List Comprehensions

List comprehensions are a feature of many modern functional programming languages. Subject to certain rules, they provide a succinct notation for generating elements in a list.

List comprehensions are analogous to set comprehensions in Zermelo-Frankel set theory and are called ZF expressions in Miranda. They are analogous to the setof and findall predicates in Prolog.

Expression is an arbitrary expression, and each Qualifier is either a generator or a filter.

Examples of List Comprehensions

We start with a simple example:

 [X || X <- [1,2,a,3,4,b,5,6], X > 3].
[a,4,b,5,6]    ]]>

This should be read as follows:

The list of X such that X is taken from the list [1,2,a,...] and X is greater than 3.

The notation is a generator and the expression X > 3 is a filter.

An additional filter can be added in order to restrict the result to integers:

 [X || X <- [1,2,a,3,4,b,5,6], integer(X), X > 3].
[4,5,6]    ]]>

Generators can be combined. For example, the Cartesian product of two lists can be written as follows:

 [{X, Y} || X <- [1,2,3], Y <- [a,b]].
[{1,a},{1,b},{2,a},{2,b},{3,a},{3,b}]    ]]>

Quick Sort

The well known quick sort routine can be written as follows:


    sort([ X || X <- T, X < Pivot]) ++
    [Pivot] ++
    sort([ X || X <- T, X >= Pivot]);
sort([]) -> [].      ]]>

The expression is the list of all elements in T, which are less than Pivot.

= Pivot]]]> is the list of all elements in T, which are greater or equal to Pivot.

To sort a list, we isolate the first element in the list and split the list into two sub-lists. The first sub-list contains all elements which are smaller than the first element in the list, the second contains all elements which are greater than or equal to the first element in the list. We then sort the sub-lists and combine the results.

Permutations

The following example generates all permutations of the elements in a list:

 [[]];
perms(L)  -> [[H|T] || H <- L, T <- perms(L--[H])].      ]]>

We take take H from L in all possible ways. The result is the set of all lists [H|T], where T is the set of all possible permutations of L with H removed.


      
> perms([b,u,g]).     
[[b,u,g],[b,g,u],[u,b,g],[u,g,b],[g,b,u],[g,u,b]]

Pythagorean Triplets

Pythagorean triplets are sets of integers {A,B,C} such that A**2 + B**2 = C**2.

The function pyth(N) generates a list of all integers {A,B,C} such that A**2 + B**2 = C**2 and where the sum of the sides is less than N.


    [ {A,B,C} ||
        A <- lists:seq(1,N),
        B <- lists:seq(1,N),
        C <- lists:seq(1,N),
        A+B+C =< N,
        A*A+B*B == C*C 
    ].      ]]>


> pyth(3).
[].
> pyth(11).
[].
>pyth(12).
[{3,4,5},{4,3,5}]
> pyth(50).
[{3,4,5},
 {4,3,5},
 {5,12,13},
 {6,8,10},
 {8,6,10},
 {8,15,17},
 {9,12,15},
 {12,5,13},
 {12,9,15},
 {12,16,20},
 {15,8,17},
 {16,12,20}]

The following code reduces the search space and is more efficient:


    [{A,B,C} ||
        A <- lists:seq(1,N),
        B <- lists:seq(1,N-A+1),
        C <- lists:seq(1,N-A-B+2),
        A+B+C =< N,
        A*A+B*B == C*C ].      ]]>

Simplifications with List Comprehensions

As an example, list comprehensions can be used to simplify some of the functions in lists.erl:

  [X || L1 <- L, X <- L1].
map(Fun, L) -> [Fun(X) || X <- L].
filter(Pred, L) -> [X || X <- L, Pred(X)].      ]]>

Variable Bindings in List Comprehensions

The scope rules for variables which occur in list comprehensions are as follows:

all variables which occur in a generator pattern are assumed to be "fresh" variables any variables which are defined before the list comprehension and which are used in filters have the values they had before the list comprehension no variables may be exported from a list comprehension.

As an example of these rules, suppose we want to write the function select, which selects certain elements from a list of tuples. We might write [Y || {X, Y} <- L].]]> with the intention of extracting all tuples from L where the first item is X.

Compiling this yields the following diagnostic:



./FileName.erl:Line: Warning: variable 'X' shadowed in generate

This diagnostic warns us that the variable X in the pattern is not the same variable as the variable X which occurs in the function head.

Evaluating select yields the following result:


> select(b,[{a,1},{b,2},{c,3},{b,7}]).
[1,2,3,7]

This result is not what we wanted. To achieve the desired effect we must write select as follows:

[Y || {X1, Y} <- L, X == X1]. ]]>

The generator now contains unbound variables and the test has been moved into the filter. This now works as expected:


> select(b,[{a,1},{b,2},{c,3},{b,7}]).
[2,7]

One consequence of the rules for importing variables into a list comprehensions is that certain pattern matching operations have to be moved into the filters and cannot be written directly in the generators. To illustrate this, do not write as follows:


    Y = ...
    [ Expression || PatternInvolving Y  <- Expr, ...]
    ...    ]]>

Instead, write as follows:


    Y = ...
    [ Expression || PatternInvolving Y1  <- Expr, Y == Y1, ...]
    ...
    ]]>