This chapter describes the GAP3 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 GAP3, given as a string of characters. How those characters enter GAP3 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 GAP3 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;
GAP3 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 GAP3 first evaluates the
right subexpression 2*3. Again to do this GAP3 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, Language 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.
The input of GAP3 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.
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).
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.
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.
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.
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
A variable is a location in a GAP3 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, GAP3 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 GAP3 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 GAP3 on the other hand each variable justs points to a value.
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 GAP3 evaluates the function-var. Usually function-var is a
variable, and GAP3 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 GAP3 tests whether the value is a
function. If it is not, GAP3 signals an error.
Next GAP3 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 GAP3 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 GAP3 allocates for each formal argument and for each formal local a new variable. Remember that a variable is a location in a GAP3 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, GAP3 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
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
+ 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
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.
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 component selection record-var.ident (see 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) ] )
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, GAP3 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
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., GAP3 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
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
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
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;
[ ]
function ( [ arg-ident {, arg-ident} ]
)
[ local loc-ident {, loc-ident} ; ]
statementsend
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 component. Later this function can be called as described in 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, GAP3 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 GAP3 is not very efficient.
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.
This section contains the definition of the GAP3 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.
\newpage
beg-tabbing
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 ; }
>| >;
end-tabbing
gap3-jm