Equality Is Hard

Posted on March 9, 2020

As the joke goes, there are two hard problems in computer science: cache invalidation and naming things.1 But I’d suggest there’s a much harder problem, namely, =. Did you miss it? The equals sign, =, is small, but I’m going to argue that the use and misuse of equals is at the root of a large number of problems in software engineering.

How Equality Should Work

I am going to show how equality in programming languages is often broken. But before I can do that, I have to talk about how it should work, and it turns out that’s not simple! When we talk about how equality “should work,” we have to say what this means in a certain context, because it turns out there are lots of different ways that equality can work, and many of them are valid in different contexts.

The heart and soul of mathematics consists of the fact that the “same” objects can be presented to us in different ways.
-Barry Mazur, When is one thing equal to some other thing


Now I said that we can have different definitions of equality in different contexts, but despite this there are some things which should always be true. These are the laws of equality.

Equals is a binary relation2 that is:

In the programming world, we need to add a law, because programmers do weird things sometimes:

Equals must be:

The above seems simple enough, although popular programming languages manage to screw up even those trivial rules. But there are more concerns about equality which are harder to state quite so concisely.

Structural Equality

One difference in how programming languages implement equality is structural equality and reference equality.

Structural equality asks if two references are the same value. This is the default in F#:

type MyString = { SomeField : string }
let  a = { SomeField = "Some value" }
let  b = { SomeField = "Some value" }
if a = b then // returns true, enters "then" block

This is not true in C#; C# uses reference equality. Reference equality asks if the two objects being compared are the same object. In other words, does the variable point at the same area of memory? A reference to two different blocks of memory will be unequal even if their contents are identical:

class MyString {
    private readonly string someField;
    public string SomeField { get; }
    public MyString(string someField) => this.someField = someField;
var a = new MyString("Some value");
var b = new MyString("Some value");
if (a == b) { // returns false, does not enter block

Other languages let you choose. Scheme, for example, provides equal? to check structural equality and eq? to check reference equality. Kotlin provides == for structural equailty and === for reference equality (don’t confuse these with JavaScript’s == and === operators which are… something else entirely).

When does it make sense to use structural equality in your programs? In the absence of mutation (changing the values of variables), nearly always! Most programming languages that I’m aware of do structural comparisons on value types such as integers. Well, except Java, which has confused generations of programmers with an int value type which does a structural comparison and an Integer reference type which, well, the best thing you can say is don’t use == on Integer. Python has similar issues with is.

Structural comparison of reference types such as objects makes sense as well. Consider a unit test, where you want to check that the object returned is equal to the value you expected. In a language with structural equality, this is trivial:

let ``The result of the calculation is the expected value``() = 
    let expected = { SomeField = "Some value"; SomeOtherField = 15; StillAnotherField = true; ... }
    let actual = calculate()
    Assert.AreEqual(expected, actual)

When a language does not have structural equality from the outset, developers will try to build it ad hoc, and you end up with this horror show, which is now permanently part of the NUnit framework.

Reference Equality

But as I hinted above, there are certainly cases where structural equality does not make sense. One example is with languages which support mutation of variables, which is most of them.3 When you can change the value of a variable, it probably does not make sense to say that variable is equal to some other variable, in general. Sure, you can say they’re (structurally) equal as of a moment in time, such as in last line of a unit test, but you can’t generally imply that they’re the same. This is a kind of subtle point, so let’s look at an example.

Let’s say I have an object which represents a person. In F#, with structural equality, I can write:

type Person = { Name : string; Age : integer; Offspring : Person list }

Now I have two friends, Jane and Sue. Both have a son named John, who is 15. They’re different people, but the sons have the same name and age. No problem!

let jane = { Name = "Jane"; Age = 47; Offspring = [ { Name = "John"; Age = 15; Offspring = [] } ] }
let sue  = { Name = "Sue";  Age = 35; Offspring = [ { Name = "John"; Age = 15; Offspring = [] } ] }

I could also have written this:

let john = { Name = "John"; Age = 15; Offspring = [] };
let jane = { Name = "Jane"; Age = 47; Offspring = [ john ] }
let sue  = { Name = "Sue";  Age = 35; Offspring = [ john ] }

The behavior of these two blocks is precisely the same. I can’t distinguish the two sons, even though I know they’re different people. That’s OK! If I needed to distinguish them, I could add a hash of their DNA or something to my Person type. But if I just need to know their name and age, it doesn’t matter if I can distinguish the two objects or not, because the values are the same, no matter how you slice it.

Imagine Jane’s son changes his name to Pat. F# doesn’t support mutating the values of variables, so I need to make a new Person instance for John and Jane:

let newJane = { Name = "Jane"; Age = 47; Offspring = [ { Name = "Pat"; Age = 15; Offspring = [] } ] }

It seems weird to have a new variable, newJane, but in practice it doesn’t create a problem. The code above is fine. Now let’s try this in C#, a language which is mutable by default:

var john = new Person("John", 15, null);
var jane = new Person("Jane", 15, new List<Person> { john });
var sue  = new Person("Sue",  15, new List<Person> { john });

Well, this code is clearly incorrect: If Jane’s son changes his name to “Pat”, I can change the reference directly:

jane.Offspring.First().Name = "Pat";

But I’ll find that Sue’s son’s name has changed as well! Therefore, even though the two sons had the same values at the start, before he changed his name, they were not equal! I should have written:

var jane = new Person("Jane", 15, new List<Person> { new Person("John", 15, null) });
var sue  = new Person("Sue",  15, new List<Person> { new Person("John", 15, null) });

…so that Jane and Sue’s offspring were reference unequal to each other. So reference equality is a sensible default in a language which supports mutation.

Another case where reference equality makes sense is when you know it’s going to give the same result as structural equality anyway. There is a certain performance overhead for testing structural equality, which is reasonable if you actually need to test structural equality. But if, for example, you create a large number of objects which you know are all different structually, it doesn’t make sense to pay the overhead of testing structural equality when you know that testing reference equality alone would give the same result.

Equivalent Representations

In the real numbers, .999… (repeating infinitely) equals 1. Note that the “real numbers” here are distinct from the “Real” type in your programming language. Real numbers in math are infinite, and real numbers in your programming language are finite. So there is no notion of .999… in your programming language, but that’s OK, because you can just use 1, which is the same value.

This is, essentially, a choice that mathematicians made when formulating the real number system. If one adds other objects, such as infinitessimals, to the system, then .999… and 1 are not equivalent.

However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.
-Timothy Gowers, Mathematics: A Very Short Introduction4

Similarly, in the rational numbers, 1/2 and 2/4 represent the same value.

Do not confuse these equivalances with the “loose” equivalence operator == found in JavaScript and PHP. Unlike those operators, these equivalences follow the laws of equality. It is important to realize that equal objects can be represented differently.

In IEEE-754 floats, -0 = 0.

Intensional vs. Extensional Equality

When is some function equal to some other function? Most programming languages will happily do a reference = comparison, and I suppose that’s fine, but what would a structural equality comparison of a function even mean? Well, if we could use reflection to look into the implementation of the function, and see if it does the same thing? But what is “the same?” Would it have to have the same variable names? Are a quicksort and a merge sort “the same function?”

Cutting to the chase, we say that functions are extensionally equal if they return the same outputs for the same inputs (regardless of internal implementation), and intensionally equal if their internal definition is the same. Of course, this is context-dependent. There may be a context where I need a constant time function and another context where the speed of the function doesn’t matter. The important point is I need to have some context for equality and use it to compare the two functions.

I don’t know of any programming language which even attempts to do anything beyond reference equality for functions. But it’s easy to come up with examples where it would be useful! (An optimizer which removes duplicate code, e.g.) You’re on your own if you need this, but I have to say that not shipping an equals comparison is preferable to shipping one that’s broken.

Equality vs. Assignment

One of the first lessons we learn when becoming a programmer is that there are two different concepts which we both call “equals.” One is assignment, the other is testing equality. In JavaScript, these look like:

const aValue = someFunction(); // Assignment
if (aValue === 3) {            // Test for equality

These are fundamentally different. Comparison returns a boolean; assignment, in an expression-oriented language such as Ruby, returns the value assigned.

So we can write Ruby code like this:

a = b = c = 3

Which does indeed assign 3 to the variables a, b, and c. Don’t try it with a reference type, though; it probably won’t do what you want!5

In a non-expression-oriented language like C#, assignment doesn’t return anything.

In math, we use the equals operator for both assignment and testing equality:

if aValue = 3 ... 
where aValue = someFunction()

(And = is sometimes used for other relations in math, such as congruence. As with all things in math, context matters; you have to carefully consider what = might mean in a certain paper or book.)

Why does math not require two separate operators whereas programming languages do? You can tell from context which one is intended, and not all programming languages require different operators. F#, for example, uses = for both assignment and testing equality. Despite overloading =, assignment and testing equality are different operations.

let aValue = someFunction(); // Assignment
if aValue = 3 then           // Test for equality

The choice of syntax is partially due to heritage: F# is based on ML, which is based on math, and JavaScript syntax is based on Java -> C -> Algol -> FORTRAN.

FORTRAN had to compile on machines where distinguishing these two cases from code syntax would be genuinely challenging, so it made sense to have two different operators. Then C took this “feature” to a high art, allowing code like:

int aValue = someFunction(); // Assignment
if (aValue = 3) {            // Also assignment!

This code, for those without previous C experience, overwrites aValue with 3 and then, since the expression aValue = 3 is equal to 3, the if test returns TRUE and execution continues inside the if block. This is frequently an error, leading many C programmers to reverse the values inside an if block out of habit to avoid making the mistake:

int aValue = someFunction(); // Assignment
if (3 == aValue) {           // Test for equality

// [...]

if (3 = aValue) {            // Syntax error: Cannot assign aValue to 3.

How Equality Should Not Work

I hope I’ve shown by now that equality is not simple, and that the “correct” implementation of equality can vary depending upon context. Despite that, programming languages often get the simple parts wrong! Very often, this is caused by the combination of equality with other language features, such as implicit type conversion.

Common Mistake: Equality Isn’t Reflexive

Recall that the reflexive law of equals requires all values to be equal to themselves, a = a.

In .NET, if you call Object.ReferenceEquals() on a value type, the arguments are separately boxed before the method runs, so it returns false even if you pass the same instance:

From the docs:

int int1 = 3;
Console.WriteLine(Object.ReferenceEquals(int1, int1)); // Prints False

This means it is not necessarily true that a = a in any .NET language, so the reflexive law is broken.

In SQL, NULL is not equal to itself, so the expression NULL = NULL (or, more probably, SOME_EXPRESSION = SOME_OTHER_EXPRESSION when both of them might be null) will return NULL, which is falsy. This leads to messes like:


Or, more likely, just a bug where the developer forgot about the special rules for NULL. If your DB server’s SQL dialect supports IS NOT DISTINCT FROM then this does what = should do. (Or should I say it does NOT not do what = should do?) Otherwise you’ll just have to live with SQL like the above. The best fix is to make your columns non-nullable when possible.

This is also true of IEEE-754 floats; the standard states that NaN != NaN. A different explanation than the one given in the link for this is that “NaN” represents some unspecified “non-number” result, not necessarily the same unspecified non-number result as that of a different calculation, so it’s incorrect to compare them. For example, square_root(-2) and infinity/infinity are both NaN, but they’re clearly not the same! Similar explanations are given for SQL’s NULL sometimes. One problem with this is that it makes the term very overloaded: Is NaN and NULL an unknown or imprecise value or the known absence of a value?

One way of handling such situations, which do not occur in routine floating point calculations, would be a union type. In F#, one could write:

type MaybeFloat = 
    | Float          of float
    | Imaginary      of real: float * imaginary: float
    | Indeterminate
    | /// ...    

… and then you could handle these terms appropriately in calculations which needed them. Use a signaling NaN to throw an exception in calculations which you don’t expect will have NaNs at all.

Rust offers the Eq and PartialEq traits. Not implementing Eq is supposed to be a signal that == is not reflexive, and floating point types in Rust do not implement it. But if you don’t implement Eq, you can still call == in your code. Implementing Eq allows your object to be used as a key in a hash map and possibly results in behavior changes in other places as well.

But there are even more significant problems with = and floats.

Common Mistake: Equals Is Too Precise

I guess many developers are familiar with the problem of comparing IEEE-754 floating point numbers, which are the “float” or “double” implementation for most programming languages. 10 * (0.1) does not equal 1, because “0.1” is actually equal to 0.100000001490116119384765625 or 0.1000000000000000055511151231257827021181583404541015625. If you’re not familar with this issue, you can go read about it, but the point is that it’s rarely safe to do an == comparison on a floating point number at all! You have to ask yourself which digits are significant and compare accordingly.

(Worse, the float type backs other types, such as TDateTime in some languages, so even in cases where equality comparisons might make sense, they don’t necessarily work.)

The correct method of comparing floating point numbers is to see if they’re “close,” and what “close” means varies depending on context. It’s not something you can cram into a == operator. If you find yourself doing this a lot (say, once), you might consider using a different data type, such as a fixed precision decimal number.

So why do programming languages offer == comparisons on a type when they can’t support it? Well, because they offer == on every type, it works on most of them, and they just shrug about the rest and chastize programmers for not knowing which language feature they should not use.

Not every programming language, mind you. Standard ML doesn’t offer = comparisons on reals. It’s a compiler error if you try!

The implementation notes state:

Deciding if real should be an equality type, and if so, what should equality mean, was also problematic. IEEE specifies that the sign of zeros be ignored in comparisons, and that equality evaluate to false if either argument is NaN. These constraints are disturbing to the SML programmer. The former implies that 0 = ~0 is true while r/0 = r/~06 is false. The latter implies such anomalies as r = r is false, or that, for a ref cell rr, we could have rr = rr but not have !rr = !rr. We accepted the unsigned comparison of zeros, but felt that the reflexive property of equality, structural equality, and the equivalence of <> and not o = ought to be preserved. Additional complications led to the decision to not have real be an equality type.

By blocking = for reals, SML forces the developer to think about what kind of comparison they actually need, which is a great feature, I think!

F# offers the [<NoEquality>] attribute to mark custom types where = should not be used. Pity they didn’t mark the float type with it!

Common Mistake: “Equals” Isn’t

PHP has two separate operators, == and ===. The documentation for ==, which is named “Equal,” states, “TRUE if $a is equal to $b after type juggling.” Unfortunately, this means that the == operator is unreliable:

  var_dump("608E-4234" == "272E-3063"); // true

Although we’re comparing strings here, PHP sees that they can be converted to a number, so it does. The numbers turn out to be very small (the first argument, for example, is 608 * 10-4234), and, as we’ve already discussed, comparing floating point numbers is hard. Casting both of these to a float(0) results in rounding them to equal values, so the comparison returns true.

Note this is different than the behavior of JavaScript, which also has similar (but not the same!) == and === operators; JavaScript would see that both sides are strings and return false for this comparison.

Fortunately, PHP has the === (“Identical”) operator, which behaves correctly in this case. I would say never use ==, but == does a structural comparison on objects, which might be something you want! Instead, I’ll say: Use extreme caution with ==, because it’s broken on primitive types.

Common Mistake: Equality Isn’t Symmetric

If you override .equals() in Java, it is your responsibility to ensure that the laws of equality hold!

It is very easy to implement a comparison which is not symmetric, that is, a.equals(b) != b.equals(a), if you’re not paying attention.

Even if we remove null from the picture (because it would be a NullPointerException in one case and the contract for .equals() allows you to do this), if you subtype a class and override .equals() then you had better be careful!

public boolean equals(Object o) {
    if (this == o)
        return true;
    if (o == null)
        return false;
    if (!o.getClass().isAssignableFrom(getClass())) // Danger! This is a mistake!
        return false;
    ThisClass thisClass = (ThisClass) o;
    // field comparison
    // ...

If ThisClass and ASubtypeOfThisClass both override .equals() with code like this, a.equals(b) may not equal b.equals(a)! The correct comparison would be:

    if (getClass() != o.getClass())
        return false;

This is not just my opinion; the contract for Object.equals() requires it.

Common Mistake: Equality Isn’t Transitive

Remember one of the laws for equals comparisons is that they should be transitive; if a = b and b = c then a = c. Unfortunately, when coupled with type coersion, languages often fail at this.

In JavaScript,

'' == 0;      // true
0  == '0';    // true
'' == '0';    // false!

Never use == in JavaScript. Use === instead.

Common Mistake: Equality Is Inconsistent

In Kotlin, == returns different values depending on the type of the variable, even for the same variable:

fun equalsFloat(a: Float, b: Float) {
  println(a == b);

fun equalsAny(a: Any, b: Any) {
  println(a == b);

fun main(args: Array<String>) {
  val a = Float.NaN;
  val b = Float.NaN;
  equalsFloat(a, b);
  equalsAny(a, b);
// prints false, true

This is an unfortunate combination of language features which results in some pretty unintuitive behavior.

Common Mistake: Using Reference Equality When Structural Equality Is Needed

Consider the following MSTest unit test in C#:

public void Calculation_Is_Correct() {
    var expected = new Result(SOME_EXPECTED_VALUE);

    var actual = _service.DoCalculation(SOME_INPUT);

    Assert.AreEqual(expected, actual);

Does this work? We can’t tell! Assert.AreEqual() will eventually call Object.Equals(), which does a reference comparison by default. Unless you’ve overridden Result.Equals() to do a structural comparison instead, the unit test is broken. The contract for Object.Equals() says that you should not override it at all if your type is mutable, which is reasonable in general but probably not what you want for a unit test. (This is because .Equals() is expected to match .GetHashCode(), and the hash code is expected not to change over the lifetime of the object.) The closest thing the .NET framework has to a “guaranteed structural” comparison for reference types is IEquatable<T>, which Assert.AreEqual() doesn’t use, even if it’s implemented.

It’s worse if you use NUnit.

(Java’s Object.hashCode, by contrast, is allowed to change when the object’s fields change.)

How to Think About Equality

Wow, I’ve now written 4000 words about the = operator and I’m not finished yet! That seems, well, out of proportion to the size of the operator. Why is this so complicated? Two reasons, basically:

Another way to divide this up is “stuff which should be fixed by programming language implementors” (the “inessential complexity” above) and “stuff which must be resolved by programming language users.

What Programming Languages Should Do

With regards to inessential complexity, the situation we find ourselves in today, with mostly-broken implementations of equivalence in nearly every mainstream programming language is just a crying shame. This “simple operation which must obey certain laws” is exactly the sort of thing we depend on programming languages to get right! But it appears to me that only SML has really considered having a lawful equality in both its semantics and its runtime/standard library, and SML isn’t entirely mainstream.

First, programming languages should make it simple to create types where equality comparison is disabled because it makes no sense (like [<NoEquality>] in F#) and they should use this feature in their standard library where needed, such as on floating point types.

Programming languages must make the difference between structural equality and reference equality crystal clear. There should never be a case where it’s unclear what you’re doing. Most programming lanuages overload == to mean both structural equality or reference equality depending on the type of reference, most commonly value types vs. reference types, and this is guaranteed to confuse developers.

Kotlin comes very close to getting this right with its === for reference equality and == for structural equality, although for some reason it translates === to == for value types, instead of just failing to compile for that. The goal should be reducing developer confusion. You want the developer to see === and think “reference equality,” not “more equals signs is better.”

F# mostly gets this right by making reference equality very hard to use.

I don’t know of any language which has mutable by default variables which handles structural comparsions in a non-confusing way. But it’s easy to imagine what it might look like! Have a reference equals and structural equals operator which is only supported in contexts where the language can reasonably expect to support it. For example, if .NET did not do the boxing funkiness with Object.ReferenceEquals and value types (it could just fail to compile if you tried) and had something akin to IEquatable<T> which would allow you to use a structural comparison operator, that seems like a pretty good solution to making it clear to developers which is which.

What Programmers Should Do

One might look at the length of this post and say, “Wow, equality is really complicated! I’m going to give up coding and become a soybean farmer.” But this post is as long as it is mostly because so many languages do equality wrong. Doing equality correctly does requre some thought, but not too much though. Certainly less than soybean farming.

When doing an equality comparison on an existing type, stop and ask yourself:

You can ask yourself similar questions when designing a custom type:

If your type is mutable, consider if you can change it to be immutable. You can do this even in a language which is mutable by default! Beyond giving you more options with respect to equality comparisons, there are many other benefits of an immutable architecture as well. The C# Roslyn compiler, which uses immutable data structures throughout, is a great example of this:

The third attribute of syntax trees is that they are immutable and thread-safe. This means that after a tree is obtained, it is a snapshot of the current state of the code, and never changes. This allows multiple users to interact with the same syntax tree at the same time in different threads without locking or duplication. Because the trees are immutable and no modifications can be made directly to a tree, factory methods help create and modify syntax trees by creating additional snapshots of the tree. The trees are efficient in the way they reuse underlying nodes, so a new version can be rebuilt fast and with little extra memory.
from the .NET Compiler Platform SDK docs


Thank you to Paul Blasucci, Jeremy Loy, Bud Marrical, Michael Perry, Skyler Tweedie, and Thomas Wheeler for reading drafts of this article and giving me feedback.


This post was inspired by Barry Mazur’s wonderful math paper, “When is one thing equal to some other thing?” which uses category theory to answer the question for math.

Thanks to Tommy Hall, who drew my attention to this 1993 paper, which, discusses many of the issues covered in this post and proposes a solution for Common Lisp.

Simon Ochsenreither has a nice series on problems with equality and fixing Haskell. Overview, Problems, Solution, Fixing Haskell, Implementation in Dora.

Hillel Wayne pointed me to this great essay, “The Semantics of Object Identity.”

Brandon Bloom provided a link to the paper “The Left Hand of Equals” which “takes a reflexive journey through fifty years of identity and equality in object-oriented languages, and ends somewhere we did not expect: a ‘left-handed’ equality relying on trust and grace.”

  1. This is probably attributable to Phil Karlson

  2. A “binary relation” deserves a little bit of explanation, but this gets into a little math, so feel free to ignore if it doesn’t help you. We have two sets A and B. (They might be the same set.) For any two members a and b of the sets, we want a rule which says whether they’re in the relation or not. So if A and B are the integers, the ordered pair (1, 2) is not in the relation “is equal to” but the ordered pair (5, 5) is in the relation. A relation is a subset of the cross product of the sets.

  3. Phil Hagelberg tells me the problem isn’t mutation of variables but data structures, which is a subtle but fair distinction.

  4. Gowers, Timothy, Mathematics: A Very Short Introduction, p. 60

  5. a = b = c = [] in Ruby assigns the same reference to a, b, and c. So if you mutate a, you’ll mutate b and c at the same time. That’s probably not what you wanted, otherwise what would be the point of having three separate references? In contrast, with a value type like Integer, mutating a will not change the value of b or c.

  6. ~ is the SML operator for unary negation, so ~0 should be read as “negative zero.”

Tags: mathematics, computer science, plt, equality