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 for 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 from those values according to rules given below. Those rules are usually called the semantics of a programming language.
The result of the evaluation is, not surprisingly, called a 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, Arithmetic Operators, Statements, Assignments, Procedure Calls, If, While, Repeat, For, Function, and the chapters describing the individual data types.
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 (see section Symbols and chapter Strings and Characters) and comments (see Whitespaces) 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 also that during the process of scanning 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
whitespace characters are equivalent to a single whitespace.
Apart from the role as separator of symbols,
whitespace characters are otherwise insignificant.
Whitespace characters may also occur inside a string,
where they are significant.
Whitespace characters 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 #
loses 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 additive 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 QUIT break rec continue
Note that (almost) all keywords are written in lowercase and that they
are case sensitive. 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
.
Note: A number of tokens that appear to be normal identifiers representing functions or literals of various kinds are actually implemented as keywords for technical reasons. The only consequence of this is that those identifiers cannot be re-assigned, and do not actually have function objects bound to them, which could be assigned to other variables or passed to functions. These keywords are:
false true IsBound Unbind TryNextMethod Info Assert SaveWorkspace fail
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 identifiers.
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.
IsValidIdentifier(
str ) F
returns true
if the string str would form a valid identifier
consisting of letters, digits and underscores; otherwise it returns
false
. It does not check whether str contains characters escaped by a
backslash \
.
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 Membership Test for Collections).
The arithmetic operators are +
, -
, *
,
/
, mod
, and ^
(see Arithmetic Operators).
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
For the precedence of operators, see Comparisons.
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 is the class of 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
Function). 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 just points to a value, and different variables can share the same value.
Unbind(
ident ) F
deletes the identifier ident. If there is no other variable pointing to
the same value as ident was, this value will be removed by the next
garbage collection. Therefore Unbind
can be used to get rid of unwanted
large objects.
For records and lists Unbind
can be used to delete components or entries,
respectively (see Chapters Records and Lists).
4.9 More About Global Variables
The vast majority of variables in GAP are defined at the outer level (the global scope). They are used to access functions and other objects created either in the GAP library or in the user's code. Certain special facilities are provided for manipulating these variables which are not available for other types of variable (such as local variables or function arguments).
First, such variables may be marked read-only. In which case attempts to change them will fail. Most of the global variables defined in the GAP library are so marked.
IsReadOnlyGlobal(
name ) F
returns true
if the global variable named by the string name is
read-only and false
otherwise (the default).
MakeReadOnlyGlobal(
name ) F
marks the global variable named by the string name as read-only.
A warning is given if name has no value bound to it or if it is already read-only.
MakeReadWriteGlobal(
name ) F
marks the global variable named by the string name as read-write.
A warning is given if name is already read-write.
gap> xx := 17; 17 gap> IsReadOnlyGlobal("xx"); false gap> xx := 15; 15 gap> MakeReadOnlyGlobal("xx"); gap> xx := 16; Variable: 'xx' is read only not in any function Entering break read-eval-print loop ... you can 'quit;' to quit to outer loop, or you can return after making it writable to continue brk> quit; gap> IsReadOnlyGlobal("xx"); true gap> MakeReadWriteGlobal("xx"); gap> xx := 16; 16 gap> IsReadOnlyGlobal("xx"); false
A group of functions are also supplied for accessing and altering the
values assigned to global variables. Use of these functions differs
from the use of assignment, Unbind
and IsBound
statements, in two
ways.
First, these functions always affect global variables, even if
local variables of the same names exist.
Second, the variable names are passed as strings,
rather than being written directly into the statements.
ValueGlobal(
name ) F
returns the value currently bound to the global variable named by the string name. An error is raised if no value is currently bound.
IsBoundGlobal(
name ) F
returns true
if a value currently bound
to the global variable named by the string name and false
otherwise.
UnbindGlobal(
name ) F
removes any value currently bound to the global variable named by the string name. Nothing is returned.
A warning is given if name was not bound. The global variable named by name must be writable, otherwise an error is raised.
BindGlobal(
name,
val ) F
sets the global variable named by the string name to the value val, provided it is writable, and makes it read-only. If name already has a value, a warning message is printed.
This is intended to be the normal way to create and set ``official'' global variables (such as Operations and Categories).
Caution should be exercised in using these functions, especially
BindGlobal
and UnbindGlobal
as unexpected changes in global
variables can be very confusing for the user.
gap> xx := 16; 16 gap> IsReadOnlyGlobal("xx"); false gap> ValueGlobal("xx"); 16 gap> IsBoundGlobal("xx"); true gap> BindGlobal("xx",17); #W BIND_GLOBAL: variable `xx' already has a value gap> xx; 17 gap> IsReadOnlyGlobal("xx"); true
Finally, there are a group of functions dealing with the global namespace.
NamesGVars() F
This function returns an immutable (see Mutability and Copyability) sorted (see Sorted Lists and Sets) list of all the global variable names known to the system. This includes names of variables which were bound but have now been unbound and some other names which have never been bound but have become known to the system by various routes.
NamesSystemGVars() F
This function returns an immutable sorted list of all the global variable names created by the GAP library when GAP was started.
NamesUserGVars() F
This function returns an immutable sorted list of the global variable names created since the library was read, to which a value is currently bound.
TemporaryGlobalVarName( [
prefix] ) F
returns a string that can be used
as the name of a global variable that is not bound at the time when
TemporaryGlobalVarName()
is called. The optional argument prefix can
specify a string with which the name of the global variable starts.
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,
such as a reference to an element of a list (see Chapter Lists)
list-var
[
int-expr]
,
or to a component of a record (see Chapter Records) 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 (see function for
examples).
Now GAP allocates for each formal argument and for each formal local
(that is, the identifiers in the local
declaration) 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> RightCosets( G, Intersection( U, V ) );; # a call to the operation `RightCosets' # where the second actual argument is another function call
function-var(
arg-expr[,
arg-expr, ...][ : [
option-expr [,
option-expr, ....]]])
As well as passing arguments to a function, providing the mathematical input to its calculation, it is sometimes useful to supply ``hints'' suggesting to GAP how the desired result may be computed more quickly, or specifying a level of tolerance for random errors in a Monte Carlo algorithm.
Such hints may be supplied to a function-call and to all subsidiary
functions called from that call using the options mechanism. Options
are separated from the actual arguments by a colon :
and have much
the same syntax as the components of a record expression. The one
exception to this is that a component name may appear without a value,
in which case the value true
is silently inserted.
gap> Size( fpgrp : hard, tcselection := "external" ); # a call to `Size' passing the options `hard' (value `true') and # `tcselection' (value the string "external")
Options supplied with function calls in this way are passed down using the global options stack described in chapter Options Stack, so that the call above is exactly equivalent to
gap> PushOptions( rec( hard := true, tcselection := "external") ); gap> Size( fpgrp ); gap> PopOptions( );
Note that any option may be passed with any function, whether or not it has any actual meaning for that function, or any function called by it. The system provides no safeguard against misspelled option names.
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 in different families (see Families) 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 kind of objects the definition of the ordering is given in the
respective chapter.
Only for the following kinds of objects, an ordering via <
of objects
in different families (see Families) is supported.
Rationals (see IsRat) are smallest,
next are cyclotomics (see IsCyclotomic),
followed by finite field elements (see IsFFE);
finite field elements in different characteristics are compared
via their characteristics,
next are permutations (see IsPerm),
followed by the boolean values true
, false
, and fail
(see IsBool),
characters (such as 'a'
, see IsChar),
and lists (see IsList) are largest;
note that two lists can be compared with <
if and only if their
elements are again objects that can be compared with <
.
For other objects, GAP does not provide an ordering via <
.
The reason for this is that a total ordering of all GAP objects
would be hard to maintain when new kinds of objects are introduced,
and such a total ordering is hardly used in its full generality.
However, for objects in the filters listed above, the ordering via <
has turned out to be useful.
For example, one can form sorted lists containing integers and nested
lists of integers, and then search in them using PositionSorted
(see Finding Positions in Lists).
Of course it would in principle be possible to define an ordering
via <
also for certain other objects,
by installing appropriate methods for the operation \<
.
But this may lead to problems at least as soon as one loads GAP code
in which the same is done, under the assumption that one is completely
free to define an ordering via <
for other objects than the ones
for which the ``official'' GAP provides already an ordering via <
.
Comparison operators, including the operator in
(see Membership Test for Lists),
are not associative,
Hence 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 Arithmetic Operators).
Thus, for example, a
*
b =
c and
d is interpreted as
((
a *
b) =
c) and
d)
.
gap> 2 * 2 + 9 = Fibonacci(7); # a comparison where the left true # operand is an expression
For the underlying operations of the operators introduced above, see Comparison Operations for Elements.
+
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 (semantics) of those operators generally depend on the types
of the operands involved, and, except for mod
, they are defined in the
various chapters describing the types. However basically the meanings are
as follows.
a
+
b denotes the addition of additive elements a and b.
a
-
b denotes the addition of a and the additive inverse of b.
a
*
b denotes the multiplication of multiplicative elements a and
b.
a
/
b denotes the multiplication of a with the multiplicative
inverse of b.
a
mod
b, for integer or rational left operand a and for non-zero
integer right operand b, is defined as follows.
If a and b are both integers,
a
mod
b is the integer r in the
integer range
0 .. |
b| - 1
satisfying a
=
r +
bq
,
for some integer q (where the operations occurring have their usual meaning
over the integers, of course).
If a is a rational number and b is a non-zero integer, and a
=
m
/
n where m and n are coprime integers with n positive, then
a
mod
b is the integer r in the integer range
0 .. |
b| - 1
such that m is congruent to r
n
modulo b, and r is called the
``modular remainder'' of a modulo b. Also,
1 /
n mod
b is
called the ``modular inverse'' of n modulo b. (A pair of integers is
said to be coprime (or relatively prime) if their gcd is 1.)
With the above definition, 4 / 6 mod 32
equals 2 / 3 mod 32
and hence
exists (and is equal to 22), despite the fact that 6 has no inverse
modulo 32.
Note.
For rational a, a
mod
b could have been defined to be the
non-negative rational c less than
|
b|
such that a
-
c is a
multiple of b. However this definition is seldom useful and not the
one chosen for GAP.
+
and -
can also be used as unary operations.
The unary +
is ignored. The unary -
returns the additive inverse of
its operand; over the integers it is equivalent to multiplication by -1
.
^
denotes powering of a multiplicative element if the right operand is
an integer, and is also used to denote the action of a group element on a
point of a set 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 Membership Test for Collections)
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
For other arithmetic operations, and for the underlying operations of the operators introduced above, see Arithmetic Operations for Elements.
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.
Using expressions as statements is treated as syntax error.
gap> if i <> 0 then k = 16/i; fi; Syntax error: := expected if i <> 0 then k = 16/i; fi; ^ gap>
As you can see from the example this warning does in particular address
those users who are used to languages where =
instead of :=
denotes
assignment.
Empty statements are permitted and have no effect.
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 simplest statement sequence is a single semicolon, which can be
used as an empty statement sequence. In fact an empty statement
sequence as in for i in [1..2] do od
is also permitted and is
silently translated into the sequence containing just a semicolon.
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.
gap> data:= rec( numbers:= [ 1, 2, 3 ] ); rec( numbers := [ 1, 2, 3 ] ) gap> data.string:= "string";; data; rec( numbers := [ 1, 2, 3 ], string := "string" ) gap> data.numbers[2]:= 4;; data; rec( numbers := [ 1, 4, 3 ], string := "string" )
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> f1:= function( x ) Print( "value: ", x, "\n" ); end;; gap> f2:= function( x ) return f1( x ); end;; gap> f2( 4 ); value: 4 Function Calls: <func> must return a value at return f1( x ); <function>( <arguments> ) called from read-eval-loop Entering break read-eval-print loop, you can 'quit;' to quit to outer loop, or you can return a value for the result to continue brk> return "hello"; "hello"
In the above example, the function f2
calls f1
with argument 4
,
and since f1
does not return a value (but only prints a line ``value:
x''), the
return
statement of f2
cannot be executed.
The error message says that it is possible to return an appropriate value,
and the returned string "hello"
is used by f2
instead of the missing
return value of f1
.
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. So we state the following conventions.
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
There are a few exceptions of GAP functions that do both return
a value and produce some effect.
An example is Sortex
which sorts a list and returns the corresponding
permutation of the entries (see Sortex).
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 of the expressions bool-expr1, bool-expr2 is evaluated
and its value is neither true
nor 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 # sum of the first i squares larger than 200
A while
loop may be left prematurely using break
, see Break.
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
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 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 # sum of the first i squares larger than 200
A repeat
loop may be left prematurely using break
, see Break.
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-expr, then with simple-var
bound to the second element of list-expr 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 over a list is exactly equivalent to
the following 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,
i.e., these variables of different for
loops do not interfere with
each other.
The list list-expr is very often a range (see Ranges).
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; [ ]
for
variable in
iterator do
statements od;
It is also possible to have a for
-loop run over an iterator
(see Iterators). In this case
the for
-loop is equivalent to
while not IsDoneIterator(
iterator) do
variable := NextIterator(
iterator)
statements
od;
for
variable in
object do
statements od;
Finally, if an object object which is not a list or an iterator appears in a
for
-loop, then GAP will attempt to evaluate the function call
Iterator(
object)
. If this is successful then the loop is taken to
run over the iterator returned.
gap> g := Group((1,2,3,4,5),(1,2)(3,4)(5,6)); Group([ (1,2,3,4,5), (1,2)(3,4)(5,6) ]) gap> count := 0;; sumord := 0;; gap> for x in g do > count := count + 1; sumord := sumord + Order(x); od; gap> count; 120 gap> sumord; 471
The effect of
for
variable in
domain do
should thus normally be the same as
for
variable in AsList(
domain) do
but may use much less storage, as the iterator may be more compact than a list of all the elements.
See Iterators for details about iterators.
A for
loop may be left prematurely using break
, see Break. This
combines especially well with a loop over an iterator, as a way of
searching through a domain for an element with some useful property.
break;
The statement break;
causes an immediate exit from the innermost
loop enclosing it. It is an error to use this statement other than
inside a loop.
gap> g := Group((1,2,3,4,5),(1,2)(3,4)(5,6)); Group([ (1,2,3,4,5), (1,2)(3,4)(5,6) ]) gap> for x in g do > if Order(x) = 3 then > break; > fi; od; gap> x; (1,4,3)(2,6,5)
gap> break; A break statement can only appear inside a loop
continue;
The statement continue;
causes the rest of the current iteration of
the innermost loop enclosing it to be skipped. The next iteration
begins immediately. It is an error to use this statement other than
inside a loop.
gap> g := Group((1,2,3),(1,2)); Group([ (1,2,3), (1,2) ]) gap> for x in g do > if Order(x) = 3 then > continue; > fi; Print(x,"\n"); od; () (2,3) (1,3) (1,2)
gap> continue; A continue statement can only appear inside a loop
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 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 about Log(
n)
steps.
As noted in Section Function Calls, the case where a function is
defined with exactly one formal argument with the name arg
, is special.
It provides a way of defining a function with a variable number of
arguments; 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
. There are two typical scenarios for wanting such a
possibility: having optional arguments and having any number of
arguments.
The following example shows one way that the function Position
(see Position) might be encoded and demonstrates the ``optional
argument'' scenario.
gap> position := function ( arg ) > local list, obj, pos; > list := arg[1]; > obj := arg[2]; > if 2 = Length(arg) then > pos := 0; > else > pos := arg[3]; > fi; > repeat > pos := pos + 1; > if pos > Length(list) then > return fail; > fi; > until list[pos] = obj; > return pos; > end; function( arg ) ... end gap> position([1, 4, 2], 4); 2 gap> position([1, 4, 2], 3); fail gap> position([1, 4, 2], 4, 2); fail
The following example demonstrates the ``any number of arguments'' scenario.
gap> sum := function ( arg ) > local total, x; > total := 0; > for x in arg do > total := total + x; > od; > return total; > end; function( arg ) ... end gap> sum(1, 2, 3); 6 gap> sum(1, 2, 3, 4); 10 gap> sum(); 0
The user should compare the above with the GAP function Sum
(see Sum) which, for example, may take a list argument and optionally
an initial element (which zero should the sum of an empty list return?).
Note that if a function f is defined as above with the single formal
argument arg
then NumberArgumentsFunction(
f)
returns -1
(see NumberArgumentsFunction).
The argument arg
when used as the single argument name of some function
f tells GAP that when it encounters f that it should form a list
out of the arguments of f. What if one wishes to do the ``opposite'':
tell GAP that a list should be ``unwrapped'' and passed as several
arguments to a function. The function CallFuncList
(see CallFuncList)
is provided for this purpose.
Also see Chapter Functions.
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 the definition of a function fun1 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.
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 ctr-C
. return
expr;
can be used to continue execution after an error. What happens with the
value expr depends on the particular error.
For examples of return
statements, see the functions fib
and Stack
in Chapter Functions.
This section contains the definition of the GAP syntax in Backus-Naur form. A few recent additions to the syntax may be missing from this definition. Also, the actual rules for identifier names implemented by the system, are somewhat more permissive than those given below (see section Identifiers).
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> {`,' <Expr> } `)' { `(' <Expr> {`,' <Expr> } `)' } |
<Ident> | := | `a'|...|`z'|`A'|...|`Z'|`_' {`a'|...|`z'|`A'|...|`Z'|`0'|...|`9'|`_'} |
<Var> | := | <Ident> |
| | <Var> `.' <Ident> |
| | <Var> `.' `(' <Expr> `)' |
| | <Var> `[' <Expr> `]' |
| | <Var> `{ <Expr> }' |
| | <Var> `(' [ <Expr> { ,<Expr> } ] `)' |
| | <Var> `!.' <Ident> |
| | <Var> `!.' `(' <Expr> `)' |
| | <Var> `![' <Expr> `]' |
<List> | := | `[' [ <Expr> ] {`,' [ <Expr> ] } `]' |
| | `[' <Expr> [, <Expr> ] `..' <Expr> `]' |
| | <List> `' <List> `' |
<Record> | := | `rec(' [ <Ident> `:=' <Expr> {`,' <Ident> `:=' <Expr> } ] `)' |
<Permutation> | := | `(' <Expr> {`,' <Expr> } `)' { `(' <Expr> {`,' <Expr> } `)' } |
<Function> | := | `function (' [ <Ident> {`,' <Ident> } ] `)' |
[ `local' <Ident> {`,' <Ident> } `;' ] |
<Statements> |
`end' |
| | <Ident> `->' <Expr> |
<Char> | := | '<any character> ' |
<String> | := | `"' { <any character> } `"' |
<Int> | := | `0'|`1'|...|`9' {`0'|`1'|...|`9'} |
<Atom> | := | <Int> |
| | <Var> |
| | `(' <Expr> `)' |
| | <Permutation> |
| | <Char> |
| | <String> |
| | <Function> |
| | <List> |
| | <Record> |
| | { `not' } `true' |
| | { `not' } `false' |
<Factor> | := | {`+'|`-'} <Atom> [ `^' {`+'|`-'} <Atom> ] |
<Term> | := | <Factor> { `*'|`/'|`mod' <Factor> } |
<Arith> | := | <Term> { `+'|`-' <Term> } |
<Rel> | := | { `not' } <Arith> [ `='|`\<>'|`\<'|`>'|`\<='|`>='|`in' <Arith> ] |
<And> | := | <Rel> { `and' <Rel> } |
<Logical> | := | <And> { `or' <And> } |
<Expr> | := | <Logical> |
| | <Var> |
<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> ] |
| | `break' |
| | `quit' |
| | `QUIT' |
| |
<Statements> | := | { <Statement> `;' } |
| | `;' |
| |
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GAP 4 manual
May 2002