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 4.2, 4.3, 4.4, 4.5, and 4.6, the semantic rules are given in sections 4.7, 4.8, 4.11, 4.12, 4.13, 4.14, 4.15, 4.16, 4.17, 4.18, 4.19, 4.20, 4.23, and the chapters describing the individual data types.
Most input of GAP consists of sequences of the following characters.
Digits, uppercase and lowercase letters, Space, Tab, Newline, Return and the special characters
" ` ( ) * + , - #
. / : ; < = > ~
[ \ ] ^ _ { } !
It is possible to use other characters in identifiers by escaping them with backslashes, but we do not recommend to use this feature. Inside strings (see section 4.3 and chapter 27) and comments (see 4.4) 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 (see 4.5). An identifier is a sequence of letters, digits and underscores (or other characters escaped by backslashes) that contains at least one non-digit and is not a keyword (see 4.6). An integer is a sequence of digits (see 14), possibly prepended by - and + sign characters. A string is a sequence of arbitrary characters enclosed in double quotes (see 27).
Operator and delimiter symbols are
+ - * / ^ ~ !.
= <> < <= > >= ![
:= . .. -> , ; !{
[ ] { } ( ) :
Note also that during the process of scanning all whitespace is removed (see 4.4).
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 list of keywords is contained in the GAPInfo.Keywords component of the GAPInfo record (see 3.5-1). We will show how to print it in a nice table, demonstrating at the same time some list manipulation techniques:
gap> keys:=SortedList( GAPInfo.Keywords );; l:=Length( keys );; gap> arr:= List( [ 0 .. Int( l/4 )-1 ], i-> keys{ 4*i + [ 1 .. 4 ] } );; gap> if l mod 4 <> 0 then Add( arr, keys{[ 4*Int(l/4) + 1 .. l ]} ); fi; gap> Length( keys ); PrintArray( arr ); 35 [ [ Assert, Info, IsBound, QUIT ], [ TryNextMethod, Unbind, and, atomic ], [ break, continue, do, elif ], [ else, end, false, fi ], [ for, function, if, in ], [ local, mod, not, od ], [ or, quit, readonly, readwrite ], [ rec, repeat, return, then ], [ true, until, while ] ]
Note that (almost) all keywords are written in lowercase and that they are case sensitive. For example 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: Several tokens from the list of keywords above may appear to be normal identifiers representing functions or literals of various kinds but 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 true, false, Assert (7.5-3), IsBound (4.8-1), Unbind (4.8-2), Info (7.4-5) and TryNextMethod (78.4-1).
Keywords atomic, readonly, readwrite are not used at the moment. They are reserved for the future version of GAP to prevent their accidental use as identifiers.
An identifier is used to refer to a variable (see 4.8). An identifier usually consists of letters, digits, underscores _, and "at"-characters @, and must contain at least one non-digit. An identifier is terminated by the first character not in this class. Note that the "at"-character @ is used to implement namespaces, see Section 4.10 for details.
Examples of valid identifiers are
a foo aLongIdentifier hello Hello HELLO x100 100x _100 some_people_prefer_underscores_to_separate_words WePreferMixedCaseToSeparateWords abc@def
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 ) | ( function ) |
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 \.
Note that the "at"-character is used to implement namespaces for global variables in packages. See 4.10 for details.
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 4.14). Expressions appear as right hand sides of assignments (see 4.15), as actual arguments in function calls (see 4.11), 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 4.8, 4.11, 14, 42, 27, 4.23, 21, 29), 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 4.12 and 30.6). The arithmetic operators are +, -, *, /, mod, and ^ (see 4.13). The logical operators are not, and, and or (see 20.4).
The following example shows a very simple expression with value 4 and a more complex expression.
gap> 2 * 2; 4 gap> 2 * 2 + 9 = Fibonacci(7) and Fibonacci(13) in Primes; true
For the precedence of operators, see 4.12.
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 4.15). 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 4.11).
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 4.6). 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 4.23). 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 most compiled programming languages.
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 some languages have 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.
In GAP on the other hand each variable just points to a value, and different variables can share the same value.
‣ IsBound( ident ) | ( function ) |
IsBound returns true if the variable ident points to a value, and false otherwise.
For records and lists IsBound can be used to check whether components or entries, respectively, are bound (see Chapters 29 and 21).
‣ Unbind( ident ) | ( function ) |
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 29 and 21).
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 packages or in the user's code.
Note that for packages there is a mechanism to implement package local namespaces on top of this global namespace. See Section 4.10 for details.
Certain special facilities are provided for manipulating global 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.
Second, a group of functions are supplied for accessing and altering the values assigned to global variables. Use of these functions differs from the use of assignment, Unbind (4.8-2) and IsBound (4.8-1) 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.
Note that the functions NamesGVars (4.9-8), NamesSystemGVars (4.9-9), NamesUserGVars (4.9-10), and TemporaryGlobalVarName (4.9-11) deal with the global namespace.
‣ IsReadOnlyGlobal( name ) | ( function ) |
returns true if the global variable named by the string name is read-only and false otherwise (the default).
‣ MakeReadOnlyGlobal( name ) | ( function ) |
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 ) | ( function ) |
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
‣ ValueGlobal( name ) | ( function ) |
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 ) | ( function ) |
returns true if a value currently bound to the global variable named by the string name and false otherwise.
‣ UnbindGlobal( name ) | ( function ) |
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 ) | ( function ) |
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 had 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 filters).
Caution should be exercised in using these functions, especially BindGlobal and UnbindGlobal (4.9-6) 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 gap> MakeReadWriteGlobal("xx"); gap> Unbind(xx);
‣ NamesGVars( ) | ( function ) |
This function returns an immutable (see 12.6) sorted (see 21.19) 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( ) | ( function ) |
This function returns an immutable sorted list of all the global variable names created by the GAP library when GAP was started.
‣ NamesUserGVars( ) | ( function ) |
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] ) | ( function ) |
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.
As mentioned in Section 4.9 above all global variables share a common namespace. This can relatively easily lead to name clashes, in particular when many GAP packages are loaded at the same time. To give package code a way to have a package local namespace without breaking backward compatibility of the GAP language, the following simple rule has been devised:
If in package code a global variable that ends with an "at"-character @ is accessed in any way, the name of the package is appended before accessing it. Here, "package code" refers to everything which is read with ReadPackage (76.3-1). As the name of the package the entry PackageName in its PackageInfo.g file is taken. As for all identifiers, this name is case sensitive.
For example, if the following is done in the code of a package with name xYz:
gap> a@ := 12;
Then actually the global variable a@xYz is assigned. Further accesses to a@ within the package code will all be redirected to a@xYz. This includes all the functions described in Section 4.9 and indeed all the functions described Section 79.18 like for example DeclareCategory (79.18-1). Note that from code in the same package it is still possible to access the same global variable via a@xYz explicitly.
All other code outside the package as well as interactive user input that wants to refer to that variable a@xYz must do so explicitly by using a@xYz.
Since in earlier releases of GAP the "at"-character @ was not a legal character (without using backslashes), this small extension of the language does not break any old code.
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 21) list-var[int-expr], or to a component of a record (see Chapter 29) 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 the function has a variable length argument list, which is denoted by adding ... after the final argument. In this case there must be at least as many actual arguments as there are formal arguments before the final argument and can be any larger number (see 4.23 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 last formal argument has the name arg. In this case the values of all the actual arguments not assigned to the other formal parameters 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 4.24) 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 4.16). 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 ); 89
The above example shows a call to the function Fibonacci (16.3-1) with actual argument 11, the following one shows a call to the operation RightCosets (39.7-2) where the second actual argument is another function call.
gap> RightCosets( G, Intersection( U, V ) );;
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.
The following example shows a call to Size (30.4-6) passing the options hard (with the value true) and tcselection (with the string "external" as value).
gap> Size( fpgrp : hard, tcselection := "external" );
Options supplied with function calls in this way are passed down using the global options stack described in chapter 8, 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. For each type of objects the definition of equality is given in the respective chapter. Objects in different families (see 13.1) 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.
Note that < implements a total ordering of objects (which can be used for example to sort a list of elements). Therefore in general < will not be compatible with any inclusion relation (which can be tested using IsSubset (30.5-1)). (For example, it is possible to compare permutation groups with < in a total ordering of all permutation groups, but this ordering is not compatible with the relation of being a subgroup.)
Only for the following kinds of objects, an ordering via < of objects in different families (see 13.1) is supported. Rationals (see IsRat (17.2-1)) are smallest, next are cyclotomics (see IsCyclotomic (18.1-3)), followed by finite field elements (see IsFFE (59.1-1)); finite field elements in different characteristics are compared via their characteristics, next are permutations (see IsPerm (42.1-1)), followed by the boolean values true, false, and fail (see IsBool (20.1-1)), characters (such as {}a{'}', see IsChar (27.1-1)), and lists (see IsList (21.1-1)) 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 21.16).
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 21.8), 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 20.4), but lower precedence than the arithmetic operators (see 4.13). Thus, for instance, a * b = c and d is interpreted as ((a * b) = c) and d).
The following example shows a comparison where the left operand is an expression.
gap> 2 * 2 + 9 = Fibonacci(7); true
For the underlying operations of the operators introduced above, see 31.11.
+ 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 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 rn 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 greatest common divisor 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 invalid 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 4.12 and 30.6) and the logical operators (see 20.4). 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 31.12.
Assignments (see 4.15), Procedure calls (see 4.16), if statements (see 4.17), while (see 4.18), repeat (see 4.19) and for loops (see 4.20), and the return statement (see 4.24) 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> i := 7;; 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. 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 4.8), a list element selection list-var[int-expr] (see 21.4) or a record component selection record-var.ident (see 29.3). 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 6.4). 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 ); called from <function>( <arguments> ) called from read-eval-loop Entering break read-eval-print loop ... you can 'quit;' to quit to outer loop, or you can supply one by 'return <value>;' 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: ..."), 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( [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 4.11). 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 (21.18-3) which sorts a list and returns the corresponding permutation of the entries.
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 6.4) 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; # the sign of i 1
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 4.19) 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 6.4) 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.
The following example shows a while loop that sums up the squares 1^2, 2^2, ... until the sum exceeds 200.
gap> i := 0;; s := 0;; gap> while s <= 200 do > i := i + 1; s := s + i^2; > od; gap> s; 204
A while loop may be left prematurely using break, see 4.21.
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 4.18) 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 6.4) 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.
The repeat loop in the following example has the same purpose as the while loop in the preceding example, namely to sum up the squares 1^2, 2^2, ... until the sum exceeds 200.
gap> i := 0;; s := 0;; gap> repeat > i := i + 1; s := s + i^2; > until s > 200; gap> s; 204
A repeat loop may be left prematurely using break, see 4.21.
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 21.22).
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 30.8). 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 30.8 for details about iterators.
A for loop may be left prematurely using break, see 4.21. 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.
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; (1,5,2)(3,4,6)
It is an error to use this statement other than inside a loop.
gap> break; Error, A break statement can only appear inside a loop not in any function
continue;
The statement continue; causes the rest of the current iteration of the innermost loop enclosing it to be skipped.
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)
It is an error to use this statement other than inside a loop.
gap> continue; Error, A continue statement can only appear inside a loop not in any function
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 4.11.
The following is an example of a function definition. It is a function to compute values of the Fibonacci sequence (see Fibonacci (16.3-1)).
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 4.11), 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 (16.3-1) only needs about Log(n) steps.
As noted in Section 4.11, the case where a function's last argument is followed by ... is special. It provides a way of defining a function with a variable number of arguments. The values of the actual arguments are computed and the first ones are assigned to the new variables corresponding to the formal arguments before the last argument, if any. The values of all the remaining actual arguments are stored in a list and this list is assigned to the new variable corresponding to the final formal argument. 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 (21.16-1) might be encoded and demonstrates the "optional argument" scenario.
gap> position := function ( list, obj, arg... ) > local pos; > if 0 = Length(arg) then > pos := 0; > else > pos := arg[1]; > fi; > repeat > pos := pos + 1; > if pos > Length(list) then > return fail; > fi; > until list[pos] = obj; > return pos; > end; function( list, obj, 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 ( l... ) > local total, x; > total := 0; > for x in l do > total := total + x; > od; > return total; > end; function( l... ) ... 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 (21.20-26) which, for example, may take a list argument and optionally an initial element (which zero should the sum of an empty list return?).
GAP will also special case a function with a single argument with the name arg as function with a variable length list of arguments, as if the user had written arg....
Note that if a function f is defined as above then NumberArgumentsFunction(f) returns minus the number of formal arguments (including the final argument) (see NumberArgumentsFunction (5.1-2)).
Using the ... notation on a function f with only a single named argument 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 (5.2-1) is provided for this purpose.
Also see Chapter 5.
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 6.4). return; has the effect that the computation continues where it was interrupted by an error or the user hitting Ctrl-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 Section 4.23.
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