distinct values, and define numeric constants TRUE and FALSE. So in fact, you can still write the following code:

enum bool b1 = TRUE;
enum bool b2 = FALSE;
enum bool b3 = 15;

And so, no matter how hard you try, your 'boolean' variable can have many, many values!

Conclusion

I think that 'how many values does a boolean have' is a strange question. Its purpose can be one of two things:

• The interviewer expected a long-form response such as this one. This is a weird expectation for a software engineering candidate - how does knowing about \(\bot\), undefined, or null help in creating software, especially if this information is irrelevant to the company's language of choice?
• The interviewer expected the simple answer. In that case, my previous observation applies: what software engineering candidate has not seen a boolean in their time programming? Surely candidates are better screened before they are offered an interview?

Despite the question's weirdness, I think that the resulting investigation of the matter -- outside of the interview setting -- is useful, and perhaps, in a way, enlightening. It may help one understand the design choices made in their language of choice, and how those choices shape the code that they write.

That's all I have! I hope that you found it interesting.