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Three Different Levels of Equality

Comparing the equality of values is a regular task in programming, usually as the base for a decision. However, deciding whether two values are “equal” isn't as trivial as one might think, and different programming languages have different approaches to this question.

Scheme has three different concepts of equality, represented by the three equality operators eq? eqv? and equal?. We will see these three again in various incarnations, as variants to different comparison operators. The exact interpretation of each level of equality is left to the discretion of the “implementation”, so everything described here is explicitly valid for Guile Scheme only and may differ from any information you might encounter regarding MIT Scheme or Racket etc.

Objects of different type are never equal in Scheme.


eq? checks for the identity of two objects, and the scope of that identity is very narrow in Scheme. Basically this predicate returns true if and only if the two arguments refer to the same object in memory, regardless of their values. This means that for a lot of comparisons eq? isn't applicable, but it is by far the most efficient predicate.

Most importantly symbols with the same name are identical, so (eq? 'my-symbol 'my-symbol) will always evaluate to #t. Strings with the same content are, on the other hand, not equal in the sense of eq?: (eq? "my-string" "my-string") will always evaluate to #f. Consequently you should try to use symbols wherever possible to name things, especially keys in association lists.

Booleans of the same value are identical, which can be used in decision processes: (eq? #t (number? 2.4)) evaluates to #t because the evaluation of the number? predicate evaluates to #t, which is “identical” to the other #t argument.

Another object that is unique and can be compared with eq? is the end-of-list element (): (eq? (cdr '(1)) (cdr '(2))) evaluates to #t because the () exists only once.

Note: Numbers and characters may be eq? to the numbers and characters with the same content, but - according to the reference - you can't rely on that fact. So depending on the context both is possible: (eq? 1 1) => #t or (eq? 1 1) => #f.


Numbers and characters should rather be compared with eqv?. This doesn't look for identity but for equivalence of the compared objects - for numbers and characters. For all other data types eqv? behaves the same as eq?, which should not be used regularly since: (eq? '(1 . 2) '(1 . 2)) evaluates to #f even if the pairs have the same content.


For all cases except the ones mentioned above equal? should be used - which is the least strict but also the most “expensive” procedure. In the case of compound data types equal? checks for equivalent content recursively, walking over the individual elements and comparing their content. So all these expressions evaluate to #t:

(equal? "a-string" "a-string")
(equal? '(1 . 2) '(1 . 2))
(equal? (list 1 2 3 4) (list 1 2 3 4))

TODO: Investigate and discuss the difference between = and eqv? or **string=? and equal?.

Last update: November 3, 2022