Funs

## 2 Funs

### 2.1 Example 1 - map

If we want to double every element in a list, we could write a function named `double`:

    double([H|T]) -> [2*H|double(T)];
    double([])    -> [].

This function obviously doubles the argument entered as input as follows:

    > double([1,2,3,4]).
    [2,4,6,8]

We now add the function `add_one`, which adds one to every element in a list:

    add_one([H|T]) -> [H+1|add_one(T)];
    add_one([])    -> [].

These functions, `double` and `add_one`, have a very similar structure. We can exploit this fact and write a function `map` which expresses this similarity:

    map(F, [H|T]) -> [F(H)|map(F, T)];
    map(F, [])    -> [].

We can now express the functions `double` and `add_one` in terms of `map` as follows:

    double(L)  -> map(fun(X) -> 2*X end, L).
    add_one(L) -> map(fun(X) -> 1 + X end, L).

`map(F, List)` is a function which takes a function `F` and a list `L` as arguments and returns the new list which is obtained by applying `F` to each of the elements in `L`.

The process of abstracting out the common features of a number of different programs is called procedural abstraction. Procedural abstraction can be used in order to write several different functions which have a similar structure, but differ only in some minor detail. This is done as follows:

1. write one function which represents the common features of these functions
2. parameterize the difference in terms of functions which are passed as arguments to the common function.

### 2.2 Example 2 - foreach

This example illustrates procedural abstraction. Initially, we show the following two examples written as conventional functions:

1. all elements of a list are printed onto a stream
2. a message is broadcast to a list of processes.

    print_list(Stream, [H|T]) ->
        io:format(Stream, "~p~n", [H]),
        print_list(Stream, T);
    print_list(Stream, []) ->
        true.

    broadcast(Msg, [Pid|Pids]) ->
        Pid ! Msg,
        broadcast(Msg, Pids);
    broadcast(_, []) ->
        true.

Both these functions have a very similar structure. They both iterate over a list doing something to each element in the list. The "something" has to be carried round as an extra argument to the function which does this.

The function `foreach` expresses this similarity:

    foreach(F, [H|T]) ->
        F(H),
        foreach(F, T);
    foreach(F, []) ->
        ok.

Using `foreach`, `print_list` becomes:

    foreach(fun(H) -> io:format(S, "~p~n",[H]) end, L)

`broadcast` becomes:

    foreach(fun(Pid) -> Pid ! M end, L)

`foreach` is evaluated for its side-effect and not its value. `foreach(Fun ,L)` calls `Fun(X)` for each element `X` in `L` and the processing occurs in the order in which the elements were defined in `L`. `map` does not define the order in which its elements are processed.

### 2.3 The Syntax of Funs

Funs are written with the syntax:

    F = fun (Arg1, Arg2, ... ArgN) ->
            ...
        end

This creates an anonymous function of `N` arguments and binds it to the variable `F`.

If we have already written a function in the same module and wish to pass this function as an argument, we can use the following syntax:

    F = fun FunctionName/Arity

With this form of function reference, the function which is referred to does not need to be exported from the module.

We can also refer to a function defined in a different module with the following syntax:

    F = {Module, FunctionName}

In this case, the function must be exported from the module in question.

The follow program illustrates the different ways of creating funs:

    -module(fun_test).
    -export([t1/0, t2/0, t3/0, t4/0, double/1]).
    -import(lists, [map/2]).

    t1() -> map(fun(X) -> 2 * X end, [1,2,3,4,5]).

    t2() -> map(fun double/1, [1,2,3,4,5]).

    t3() -> map({?MODULE, double}, [1,2,3,4,5]).

    double(X) -> X * 2.

We can evaluate the fun `F` with the syntax:

    F(Arg1, Arg2, ..., Argn)

To check whether a term is a fun, use the test `is_function/1` in a guard. Example:

    f(F, Args) when is_function(F) ->
       apply(F, Args);
    f(N, _) when is_integer(N) ->
       N.

Funs are a distinct type. The BIFs erlang:fun_info/1,2 can be used to retrieve information about a fun, and the BIF erlang:fun_to_list/1 returns a textual representation of a fun. The check_process_code/2 BIF returns true if the process contains funs that depend on the old version of a module.

| In OTP R5 and earlier releases, funs were represented using tuples.

### 2.4 Variable Bindings Within a Fun

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

- All variables which occur in the head of a fun are assumed to be "fresh" variables.
- Variables which are defined before the fun, and which occur in function calls or guard tests within the fun, have the values they had outside the fun.
- No variables may be exported from a fun.

The following examples illustrate these rules:

    print_list(File, List) ->
        {ok, Stream} = file:open(File, write),
        foreach(fun(X) -> io:format(Stream,"~p~n",[X]) end, List),
        file:close(Stream).

In the above example, the variable `X` which is defined in the head of the fun is a new variable. The value of the variable `Stream` which is used within within the fun gets its value from the `file:open` line.

Since any variable which occurs in the head of a fun is considered a new variable it would be equally valid to write:

    print_list(File, List) ->
        {ok, Stream} = file:open(File, write),
        foreach(fun(File) ->
                    io:format(Stream,"~p~n",[File])
                end, List),
        file:close(Stream).

In this example, `File` is used as the new variable instead of `X`. This is rather silly since code in the body of the fun cannot refer to the variable `File` which is defined outside the fun. Compiling this example will yield the diagnostic:

    ./FileName.erl:Line: Warning: variable 'File'
          shadowed in 'lambda head'

This reminds us that the variable `File` which is defined inside the fun collides with the variable `File` which is defined outside the fun.

The rules for importing variables into a fun has the consequence that certain pattern matching operations have to be moved into guard expressions and cannot be written in the head of the fun. For example, we might write the following code if we intend the first clause of `F` to be evaluated when the value of its argument is `Y`:

    f(...) ->
        Y = ...
        map(fun(X) when X == Y ->
                 ;
               (_) ->
                 ...
            end, ...)
        ...

instead of

    f(...) ->
        Y = ...
        map(fun(Y) ->
                 ;
               (_) ->
                 ...
            end, ...)
        ...

### 2.5 Funs and the Module Lists

The following examples show a dialogue with the Erlang shell. All the higher order functions discussed are exported from the module `lists`.

#### 2.5.1 map

    map(F, [H|T]) -> [F(H)|map(F, T)];
    map(F, [])    -> [].

`map` takes a function of one argument and a list of terms. It returns the list obtained by applying the function to every argument in the list.

    > Double = fun(X) -> 2 * X end.
    #Fun<erl_eval.6.72228031>
    > lists:map(Double, [1,2,3,4,5]).
    [2,4,6,8,10]

When a new fun is defined in the shell, the value of the Fun is printed as `Fun#<erl_eval>`.

#### 2.5.2 any

    any(Pred, [H|T]) ->
        case Pred(H) of
            true  ->  true;
            false ->  any(Pred, T)
        end;
    any(Pred, []) ->
        false.

`any` takes a predicate `P` of one argument and a list of terms. A predicate is a function which returns `true` or `false`. `any` is true if there is a term `X` in the list such that `P(X)` is `true`.

We define a predicate `Big(X)` which is `true` if its argument is greater that 10.

    > Big = fun(X) -> if X > 10 -> true; true -> false end end.
    #Fun<erl_eval.6.72228031>
    > lists:any(Big, [1,2,3,4]).
    false
    > lists:any(Big, [1,2,3,12,5]).
    true

#### 2.5.3 all

    all(Pred, [H|T]) ->
        case Pred(H) of
            true  ->  all(Pred, T);
            false ->  false
        end;
    all(Pred, []) ->
        true.

`all` has the same arguments as `any`. It is true if the predicate applied to all elements in the list is true.

    > lists:all(Big, [1,2,3,4,12,6]).
    false
    > lists:all(Big, [12,13,14,15]).
    true

#### 2.5.4 foreach

    foreach(F, [H|T]) ->
        F(H),
        foreach(F, T);
    foreach(F, []) ->
        ok.

`foreach` takes a function of one argument and a list of terms. The function is applied to each argument in the list. `foreach` returns `ok`. It is used for its side-effect only.

    > lists:foreach(fun(X) -> io:format("~w~n",[X]) end, [1,2,3,4]).
    1
    2
    3
    4
    ok

#### 2.5.5 foldl

    foldl(F, Accu, [Hd|Tail]) ->
        foldl(F, F(Hd, Accu), Tail);
    foldl(F, Accu, []) -> Accu.

`foldl` takes a function of two arguments, an accumulator and a list. The function is called with two arguments. The first argument is the successive elements in the list, the second argument is the accumulator. The function must return a new accumulator which is used the next time the function is called.

If we have a list of lists `L = ["I","like","Erlang"]`, then we can sum the lengths of all the strings in `L` as follows:

    > L = ["I","like","Erlang"].
    ["I","like","Erlang"]
    10> lists:foldl(fun(X, Sum) -> length(X) + Sum end, 0, L).
    11

`foldl` works like a `while` loop in an imperative language:

    L =  ["I","like","Erlang"],
    Sum = 0,
    while( L != []){
        Sum += length(head(L)),
        L = tail(L)
    end

#### 2.5.6 mapfoldl

    mapfoldl(F, Accu0, [Hd|Tail]) ->
        {R,Accu1} = F(Hd, Accu0),
        {Rs,Accu2} = mapfoldl(F, Accu1, Tail),
        {[R|Rs], Accu2};
    mapfoldl(F, Accu, []) -> {[], Accu}.

`mapfoldl` simultaneously maps and folds over a list. The following example shows how to change all letters in `L` to upper case and count them.

First upcase:

    > Upcase = fun(X) when $a =< X, X =< $z -> X + $A - $a;
                     (X) -> X
                  end.
    #Fun<erl_eval.6.72228031>
    > Upcase_word =
          fun(X) ->
            lists:map(Upcase, X)
          end.
    #Fun<erl_eval.6.72228031>
    > Upcase_word("Erlang").
    "ERLANG"
    > lists:map(Upcase_word, L).
    ["I","LIKE","ERLANG"]

Now we can do the fold and the map at the same time:

    > lists:mapfoldl(fun(Word, Sum) ->
                        {Upcase_word(Word), Sum + length(Word)}
                     end, 0, L).
    {["I","LIKE","ERLANG"],11}

#### 2.5.7 filter

    filter(F, [H|T]) ->
        case F(H) of
            true  -> [H|filter(F, T)];
            false -> filter(F, T)
        end;
    filter(F, []) -> [].

`filter` takes a predicate of one argument and a list and returns all element in the list which satisfy the predicate.

    > lists:filter(Big, [500,12,2,45,6,7]).
    [500,12,45]

When we combine maps and filters we can write very succinct code. For example, suppose we want to define a set difference function. We want to define `diff(L1, L2)` to be the difference between the lists `L1` and `L2`. This is the list of all elements in L1 which are not contained in L2. This code can be written as follows:

    diff(L1, L2) ->
        filter(fun(X) -> not member(X, L2) end, L1).

The AND intersection of the list `L1` and `L2` is also easily defined:

    intersection(L1,L2) -> filter(fun(X) -> member(X,L1) end, L2).

#### 2.5.8 takewhile

    takewhile(Pred, [H|T]) ->
        case Pred(H) of
            true  -> [H|takewhile(Pred, T)];
            false -> []
        end;
    takewhile(Pred, []) ->
        [].

`takewhile(P, L)` takes elements `X` from a list `L` as long as the predicate `P(X)` is true.

    > lists:takewhile(Big, [200,500,45,5,3,45,6]).
    [200,500,45]

#### 2.5.9 dropwhile

    dropwhile(Pred, [H|T]) ->
        case Pred(H) of
            true  -> dropwhile(Pred, T);
            false -> [H|T]
        end;
    dropwhile(Pred, []) ->
        [].

`dropwhile` is the complement of `takewhile`.

    > lists:dropwhile(Big, [200,500,45,5,3,45,6]).
    [5,3,45,6]

#### 2.5.10 splitwith

    splitwith(Pred, L) ->
        splitwith(Pred, L, []).

    splitwith(Pred, [H|T], L) ->
        case Pred(H) of
            true  -> splitwith(Pred, T, [H|L]);
            false -> {reverse(L), [H|T]}
        end;
    splitwith(Pred, [], L) ->
        {reverse(L), []}.

`splitwith(P, L)` splits the list `L` into the two sub-lists `{L1, L2}`, where `L = takewhile(P, L)` and `L2 = dropwhile(P, L)`.

    > lists:splitwith(Big, [200,500,45,5,3,45,6]).
    {[200,500,45],[5,3,45,6]}

### 2.6 Funs Which Return Funs

So far, this section has only described functions which take funs as arguments. It is also possible to write more powerful functions which themselves return funs. The following examples illustrate these type of functions.

#### 2.6.1 Simple Higher Order Functions

`Adder(X)` is a function which, given `X`, returns a new function `G` such that `G(K)` returns `K + X`.

    > Adder = fun(X) -> fun(Y) -> X + Y end end.
    #Fun<erl_eval.6.72228031>
    > Add6 = Adder(6).
    #Fun<erl_eval.6.72228031>
    > Add6(10).
    16

#### 2.6.2 Infinite Lists

The idea is to write something like:

    -module(lazy).
    -export([ints_from/1]).
    ints_from(N) ->
        fun() ->
                [N|ints_from(N+1)]
        end.

Then we can proceed as follows:

    > XX = lazy:ints_from(1).
    #Fun<lazy.0.29874839>
    > XX().
    [1|#Fun<lazy.0.29874839>]
    > hd(XX()).
    1
    > Y = tl(XX()).
    #Fun<lazy.0.29874839>
    > hd(Y()).
    2

etc. - this is an example of "lazy embedding".

#### 2.6.3 Parsing

The following examples show parsers of the following type:

    Parser(Toks) -> {ok, Tree, Toks1} | fail

`Toks` is the list of tokens to be parsed. A successful parse returns `{ok, Tree, Toks1}`, where `Tree` is a parse tree and `Toks1` is a tail of `Tree` which contains symbols encountered after the structure which was correctly parsed. Otherwise `fail` is returned.

The example which follows illustrates a simple, functional parser which parses the grammar:

    (a | b) & (c | d)

The following code defines a function `pconst(X)` in the module `funparse`, which returns a fun which parses a list of tokens.

    pconst(X) ->
        fun (T) ->
           case T of
               [X|T1] -> {ok, {const, X}, T1};
               _      -> fail
           end
        end.

This function can be used as follows:

    > P1 = funparse:pconst(a).
    #Fun<funparse.0.22674075>
    > P1([a,b,c]).
    {ok,{const,a},[b,c]}
    > P1([x,y,z]).
    fail

Next, we define the two higher order functions `pand` and `por` which combine primitive parsers to produce more complex parsers. Firstly `pand`:

    pand(P1, P2) ->
        fun (T) ->
            case P1(T) of
                {ok, R1, T1} ->
                    case P2(T1) of
                        {ok, R2, T2} ->
                            {ok, {'and', R1, R2}};
                        fail ->
                            fail
                    end;
                fail ->
                    fail
            end
        end.

Given a parser `P1` for grammar `G1`, and a parser `P2` for grammar `G2`, `pand(P1, P2)` returns a parser for the grammar which consists of sequences of tokens which satisfy `G1` followed by sequences of tokens which satisfy `G2`.

`por(P1, P2)` returns a parser for the language described by the grammar `G1` or `G2`.

    por(P1, P2) ->
        fun (T) ->
            case P1(T) of
                {ok, R, T1} ->
                    {ok, {'or',1,R}, T1};
                fail ->
                    case P2(T) of
                        {ok, R1, T1} ->
                            {ok, {'or',2,R1}, T1};
                        fail ->
                            fail
                    end
            end
        end.

The original problem was to parse the grammar `(a | b) & (c | d)`. The following code addresses this problem:

    grammar() ->
        pand(
             por(pconst(a), pconst(b)),
             por(pconst(c), pconst(d))).

The following code adds a parser interface to the grammar:

    parse(List) ->
        (grammar())(List).

We can test this parser as follows:

    > funparse:parse([a,c]).
    {ok,{'and',{'or',1,{const,a}},{'or',1,{const,c}}}}
    > funparse:parse([a,d]).
    {ok,{'and',{'or',1,{const,a}},{'or',2,{const,d}}}}
    > funparse:parse([b,c]).
    {ok,{'and',{'or',2,{const,b}},{'or',1,{const,c}}}}
    > funparse:parse([b,d]).
    {ok,{'and',{'or',2,{const,b}},{'or',2,{const,d}}}}
    > funparse:parse([a,b]).
    fail

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