2 The Programming Language

This chapter describes the GAP programming language. It should allow you in principle to predict the result of each and every input. In order to know what we are talking about, we first have to look more closely at the process of interpretation and the various representations of data involved.

First we have the input to GAP, given as a string of characters. How those characters enter GAP is operating system dependent, e.g., they might be entered at a terminal, pasted with a mouse into a window, or read from a file. The mechanism does not matter. This representation of expressions by characters is called the external representation of the expression. Every expression has at least one external representation that can be entered to get exactly this expression.

The input, i.e., the external representation, is transformed in a process called reading to an internal representation. At this point the input is analyzed and inputs that are not legal external representations, according to the rules given below, are rejected as errors. Those rules are usually called the syntax of a programming language.

The internal representation created by reading is called either an expression or a statement. Later we will distinguish between those two terms, however now we will use them interchangeably. The exact form of the internal representation does not matter. It could be a string of characters equal to the external representation, in which case the reading would only need to check for errors. It could be a series of machine instructions for the processor on which GAP is running, in which case the reading would more appropriately be called compilation. It is in fact a tree--like structure.

After the input has been read it is again transformed in a process called evaluation or execution. Later we will distinguish between those two terms too, but for the moment we will use them interchangeably. The name hints at the nature of this process, it replaces an expression with the value of the expression. This works recursively, i.e., to evaluate an expression first the subexpressions are evaluated and then the value of the expression is computed according to rules given below from those values. Those rules are usually called the semantics of a programming language.

The result of the evaluation is, not surprisingly, called a value. The set of values is of course a much smaller set than the set of expressions; for every value there are several expressions that will evaluate to this value. Again the form in which such a value is represented internally does not matter. It is in fact a tree--like structure again.

The last process is called printing. It takes the value produced by the evaluation and creates an external representation, i.e., a string of characters again. What you do with this external representation is up to you. You can look at it, paste it with the mouse into another window, or write it to a file.

Lets look at an example to make this more clear. Suppose you type in the following string of 8 characters

1 + 2 * 3;

GAP takes this external representation and creates a tree like internal representation, which we can picture as follows

       +
      / \
     1   *
        / \
       2   3 

This expression is then evaluated. To do this GAP first evaluates the right subexpression 2*3. Again to do this GAP first evaluates its subexpressions 2 and 3. However they are so simple that they are their own value, we say that they are self--evaluating. After this has been done, the rule for * tells us that the value is the product of the values of the two subexpressions, which in this case is clearly 6. Combining this with the value of the left operand of the +, which is self--evaluating too gives us the value of the whole expression 7. This is then printed, i.e., converted into the external representation consisting of the single character 7.

In this fashion we can predict the result of every input when we know the syntactic rules that govern the process of reading and the semantic rules that tell us for every expression how its value is computed in terms of the values of the subexpressions. The syntactic rules are given in sections Lexical Structure, Symbols, Whitespaces, Keywords, Identifiers, and The Syntax in BNF, the semantic rules are given in sections Expressions, Variables, Function Calls, Comparisons, Operations, Statements, Assignments, Procedure Calls, If, While, Repeat, For, Functions, and the chapters describing the individual data types.

Subsections

  1. Lexical Structure
  2. Symbols
  3. Whitespaces
  4. Keywords
  5. Identifiers
  6. Expressions
  7. Variables
  8. Function Calls
  9. Comparisons
  10. Operations
  11. Statements
  12. Assignments
  13. Procedure Calls
  14. If
  15. While
  16. Repeat
  17. For
  18. Functions
  19. Return
  20. The Syntax in BNF

2.1 Lexical Structure

The input of GAP consists of sequences of the following characters.

Digits, uppercase and lowercase letters, space, tab, newline, and the special characters

    "       '       (       )       *       +       ,       _
    .       /       :       ;       <       =       >       ~
    [       \       ]       ^       _       {       }       # 

Other characters will be signalled as illegal. Inside strings and comments the full character set supported by the computer is allowed.

2.2 Symbols

The process of reading, i.e., of assembling the input into expressions, has a subprocess, called scanning, that assembles the characters into symbols. A symbol is a sequence of characters that form a lexical unit. The set of symbols consists of keywords, identifiers, strings, integers, and operator and delimiter symbols.

A keyword is a reserved word consisting entirely of lowercase letters (see Keywords). An identifier is a sequence of letters and digits that contains at least one letter and is not a keyword (see Identifiers). An integer is a sequence of digits (see Integers). A string is a sequence of arbitrary characters enclosed in double quotes (see Strings and Characters).

Operator and delimiter symbols are

    +       -       *       /       ^       ~
    =       <>      <       <=      >       >=
    :=      .       ..      ->      ,       ;
    [       ]       {       }       (       )  

Note that during the process of scanning also all whitespace is removed (see Whitespaces).

2.3 Whitespaces

The characters space, tab, newline, and return are called whitespace characters. Whitespace is used as necessary to separate lexical symbols, such as integers, identifiers, or keywords. For example Thorondor is a single identifier, while Th or ondor is the keyword or between the two identifiers Th and ondor. Whitespace may occur between any two symbols, but not within a symbol. Two or more adjacent whitespaces are equivalent to a single whitespace. Apart from the role as separator of symbols, whitespaces are otherwise insignificant. Whitespaces may also occur inside a string, where they are significant. Whitespaces should also be used freely for improved readability.

A comment starts with the character #, which is sometimes called sharp or hatch, and continues to the end of the line on which the comment character appears. The whole comment, including # and the newline character is treated as a single whitespace. Inside a string, the comment character # looses its role and is just an ordinary character.

For example, the following statement

if i<0 then a:=-i;else a:=i;fi;

is equivalent to

    if i < 0  then      # if i is negative
        a := -i;        #     take its inverse
    else                # otherwise
        a := i;         #     take itself
    fi; 

(which by the way shows that it is possible to write superfluous comments). However the first statement is not equivalent to

ifi<0thena:=-i;elsea:=i;fi;

since the keyword if must be separated from the identifier i by a whitespace, and similarly then and a, and else and a must be separated.

2.4 Keywords

Keywords are reserved words that are used to denote special operations or are part of statements. They must not be used as identifiers. The keywords are

    and       do        elif      else      end       fi
    for       function  if        in        local     mod
    not       od        or        repeat    return    then
    until     while     quit 

Note that all keywords are written in lowercase. For example only else is a keyword; Else, eLsE, ELSE and so forth are ordinary identifiers. Keywords must not contain whitespace, for example el if is not the same as elif.

2.5 Identifiers

An identifier is used to refer to a variable (see Variables). An identifier consists of letters, digits, and underscores _, and must contain at least one letter or underscore. An identifier is terminated by the first character not in this class. Examples of valid identifiers are

    a                   foo                 aLongIdentifier
    hello               Hello               HELLO
    x100                100x                _100
    some_people_prefer_underscores_to_separate_words
    WePreferMixedCaseToSeparateWords 

Note that case is significant, so the three identifiers in the second line are distinguished.

The backslash \ can be used to include other characters in identifiers; a backslash followed by a character is equivalent to the character, except that this escape sequence is considered to be an ordinary letter. For example G\(2\,5\) is an identifier, not a call to a function G.

An identifier that starts with a backslash is never a keyword, so for example \* and \mod are identifier.

The length of identifiers is not limited, however only the first 1023 characters are significant. The escape sequence \newline is ignored, making it possible to split long identifiers over multiple lines.

2.6 Expressions

An expression is a construct that evaluates to a value. Syntactic constructs that are executed to produce a side effect and return no value are called statements (see Statements). Expressions appear as right hand sides of assignments (see Assignments), as actual arguments in function calls (see Function Calls), and in statements.

Note that an expression is not the same as a value. For example 1 + 11 is an expression, whose value is the integer 12. The external representation of this integer is the character sequence 12, i.e., this sequence is output if the integer is printed. This sequence is another expression whose value is the integer 12. The process of finding the value of an expression is done by the interpreter and is called the evaluation of the expression.

Variables, function calls, and integer, permutation, string, function, list, and record literals (see Variables, Function Calls, Integers, Permutations, Strings and Characters, Functions, Lists, Records), are the simplest cases of expressions.

Expressions, for example the simple expressions mentioned above, can be combined with the operators to form more complex expressions. Of course those expressions can then be combined further with the operators to form even more complex expressions. The operators fall into three classes. The comparisons are =, <, <=, , =, and in (see Comparisons and In). The arithmetic operators are +, -, *, /, mod, and ^ (see Operations). The logical operators are not, and, and or (see Operations for Booleans).

    gap> 2 * 2;    # a very simple expression with value
    4
    gap> 2 * 2 + 9 = Fibonacci(7) and  Fibonacci(13) in Primes;
    true            # a more complex expression 

2.7 Variables

A variable is a location in a GAP program that points to a value. We say the variable is bound to this value. If a variable is evaluated it evaluates to this value.

Initially an ordinary variable is not bound to any value. The variable can be bound to a value by assigning this value to the variable (see Assignments). Because of this we sometimes say that a variable that is not bound to any value has no assigned value. Assignment is in fact the only way by which a variable, which is not an argument of a function, can be bound to a value. After a variable has been bound to a value an assignment can also be used to bind the variable to another value.

A special class of variables are arguments of functions. They behave similarly to other variables, except they are bound to the value of the actual arguments upon a function call (see Function Calls).

Each variable has a name that is also called its identifier. This is because in a given scope an identifier identifies a unique variable (see Identifiers). A scope is a lexical part of a program text. There is the global scope that encloses the entire program text, and there are local scopes that range from the function keyword, denoting the beginning of a function definition, to the corresponding end keyword. A local scope introduces new variables, whose identifiers are given in the formal argument list and the local declaration of the function (see Functions). Usage of an identifier in a program text refers to the variable in the innermost scope that has this identifier as its name. Because this mapping from identifiers to variables is done when the program is read, not when it is executed, GAP is said to have lexical scoping. The following example shows how one identifier refers to different variables at different points in the program text.

     g := 0;            # global variable g
     x := function ( a, b, c )
        local   y;
        g := c;         # c refers to argument c of function x
        y := function ( y )
            local  d, e, f;
            d := y;     # y refers to argument y of function y
            e := b;     # b refers to argument b of function x
            f := g;     # g refers to global variable g
            return d + e + f;
        end;
        return y( a );  # y refers to local y of function x
    end; 

It is important to note that the concept of a variable in GAP is quite different from the concept of a variable in programming languages like PASCAL. In those languages a variable denotes a block of memory. The value of the variable is stored in this block. So in those languages two variables can have the same value, but they can never have identical values, because they denote different blocks of memory. (Note that PASCAL has the concept of a reference argument. It seems as if such an argument and the variable used in the actual function call have the same value, since changing the argument's value also changes the value of the variable used in the actual function call. But this is not so; the reference argument is actually a pointer to the variable used in the actual function call, and it is the compiler that inserts enough magic to make the pointer invisible.) In order for this to work the compiler needs enough information to compute the amount of memory needed for each variable in a program, which is readily available in the declarations PASCAL requires for every variable.

In GAP on the other hand each variable justs points to a value.

2.8 Function Calls

function-var()
function-var( arg-expr {, arg-expr} )

The function call has the effect of calling the function function-var. The precise semantics are as follows.

First GAP evaluates the function-var. Usually function-var is a variable, and GAP does nothing more than taking the value of this variable. It is allowed though that function-var is a more complex expression, namely it can for example be a selection of a list element list-var[int-expr], or a selection of a record component record-var.ident. In any case GAP tests whether the value is a function. If it is not, GAP signals an error.

Next GAP checks that the number of actual arguments arg-exprs agrees with the number of formal arguments as given in the function definition. If they do not agree GAP signals an error. An exception is the case when there is exactly one formal argument with the name arg, in which case any number of actual arguments is allowed.

Now GAP allocates for each formal argument and for each formal local a new variable. Remember that a variable is a location in a GAP program that points to a value. Thus for each formal argument and for each formal local such a location is allocated.

Next the arguments arg-exprs are evaluated, and the values are assigned to the newly created variables corresponding to the formal arguments. Of course the first value is assigned to the new variable corresponding to the first formal argument, the second value is assigned to the new variable corresponding to the second formal argument, and so on. However, GAP does not make any guarantee about the order in which the arguments are evaluated. They might be evaluated left to right, right to left, or in any other order, but each argument is evaluated once. An exception again occurs if the function has only one formal argument with the name arg. In this case the values of all the actual arguments are stored in a list and this list is assigned to the new variable corresponding to the formal argument arg.

The new variables corresponding to the formal locals are initially not bound to any value. So trying to evaluate those variables before something has been assigned to them will signal an error.

Now the body of the function, which is a statement, is executed. If the identifier of one of the formal arguments or formal locals appears in the body of the function it refers to the new variable that was allocated for this formal argument or formal local, and evaluates to the value of this variable.

If during the execution of the body of the function a return statement with an expression (see Return) is executed, execution of the body is terminated and the value of the function call is the value of the expression of the return. If during the execution of the body a return statement without an expression is executed, execution of the body is terminated and the function call does not produce a value, in which case we call this call a procedure call (see Procedure Calls). If the execution of the body completes without execution of a return statement, the function call again produces no value, and again we talk about a procedure call.

    gap> Fibonacci( 11 );
        # a call to the function 'Fibonacci' with actual argument '11'
    89 
    gap> G.operations.RightCosets( G, Intersection( U, V ) );;
        # a call to the function in 'G.operations.RightCosets'
        # where the second actual argument is another function call 

2.9 Comparisons

left-expr = right-expr
left-expr < right-expr

The operator = tests for equality of its two operands and evaluates to true if they are equal and to false otherwise. Likewise < tests for inequality of its two operands. Note that any two objects can be compared, i.e., = and < will never signal an error. For each type of objects the definition of equality is given in the respective chapter. Objects of different types are never equal, i.e., = evaluates in this case to false, and < evaluates to true.

left-expr < right-expr
left-expr right-expr
left-expr <= right-expr
left-expr = right-expr

< denotes less than, <= less than or equal, greater than, and = greater than or equal of its two operands. For each type of objects the definition of the ordering is given in the respective chapter. The ordering of objects of different types is as follows. Rationals are smallest, next are cyclotomics, followed by finite field elements, permutations, words, words in solvable groups, boolean values, functions, lists, and records are largest.

Comparison operators, which includes the operator in (see In) are not associative, i.e., it is not allowed to write a = b < c = d, you must use (a = b) < (c = d) instead. The comparison operators have higher precedence than the logical operators (see Operations for Booleans), but lower precedence than the arithmetic operators (see Operations). Thus, for example, a * b = c and d is interpreted, ((a * b) = c) and d).

    gap> 2 * 2 + 9 = Fibonacci(7);    # a comparison where the left
    true                              # operand is an expression 

2.10 Operations

+ right-expr
- right-expr
left-expr + right-expr
left-expr - right-expr
left-expr * right-expr
left-expr / right-expr
left-expr mod right-expr
left-expr ^ right-expr

The arithmetic operators are +, -, *, /, mod, and ^. The meanings (semantic) of those operators generally depend on the types of the operands involved, and they are defined in the various chapters describing the types. However basically the meanings are as follows.

+ denotes the addition, and - the subtraction of ring and field elements. * is the multiplication of group elements, / is the multiplication of the left operand with the inverse of the right operand. mod is only defined for integers and rationals and denotes the modulo operation. + and - can also be used as unary operations. The unary + is ignored and unary - is equivalent to multiplication by -1. ^ denotes powering of a group element if the right operand is an integer, and is also used to denote operation if the right operand is a group element.

The precedence of those operators is as follows. The powering operator ^ has the highest precedence, followed by the unary operators + and -, which are followed by the multiplicative operators *, /, and mod, and the additive binary operators + and - have the lowest precedence. That means that the expression -2 ^ -2 * 3 + 1 is interpreted as (-(2 ^ (-2)) * 3) + 1. If in doubt use parentheses to clarify your intention.

The associativity of the arithmetic operators is as follows.^ is not associative, i.e., it is illegal to write 2^3^4, use parentheses to clarify whether you mean (2^3) ^ 4 or 2 ^ (3^4). The unary operators + and - are right associative, because they are written to the left of their operands. *, /, mod, +, and - are all left associative, i.e., 1-2-3 is interpreted as (1-2)-3 not as 1-(2-3). Again, if in doubt use parentheses to clarify your intentions.

The arithmetic operators have higher precedence than the comparison operators (see Comparisons and In) and the logical operators (see Operations for Booleans). Thus, for example, a * b = c and d is interpreted, ((a * b) = c) and d.

    gap> 2 * 2 + 9;    # a very simple arithmetic expression
    13 

2.11 Statements

Assignments (see Assignments), Procedure calls (see Procedure Calls), if statements (see If), while (see While), repeat (see Repeat) and for loops (see For), and the return statement (see Return) are called statements. They can be entered interactively or be part of a function definition. Every statement must be terminated by a semicolon.

Statements, unlike expressions, have no value. They are executed only to produce an effect. For example an assignment has the effect of assigning a value to a variable, a for loop has the effect of executing a statement sequence for all elements in a list and so on. We will talk about evaluation of expressions but about execution of statements to emphasize this difference.

It is possible to use expressions as statements. However this does cause a warning.

    gap> if i <> 0  then  k = 16/i;  fi;
    Syntax error: warning, this statement has no effect
    if i <> 0  then  k = 16/i;  fi;
                             ^ 

As you can see from the example this is useful for those users who are used to languages where = instead of := denotes assignment.

A sequence of one or more statements is a statement sequence, and may occur everywhere instead of a single statement. There is nothing like PASCAL's BEGIN-END, instead each construct is terminated by a keyword. The most simple statement sequence is a single semicolon, which can be used as an empty statement sequence.

2.12 Assignments

var := expr;

The assignment has the effect of assigning the value of the expressions expr to the variable var.

The variable var may be an ordinary variable (see Variables), a list element selection list-var[int-expr] (see List Assignment) or a Record Assignment). Since a list element or a record component may itself be a list or a record the left hand side of an assignment may be arbitrarily complex.

Note that variables do not have a type. Thus any value may be assigned to any variable. For example a variable with an integer value may be assigned a permutation or a list or anything else.

If the expression expr is a function call then this function must return a value. If the function does not return a value an error is signalled and you enter a break loop (see Break Loops). As usual you can leave the break loop with quit;. If you enter return return-expr; the value of the expression return-expr is assigned to the variable, and execution continues after the assignment.

    gap> S6 := rec( size := 720 );; S6;
    rec(
      size := 720 )
    gap> S6.generators := [ (1,2), (1,2,3,4,5) ];; S6;
    rec(
      size := 720,
      generators := [ (1,2), (1,2,3,4,5) ] )
    gap> S6.generators[2] := (1,2,3,4,5,6);; S6;
    rec(
      size := 720,
      generators := [ (1,2), (1,2,3,4,5,6) ] ) 

2.13 Procedure Calls

procedure-var();
procedure-var( arg-expr {, arg-expr} );

The procedure call has the effect of calling the procedure procedure-var. A procedure call is done exactly like a function call (see Function Calls). The distinction between functions and procedures is only for the sake of the discussion, GAP does not distinguish between them.

A function does return a value but does not produce a side effect. As a convention the name of a function is a noun, denoting what the function returns, e.g., Length, Concatenation and Order.

A procedure is a function that does not return a value but produces some effect. Procedures are called only for this effect. As a convention the name of a procedure is a verb, denoting what the procedure does, e.g., Print, Append and Sort.

    gap> Read( "myfile.g" );     # a call to the procedure Read
    gap> l := [ 1, 2 ];;
    gap> Append( l, [3,4,5] );    # a call to the procedure Append 

2.14 If

if bool-expr1 then statements1
{ elif bool-expr2 then statements2 }
[ else statements3 ]
fi;

The if statement allows one to execute statements depending on the value of some boolean expression. The execution is done as follows.

First the expression bool-expr1 following the if is evaluated. If it evaluates to true the statement sequence statements1 after the first then is executed, and the execution of the if statement is complete.

Otherwise the expressions bool-expr2 following the elif are evaluated in turn. There may be any number of elif parts, possibly none at all. As soon as an expression evaluates to true the corresponding statement sequence statements2 is executed and execution of the if statement is complete.

If the if expression and all, if any, elif expressions evaluate to false and there is an else part, which is optional, its statement sequence statements3 is executed and the execution of the if statement is complete. If there is no else part the if statement is complete without executing any statement sequence.

Since the if statement is terminated by the fi keyword there is no question where an else part belongs, i.e., GAP has no dangling else.
In if expr1 then if expr2 then stats1 else stats2 fi; fi;
the else part belongs to the second if statement, whereas in
if expr1 then if expr2 then stats1 fi; else stats2 fi;
the else part belongs to the first if statement.

Since an if statement is not an expression it is not possible to write

abs := if x > 0 then x; else -x; fi;

which would, even if legal syntax, be meaningless, since the if statement does not produce a value that could be assigned to abs.

If one expression evaluates neither to true nor to false an error is signalled and a break loop (see Break Loops) is entered. As usual you can leave the break loop with quit;. If you enter return true;, execution of the if statement continues as if the expression whose evaluation failed had evaluated to true. Likewise, if you enter return false;, execution of the if statement continues as if the expression whose evaluation failed had evaluated to false.

    gap> i := 10;;
    gap> if 0 < i  then
    >        s := 1;
    >    elif i < 0  then
    >        s := -1;
    >    else
    >        s := 0;
    >    fi;
    gap> s;
    1        # the sign of i 

2.15 While

while bool-expr do statements od;

The while loop executes the statement sequence statements while the condition bool-expr evaluates to true.

First bool-expr is evaluated. If it evaluates to false execution of the while loop terminates and the statement immediately following the while loop is executed next. Otherwise if it evaluates to true the statements are executed and the whole process begins again.

The difference between the while loop and the repeat until loop (see Repeat) is that the statements in the repeat until loop are executed at least once, while the statements in the while loop are not executed at all if bool-expr is false at the first iteration.

If bool-expr does not evaluate to true or false an error is signalled and a break loop (see Break Loops) is entered. As usual you can leave the break loop with quit;. If you enter return false;, execution continues with the next statement immediately following the while loop. If you enter return true;, execution continues at statements, after which the next evaluation of bool-expr may cause another error.

    gap> i := 0;;  s := 0;;
    gap> while s <= 200  do
    >        i := i + 1;  s := s + i^2;
    >    od;
    gap> s;
    204        # first sum of the first i squares larger than 200 

2.16 Repeat

repeat statements until bool-expr;

The repeat loop executes the statement sequence statements until the condition bool-expr evaluates to true.

First statements are executed. Then bool-expr is evaluated. If it evaluates to true the repeat loop terminates and the statement immediately following the repeat loop is executed next. Otherwise if it evaluates to false the whole process begins again with the execution of the statements.

The difference between the while loop (see While) and the repeat until loop is that the statements in the repeat until loop are executed at least once, while the statements in the while loop are not executed at all if bool-expr is false at the first iteration.

If bool-expr does not evaluate to true or false a error is signalled and a break loop (see Break Loops) is entered. As usual you can leave the break loop with quit;. If you enter return true;, execution continues with the next statement immediately following the repeat loop. If you enter return false;, execution continues at statements, after which the next evaluation of bool-expr may cause another error.

    gap> i := 0;;  s := 0;;
    gap> repeat
    >        i := i + 1;  s := s + i^2;
    >    until s > 200;
    gap> s;
    204        # first sum of the first i squares larger than 200 

2.17 For

for simple-var in list-expr do statements od;

The for loop executes the statement sequence statements for every element of the list list-expr.

The statement sequence statements is first executed with simple-var bound to the first element of the list list, then with simple-var bound to the second element of list and so on. simple-var must be a simple variable, it must not be a list element selection list-var[int-expr] or a record component selection record-var.ident.

The execution of the for loop is exactly equivalent to the while loop

loop-list := list

;
    
loop-index
 := 1;
    while 
loop-index <= Length(loop-list
) do
        
variable := loop-list[loop-index
];
        
statements
        
loop-index := loop-index
 + 1;
    od; 

with the exception that loop-list and loop-index are different variables for each for loop that do not interfere with each other.

The list list is very often a range.
for variable in [from..to] do statements od;
corresponds to the more common
for variable from from to to do statements od;
in other programming languages.

    gap> s := 0;;
    gap> for i  in [1..100]  do
    >        s := s + i;
    > od;
    gap> s;
    5050 

Note in the following example how the modification of the list in the loop body causes the loop body also to be executed for the new values

    gap> l := [ 1, 2, 3, 4, 5, 6 ];;
    gap> for i  in l  do
    >        Print( i, " " );
    >        if i mod 2 = 0  then Add( l, 3 * i / 2 );  fi;
    > od;  Print( "\n" );
    1 2 3 4 5 6 3 6 9 9
    gap> l;
    [ 1, 2, 3, 4, 5, 6, 3, 6, 9, 9 ] 

Note in the following example that the modification of the variable that holds the list has no influence on the loop

    gap> l := [ 1, 2, 3, 4, 5, 6 ];;
    gap> for i  in l  do
    >        Print( i, " " );
    >        l := [];
    > od;  Print( "\n" );
    1 2 3 4 5 6
    gap> l;
    [  ] 

2.18 Functions

function ( [ arg-ident {, arg-ident} ]

)
    
[ local loc-ident {, loc-ident} ; ]
    
statements
end

A function is in fact a literal and not a statement. Such a function literal can be assigned to a variable or to a list element or a record Function Calls.

The following is an example of a function definition. It is a function to compute values of the Fibonacci sequence (see Fibonacci)

    gap> fib := function ( n )
    >         local  f1,  f2,  f3,  i;
    >         f1 := 1;  f2 := 1;
    >         for i  in [3..n]  do
    >             f3 := f1 + f2;
    >             f1 := f2;
    >             f2 := f3;
    >         od;
    >         return f2;
    >     end;;
    gap> List( [1..10], fib );
    [ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ] 

Because for each of the formal arguments arg-ident and for each of the formal locals loc-ident a new variable is allocated when the function is called (see Function Calls), it is possible that a function calls itself. This is usually called recursion. The following is a recursive function that computes values of the Fibonacci sequence

    gap> fib := function ( n )
    >         if n < 3  then
    >             return 1;
    >         else
    >             return fib(n-1) + fib(n-2);
    >         fi;
    >     end;;
    gap> List( [1..10], fib );
    [ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ] 

Note that the recursive version needs 2 * fib(n)-1 steps to compute fib(n), while the iterative version of fib needs only n-2 steps. Both are not optimal however, the library function Fibonacci only needs on the order of Log(n) steps.

arg-ident - expr

This is a shorthand for
function ( arg-ident ) return expr; end.
arg-ident must be a single identifier, i.e., it is not possible to write functions of several arguments this way. Also arg is not treated specially, so it is also impossible to write functions that take a variable number of arguments this way.

The following is an example of a typical use of such a function

    gap> Sum( List( [1..100], x -> x^2 ) );
    338350 

When a function fun1 definition is evaluated inside another function fun2, GAP binds all the identifiers inside the function fun1 that are identifiers of an argument or a local of fun2 to the corresponding variable. This set of bindings is called the environment of the function fun1. When fun1 is called, its body is executed in this environment. The following implementation of a simple stack uses this. Values can be pushed onto the stack and then later be popped off again. The interesting thing here is that the functions push and pop in the record returned by Stack access the local variable stack of Stack. When Stack is called a new variable for the identifier stack is created. When the function definitions of push and pop are then evaluated (as part of the return statement) each reference to stack is bound to this new variable. Note also that the two stacks A and B do not interfere, because each call of Stack creates a new variable for stack.

    gap> Stack := function ()
    >         local   stack;
    >         stack := [];
    >         return rec(
    >             push := function ( value )
    >                 Add( stack, value );
    >             end,
    >             pop := function ()
    >                 local   value;
    >                 value := stack[Length(stack)];
    >                 Unbind( stack[Length(stack)] );
    >                 return value;
    >             end
    >         );
    >    end;;
    gap> A := Stack();;
    gap> B := Stack();;
    gap> A.push( 1 );  A.push( 2 );  A.push( 3 );
    gap> B.push( 4 );  B.push( 5 );  B.push( 6 );
    gap> A.pop();  A.pop();  A.pop();
    3
    2
    1
    gap> B.pop();  B.pop();  B.pop();
    6
    5
    4 

This feature should be used rarely, since its implementation in GAP is not very efficient.

2.19 Return

return;

In this form return terminates the call of the innermost function that is currently executing, and control returns to the calling function. An error is signalled if no function is currently executing. No value is returned by the function.

return expr;

In this form return terminates the call of the innermost function that is currently executing, and returns the value of the expression expr. Control returns to the calling function. An error is signalled if no function is currently executing.

Both statements can also be used in break loops (see Break Loops). return; has the effect that the computation continues where it was interrupted by an error or the user hitting ctrC. return expr; can be used to continue execution after an error. What happens with the value expr depends on the particular error.

2.20 The Syntax in BNF

This section contains the definition of the GAP syntax in Backus-Naur form.

A BNF is a set of rules, whose left side is the name of a syntactical construct. Those names are enclosed in angle brackets and written in italics. The right side of each rule contains a possible form for that syntactic construct. Each right side may contain names of other syntactic constructs, again enclosed in angle brackets and written in italics, or character sequences that must occur literally; they are written in typewriter style.

Furthermore each righthand side can contain the following metasymbols written in boldface. If the right hand side contains forms separated by a pipe symbol (|) this means that one of the possible forms can occur. If a part of a form is enclosed in square brackets ([ ]) this means that this part is optional, i.e. might be present or missing. If part of the form is enclosed in curly braces ({ }) this means that the part may occur arbitrarily often, or possibly be missing.

Permutation := Expr kill Ident := a|...|z|A|...|Z|_ {a|...|z|A|...|Z|0|...|9|_}
Var := Ident
| Var . Ident
| Var . ( Expr )
| Var [ Expr ]
| Var { Expr }
| Var ( [ Expr { , Expr } ] )
List := [ [ Expr ] {, [ Expr ] } ]
| [ Expr [, Expr ] .. Expr ]
Record := rec( [ Ident := Expr {, Ident := Expr } ] )
Permutation := ( Expr {, Expr } ) { ( Expr {, Expr } ) }
Function := function ( [ Ident {, Ident } ] )
[ local Ident {, Ident } ; ]
Statements
end
Char := ' any character '
String := " { any character } "
Int := 0|1|...|9 { 0|1|...|9 }
Atom := Int
| Var
| ( Expr )
| Permutation
| Char
| String
| Function
| List
| Record
Factor := {+|-} Atom [ ^ {+|-} Atom ]
Term := Factor { *|/|mod Factor }
Arith := Term { +|- Term }
Rel := { not } Arith { =|<>|<|>|<=|>=|in Arith }
And := Rel { and Rel }
Log := And { or And }
Expr := Log
| Var [ - Log ]
Statement := Expr
| Var := Expr
| if Expr then Statements
{ elif Expr then Statements }
[ else Statements ] fi
| for Var in Expr do Statements od
| while Expr do Statements od
| repeat Statements until Expr
| return [ Expr ]
| quit
Statements := { Statement ; }
| ;

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GAP 3.4.4
April 1997