### Add draft of boolean values post.

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 @ -0,0 +1,220 @@ --- 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_ ints!   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;