What is the difference between eq?, eqv?, equal?, and = in Scheme?

I wonder what the difference is between those operations in Scheme. I have seen similar questions in Stack Overflow but they are about Lisp, and there is not a comparison between three of those operators.

I am writing the different types of commands in Scheme, and I get the following outputs:

(eq? 5 5) -->#t
(eq? 2.5 2.5) -->#f
(equal? 2.5 2.5) --> #t
(= 2.5 2.5) --> #t

Why is this the case?

Solution 1:

I'll answer this question incrementally. Let's start with the = equivalence predicate. The = predicate is used to check whether two numbers are equal. If you supply it anything else but a number then it will raise an error:

(= 2 3)     => #f
(= 2.5 2.5) => #t
(= '() '()) => error

The eq? predicate is used to check whether its two parameters respresent the same object in memory. For example:

(define x '(2 3))
(define y '(2 3))
(eq? x y)         => #f
(define y x)
(eq? x y)         => #t

Note however that there's only one empty list '() in memory (actually the empty list doesn't exist in memory, but a pointer to the memory location 0 is considered as the empty list). Hence when comparing empty lists eq? will always return #t (because they represent the same object in memory):

(define x '())
(define y '())
(eq? x y)      => #t

Now depending upon the implementation eq? may or may not return #t for primitive values such as numbers, strings, etc. For example:

(eq? 2 2)     => depends upon the implementation
(eq? "a" "a") => depends upon the implementation

This is where the eqv? predicate comes into picture. The eqv? is exactly the same as the eq? predicate, except that it will always return #t for same primitive values. For example:

(eqv? 2 2)     => #t
(eqv? "a" "a") => depends upon the implementation

Hence eqv? is a superset of eq? and for most cases you should use eqv? instead of eq?.

Finally we come to the equal? predicate. The equal? predicate is exactly the same as the eqv? predicate, except that it can also be used to test whether two lists, vectors, etc. have corresponding elements which satisfy the eqv? predicate. For example:

(define x '(2 3))
(define y '(2 3))
(equal? x y)      => #t
(eqv? x y)        => #f

In general:

Use the = predicate when you wish to test whether two numbers are equivalent.
Use the eqv? predicate when you wish to test whether two non-numeric values are equivalent.
Use the equal? predicate when you wish to test whether two lists, vectors, etc. are equivalent.
Don't use the eq? predicate unless you know exactly what you're doing.

Solution 2:

There are a full two pages in the RnRS specification related to eq?, eqv?, equal? and =. Here is the Draft R7RS Specification. Check it out!

Explanation:

= compares numbers, 2.5 and 2.5 are numerically equal.
equal? for numbers reduces to =, 2.5 and 2.5 are numerically equal.
eq? compares 'pointers'. The number 5, in your Scheme implementation, is implemented as an 'immediate' (likely), thus 5 and 5 are identical. The number 2.5 may require an allocation of a 'floating point record' in your Scheme implementation, the two pointers are not identical.

Solution 3:

eq? is #t when it is the same address/object. Normally one could expect #t for same symbol, boolean and object and #f for values that is of different type, with different values, or not the same structure Scheme/Lisp-implementations has a tradition to embed type in their pointers and to embed values in the same space if it's enough space. Thus some pointers really are not addresses but values, like the char R or the Fixnum 10. These will be eq? since the "address" is an embedded type+value. Some implementations also reuse immutable constants. (eq? '(1 2 3) '(1 2 3)) might be #f when interpreted but #t when compiled since it might get the same address. (Like the constant String pool in Java). Because of this, many expresions involving eq? are unspecified, thus wether it evaluates to #t or #f is implementation dependent.

eqv? are #t for the same things as eq?. It is also #t if it's a number or character and it's value is the same, even when the data is too big to fit into a pointer. Thus for those eqv? does the extra work of checking that type is one of the supported, that both are the same type and it's target objects have the same data value.

equal? is #t for the same things as eqv? and if it's a compound type like pair, vector, string, and bytevector it recursively does equal? with the parts. In practice it will return #t if the two objects looks the same. Prior to R6RS, it's unsafe to use equal? on circular structures.

= is like eqv? but it only works for numeric types. It might be more efficient.

string=? is like equal?, but it only works for strings. It might be more efficient.

Solution 4:

equal? recursively compares two objects (of any type) for equality.

Note this could be expensive for a large data structure since potentially the entire list, string, vector, etc must be traversed.
If the object just contains a single element (EG: number, character, etc), this is the same as eqv?.

eqv? tests two objects to determine if both are "normally regarded as the same object".

eqv? and eq? are very similar operations, and the differences between them are going to be somewhat implementation specific.

eq? is the same as eqv? but may be able to discern finer distinctions, and may be implemented more efficiently.

According to the spec, this might be implemented as a fast and efficient pointer comparison, as opposed to a more complicated operation for eqv?.

= compares numbers for numerical equality.

Note that more than two numbers can be provided, eg: (= 1 1.0 1/1 2/2)

Solution 5:

You don't mention a scheme implementation, but in Racket, eq? only returns true if the arguments refer to the same object. Your second example is yielding #f because the system is creating a new floating point number for each argument; they're not the same object.

equal? and = are checking for value equivalence, but = is only applicable to numbers.

If you're using Racket, check here for more information. Otherwise, check the documentation of your scheme implementation.