%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A language.tex GAP documentation Martin Schoenert %% %A @(#)$Id: language.tex,v 3.11 1993/03/11 17:54:26 fceller Rel $ %% %Y Copyright 1990-1992, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %% This file describes the {\GAP} programming language. %% %H $Log: language.tex,v $ %H Revision 3.11 1993/03/11 17:54:26 fceller %H strings are now lists %H %H Revision 3.10 1993/02/19 10:48:42 gap %H adjustments in line length and spelling %H %H Revision 3.9 1993/02/18 15:52:09 felsch %H index entries added %H %H Revision 3.8 1993/02/17 07:43:50 felsch %H examples fixed %H %H Revision 3.7 1993/02/11 12:20:47 martin %H changed reference "Strings" to "Strings and Characters" %H %H Revision 3.6 1993/02/10 21:10:03 martin %H fixed various typos, extended description for 3.2 %H %H Revision 3.5 1992/04/30 12:07:10 martin %H changed a few sentences to avoid bold non-roman fonts %H %H Revision 3.4 1992/04/09 11:36:01 martin %H made a few changes so that two LaTeX passes suffice %H %H Revision 3.3 1992/04/02 20:58:24 martin %H fixed a wrong example in "While" %H %H Revision 3.2 1992/03/11 15:59:22 martin %H fixed several typos %H %H Revision 3.1 1992/01/14 14:42:24 martin %H changed the BNF for nicer online viewing %H %H Revision 3.0 1991/12/27 16:10:27 martin %H initial revision under RCS %H %% \Chapter{The Programming Language} This chapter describes the {\GAP} programming language. It should allow you in principle to predict the result of each and every input. In order to know what we are talking about, we first have to look more closely at the process of interpretation and the various representations of data involved. First we have the input to {\GAP}, given as a string of characters. How those characters enter {\GAP} is operating system dependent, e.g., they might be entered at a terminal, pasted with a mouse into a window, or read from a file. The mechanism does not matter. This representation of expressions by characters is called the *external representation* of the expression. Every expression has at least one external representation that can be entered to get exactly this expression. The input, i.e., the external representation, is transformed in a process called *reading* to an internal representation. At this point the input is analyzed and inputs that are not legal external representations, according to the rules given below, are rejected as errors. Those rules are usually called the *syntax* of a programming language. The internal representation created by reading is called either an *expression* or a *statement*. Later we will distinguish between those two terms, however now we will use them interchangeably. The exact form of the internal representation does not matter. It could be a string of characters equal to the external representation, in which case the reading would only need to check for errors. It could be a series of machine instructions for the processor on which {\GAP} is running, in which case the reading would more appropriately be called compilation. It is in fact a tree--like structure. After the input has been read it is again transformed in a process called *evaluation* or *execution*. Later we will distinguish between those two terms too, but for the moment we will use them interchangeably. The name hints at the nature of this process, it replaces an expression with the value of the expression. This works recursively, i.e., to evaluate an expression first the subexpressions are evaluated and then the value of the expression is computed according to rules given below from those values. Those rules are usually called the *semantics* of a programming language. The result of the evaluation is, not surprisingly, called a *value*. The set of values is of course a much smaller set than the set of expressions; for every value there are several expressions that will evaluate to this value. Again the form in which such a value is represented internally does not matter. It is in fact a tree--like structure again. The last process is called *printing*. It takes the value produced by the evaluation and creates an external representation, i.e., a string of characters again. What you do with this external representation is up to you. You can look at it, paste it with the mouse into another window, or write it to a file. Lets look at an example to make this more clear. Suppose you type in the following string of 8 characters | 1 + 2 * 3;| {\GAP} takes this external representation and creates a tree like internal representation, which we can picture as follows | + / \ 1 * / \ 2 3 | This expression is then evaluated. To do this {\GAP} first evaluates the right subexpression '2\*3'. Again to do this {\GAP} first evaluates its subexpressions 2 and 3. However they are so simple that they are their own value, we say that they are self--evaluating. After this has been done, the rule for '\*' tells us that the value is the product of the values of the two subexpressions, which in this case is clearly 6. Combining this with the value of the left operand of the '+', which is self--evaluating too gives us the value of the whole expression 7. This is then printed, i.e., converted into the external representation consisting of the single character '7'. In this fashion we can predict the result of every input when we know the syntactic rules that govern the process of reading and the semantic rules that tell us for every expression how its value is computed in terms of the values of the subexpressions. The syntactic rules are given in sections "Lexical Structure", "Symbols", "Whitespaces", "Keywords", "Identifiers", and "The Syntax in BNF", the semantic rules are given in sections "Expressions", "Variables", "Function Calls", "Comparisons", "Operations", "Statements", "Assignments", "Procedure Calls", "If", "While", "Repeat", "For", "Functions", and the chapters describing the individual data types. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Lexical Structure} The input of {\GAP} consists of sequences of the following characters. Digits, uppercase and lowercase letters, , , , and the special characters | " ' ( ) * + , _ . / : ; < = > ~ [ \ ] ^ _ { } & | Other characters will be signalled as illegal. Inside strings and comments the full character set supported by the computer is allowed. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Symbols} The process of reading, i.e., of assembling the input into expressions, has a subprocess, called *scanning*, that assembles the characters into symbols. A *symbol* is a sequence of characters that form a lexical unit. The set of symbols consists of keywords, identifiers, strings, integers, and operator and delimiter symbols. A keyword is a reserved word consisting entirely of lowercase letters (see "Keywords"). An identifier is a sequence of letters and digits that contains at least one letter and is not a keyword (see "Identifiers"). An integer is a sequence of digits (see "Integers"). A string is a sequence of arbitrary characters enclosed in double quotes (see "Strings and Characters"). Operator and delimiter symbols are | + - * / ^ ~ = <> < <= > >= := . .. -> , ; [ ] { } ( ) | Note that during the process of scanning also all whitespace is removed (see "Whitespaces"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Whitespaces}% \index{space}% \index{blank}\index{tabulator}\index{newline}\index{comments} The characters , , , and are called *whitespace characters*. Whitespace is used as necessary to separate lexical symbols, such as integers, identifiers, or keywords. For example 'Thorondor' is a single identifier, while 'Th or ondor' is the keyword 'or' between the two identifiers 'Th' and 'ondor'. Whitespace may occur between any two symbols, but not within a symbol. Two or more adjacent whitespaces are equivalent to a single whitespace. Apart from the role as separator of symbols, whitespaces are otherwise insignificant. Whitespaces may also occur inside a string, where they are significant. Whitespaces should also be used freely for improved readability. A *comment* starts with the character '\#', which is sometimes called sharp or hatch, and continues to the end of the line on which the comment character appears. The whole comment, including '\#' and the character is treated as a single whitespace. Inside a string, the comment character '\#' looses its role and is just an ordinary character. For example, the following statement | if i<0 then a:=-i;else a:=i;fi; | is equivalent to | if i < 0 then & if i is negative a := -i; & take its inverse else & otherwise a := i; & take itself fi; | (which by the way shows that it is possible to write superfluous comments). However the first statement is not equivalent to | ifi<0thena:=-i;elsea:=i;fi; | since the keyword 'if' must be separated from the identifier 'i' by a whitespace, and similarly 'then' and 'a', and 'else' and 'a' must be separated. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Keywords} *Keywords* are reserved words that are used to denote special operations or are part of statements. They must not be used as identifiers. The keywords are | and do elif else end fi for function if in local mod not od or repeat return then until while quit | Note that all keywords are written in lowercase. For example only 'else' is a keyword; 'Else', 'eLsE', 'ELSE' and so forth are ordinary identifiers. Keywords must not contain whitespace, for example 'el if' is not the same as 'elif'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Identifiers} An identifier is used to refer to a variable (see "Variables"). An identifier consists of letters, digits, and underscores '\_', and must contain at least one letter or underscore. An identifier is terminated by the first character not in this class. Examples of valid identifiers are | a foo aLongIdentifier hello Hello HELLO x100 100x _100 some_people_prefer_underscores_to_separate_words WePreferMixedCaseToSeparateWords | Note that case is significant, so the three identifiers in the second line are distinguished. The backslash |\| can be used to include other characters in identifiers; a backslash followed by a character is equivalent to the character, except that this escape sequence is considered to be an ordinary letter. For example |G\(2\,5\)| is an identifier, not a call to a function 'G'. An identifier that starts with a backslash is never a keyword, so for example |\*| and |\mod| are identifier. The length of identifiers is not limited, however only the first 1023 characters are significant. The escape sequence |\| is ignored, making it possible to split long identifiers over multiple lines. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Expressions}% \index{evaluation} An *expression* is a construct that evaluates to a value. Syntactic constructs that are executed to produce a side effect and return no value are called *statements* (see "Statements"). Expressions appear as right hand sides of assignments (see "Assignments"), as actual arguments in function calls (see "Function Calls"), and in statements. Note that an expression is not the same as a value. For example '1 + 11' is an expression, whose value is the integer 12. The external representation of this integer is the character sequence '12', i.e., this sequence is output if the integer is printed. This sequence is another expression whose value is the integer 12. The process of finding the value of an expression is done by the interpreter and is called the *evaluation* of the expression. Variables, function calls, and integer, permutation, string, function, list, and record literals (see "Variables", "Function Calls", "Integers", "Permutations", "Strings and Characters", "Functions", "Lists", "Records"), are the simplest cases of expressions. Expressions, for example the simple expressions mentioned above, can be combined with the operators to form more complex expressions. Of course those expressions can then be combined further with the operators to form even more complex expressions. The *operators* fall into three classes. The *comparisons* are '=', '\<>', '\<=', '>', '>=', and 'in' (see "Comparisons" and "In"). The *arithmetic operators* are '+', '-', '\*', '/', 'mod', and '\^' (see "Operations"). The *logical operators* are 'not', 'and', and 'or' (see "Operations for Booleans"). | gap> 2 * 2; # a very simple expression with value 4 gap> 2 * 2 + 9 = Fibonacci(7) and Fibonacci(13) in Primes; true # a more complex expression | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Variables}% \index{scope}\index{bound} A *variable* is a location in a {\GAP} program that points to a value. We say the variable is *bound* to this value. If a variable is evaluated it evaluates to this value. Initially an ordinary variable is not bound to any value. The variable can be bound to a value by *assigning* this value to the variable (see "Assignments"). Because of this we sometimes say that a variable that is not bound to any value has no assigned value. Assignment is in fact the only way by which a variable, which is not an argument of a function, can be bound to a value. After a variable has been bound to a value an assignment can also be used to bind the variable to another value. A special class of variables are *arguments* of functions. They behave similarly to other variables, except they are bound to the value of the actual arguments upon a function call (see "Function Calls"). Each variable has a name that is also called its *identifier*. This is because in a given scope an identifier identifies a unique variable (see "Identifiers"). A *scope* is a lexical part of a program text. There is the global scope that encloses the entire program text, and there are local scopes that range from the 'function' keyword, denoting the beginning of a function definition, to the corresponding 'end' keyword. A local scope introduces new variables, whose identifiers are given in the formal argument list and the 'local' declaration of the function (see "Functions"). Usage of an identifier in a program text refers to the variable in the innermost scope that has this identifier as its name. Because this mapping from identifiers to variables is done when the program is read, not when it is executed, {\GAP} is said to have lexical scoping. The following example shows how one identifier refers to different variables at different points in the program text. | g := 0; & global variable g x := function ( a, b, c ) local y; g := c; & c refers to argument c of function x y := function ( y ) local d, e, f; d := y; & y refers to argument y of function y e := b; & b refers to argument b of function x f := g; & g refers to global variable g return d + e + f; end; return y( a ); & y refers to local y of function x end; | It is important to note that the concept of a variable in {\GAP} is quite different from the concept of a variable in programming languages like PASCAL. In those languages a variable denotes a block of memory. The value of the variable is stored in this block. So in those languages two variables can have the same value, but they can never have identical values, because they denote different blocks of memory. (Note that PASCAL has the concept of a reference argument. It seems as if such an argument and the variable used in the actual function call have the same value, since changing the argument\'s value also changes the value of the variable used in the actual function call. But this is not so; the reference argument is actually a pointer to the variable used in the actual function call, and it is the compiler that inserts enough magic to make the pointer invisible.) In order for this to work the compiler needs enough information to compute the amount of memory needed for each variable in a program, which is readily available in the declarations PASCAL requires for every variable. In {\GAP} on the other hand each variable justs points to a value. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Function Calls} '()' \\ '( \{',' \} )' The function call has the effect of calling the function . The precise semantics are as follows. First {\GAP} evaluates the . Usually is a variable, and {\GAP} does nothing more than taking the value of this variable. It is allowed though that is a more complex expression, namely it can for example be a selection of a list element '[]', or a selection of a record component '.'. 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 s agrees with the number of formal arguments as given in the function definition. If they do not agree {\GAP} signals an error. An exception is the case when there is exactly one formal argument with the name 'arg', in which case any number of actual arguments is allowed. Now {\GAP} allocates for each formal argument and for each formal local a new variable. Remember that a variable is a location in a {\GAP} program that points to a value. Thus for each formal argument and for each formal local such a location is allocated. Next the arguments s are evaluated, and the values are assigned to the newly created variables corresponding to the formal arguments. Of course the first value is assigned to the new variable corresponding to the first formal argument, the second value is assigned to the new variable corresponding to the second formal argument, and so on. However, {\GAP} does not make any guarantee about the order in which the arguments are evaluated. They might be evaluated left to right, right to left, or in any other order, but each argument is evaluated once. An exception again occurs if the function has only one formal argument with the name 'arg'. In this case the values of all the actual arguments are stored in a list and this list is assigned to the new variable corresponding to the formal argument 'arg'. The new variables corresponding to the formal locals are initially not bound to any value. So trying to evaluate those variables before something has been assigned to them will signal an error. Now the body of the function, which is a statement, is executed. If the identifier of one of the formal arguments or formal locals appears in the body of the function it refers to the new variable that was allocated for this formal argument or formal local, and evaluates to the value of this variable. If during the execution of the body of the function a 'return' statement with an expression (see "Return") is executed, execution of the body is terminated and the value of the function call is the value of the expression of the 'return'. If during the execution of the body a 'return' statement without an expression is executed, execution of the body is terminated and the function call does not produce a value, in which case we call this call a procedure call (see "Procedure Calls"). If the execution of the body completes without execution of a 'return' statement, the function call again produces no value, and again we talk about a procedure call. | gap> Fibonacci( 11 ); # a call to the function 'Fibonacci' with actual argument '11' 89 | | gap> G.operations.RightCosets( G, Intersection( U, V ) );; # a call to the function in 'G.operations.RightCosets' # where the second actual argument is another function call | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Comparisons} ' = ' \\ ' \<> ' The operator '=' tests for equality of its two operands and evaluates to 'true' if they are equal and to 'false' otherwise. Likewise '\<>' tests for inequality of its two operands. Note that any two objects can be compared, i.e., '=' and '\<>' will never signal an error. For each type of objects the definition of equality is given in the respective chapter. Objects of different types are never equal, i.e., '=' evaluates in this case to 'false', and '\<>' evaluates to 'true'. ' \<\ ' \\ ' > ' \\ ' \<= ' \\ ' >= ' '\<' denotes less than, '\<=' less than or equal, '>' greater than, and '>=' greater than or equal of its two operands. For each type of objects the definition of the ordering is given in the respective chapter. The ordering of objects of different types is as follows. Rationals are smallest, next are cyclotomics, followed by finite field elements, permutations, words, words in solvable groups, boolean values, functions, lists, and records are largest. Comparison operators, which includes the operator 'in' (see "In") are not associative, i.e., it is not allowed to write ' = \<> = ', you must use '( = ) \<> ( = )' instead. The comparison operators have higher precedence than the logical operators (see "Operations for Booleans"), but lower precedence than the arithmetic operators (see "Operations"). Thus, for example, ' \*\ = and ' is interpreted, '(( \*\ ) = ) and )'. | gap> 2 * 2 + 9 = Fibonacci(7); # a comparison where the left true # operand is an expression | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Operations}% \index{precedence}\index{associativity} '+ ' \\ '- ' \\ ' + ' \\ ' - ' \\ ' \*\ ' \\ ' / ' \\ ' mod ' \\ ' \^\ ' The arithmetic operators are '+', '-', '\*', '/', 'mod', and '\^'. The meanings (semantic) of those operators generally depend on the types of the operands involved, and they are defined in the various chapters describing the types. However basically the meanings are as follows. '+' denotes the addition, and '-' the subtraction of ring and field elements. '\*' is the multiplication of group elements, '/' is the multiplication of the left operand with the inverse of the right operand. 'mod' is only defined for integers and rationals and denotes the modulo operation. '+' and '-' can also be used as unary operations. The unary '+' is ignored and unary '-' is equivalent to multiplication by -1. '\^' denotes powering of a group element if the right operand is an integer, and is also used to denote operation if the right operand is a group element. The *precedence* of those operators is as follows. The powering operator '\^' has the highest precedence, followed by the unary operators '+' and '-', which are followed by the multiplicative operators '\*', '/', and 'mod', and the additive binary operators '+' and '-' have the lowest precedence. That means that the expression '-2 \^\ -2 \*\ 3 + 1' is interpreted as '(-(2 \^\ (-2)) \*\ 3) + 1'. If in doubt use parentheses to clarify your intention. The *associativity* of the arithmetic operators is as follows.'\^' is not associative, i.e., it is illegal to write '2\^3\^4', use parentheses to clarify whether you mean '(2\^3) \^\ 4' or '2 \^\ (3\^4)'. The unary operators '+' and '-' are right associative, because they are written to the left of their operands. '\*', '/', 'mod', '+', and '-' are all left associative, i.e., '1-2-3' is interpreted as '(1-2)-3' not as '1-(2-3)'. Again, if in doubt use parentheses to clarify your intentions. The arithmetic operators have higher precedence than the comparison operators (see "Comparisons" and "In") and the logical operators (see "Operations for Booleans"). Thus, for example, ' \*\ = and ' is interpreted, '(( \*\ ) = ) and '. | gap> 2 * 2 + 9; & a very simple arithmetic expression 13 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Statements}% \index{execution} Assignments (see "Assignments"), Procedure calls (see "Procedure Calls"), 'if' statements (see "If"), 'while' (see "While"), 'repeat' (see "Repeat") and 'for' loops (see "For"), and the 'return' statement (see "Return") are called statements. They can be entered interactively or be part of a function definition. Every statement must be terminated by a semicolon. Statements, unlike expressions, have no value. They are executed only to produce an effect. For example an assignment has the effect of assigning a value to a variable, a 'for' loop has the effect of executing a statement sequence for all elements in a list and so on. We will talk about *evaluation* of expressions but about *execution* of statements to emphasize this difference. It is possible to use expressions as statements. However this does cause a warning. | gap> if i <> 0 then k = 16/i; fi; Syntax error: warning, this statement has no effect if i <> 0 then k = 16/i; fi; ^ | As you can see from the example this is useful for those users who are used to languages where '=' instead of '\:=' denotes assignment. A sequence of one or more statements is a statement sequence, and may occur everywhere instead of a single statement. There is nothing like PASCAL\'s BEGIN-END, instead each construct is terminated by a keyword. The most simple statement sequence is a single semicolon, which can be used as an empty statement sequence. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Assignments}% \index{assignment!variable} ' \:= ;' The *assignment* has the effect of assigning the value of the expressions to the variable . The variable may be an ordinary variable (see "Variables"), a list element selection '[]' (see "List Assignment") or a record component selection '.' (see "Record Assignment"). Since a list element or a record component may itself be a list or a record the left hand side of an assignment may be arbitrarily complex. Note that variables do not have a type. Thus any value may be assigned to any variable. For example a variable with an integer value may be assigned a permutation or a list or anything else. If the expression 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 ;' the value of the expression is assigned to the variable, and execution continues after the assignment. | gap> S6 := rec( size := 720 );; S6; rec( size := 720 ) gap> S6.generators := [ (1,2), (1,2,3,4,5) ];; S6; rec( size := 720, generators := [ (1,2), (1,2,3,4,5) ] ) gap> S6.generators[2] := (1,2,3,4,5,6);; S6; rec( size := 720, generators := [ (1,2), (1,2,3,4,5,6) ] ) | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Procedure Calls} '();' \\ '( \{',' \} );' The procedure call has the effect of calling the procedure . A procedure call is done exactly like a function call (see "Function Calls"). The distinction between functions and procedures is only for the sake of the discussion, {\GAP} does not distinguish between them. A *function* does return a value but does not produce a side effect. As a convention the name of a function is a noun, denoting what the function returns, e.g., 'Length', 'Concatenation' and 'Order'. A *procedure* is a function that does not return a value but produces some effect. Procedures are called only for this effect. As a convention the name of a procedure is a verb, denoting what the procedure does, e.g., 'Print', 'Append' and 'Sort'. | gap> Read( "myfile.g" ); & a call to the procedure Read gap> l := [ 1, 2 ];; gap> Append( l, [3,4,5] ); & a call to the procedure Append | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{If}% \index{if}\index{if statement}% \index{fi}\index{then}\index{else}\index{elif} 'if then \\ \{ elif then \} \\ {}[ else ] \\ 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 following the 'if' is evaluated. If it evaluates to 'true' the statement sequence after the first 'then' is executed, and the execution of the 'if' statement is complete. Otherwise the expressions 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 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 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 then if then else fi; fi;'\\ the 'else' part belongs to the second 'if' statement, whereas in\\ 'if then if then fi; else fi;'\\ the 'else' part belongs to the first 'if' statement. Since an if statement is not an expression it is not possible to write | abs := if x > 0 then x; else -x; fi;| which would, even if legal syntax, be meaningless, since the 'if' statement does not produce a value that could be assigned to 'abs'. If one expression evaluates neither to 'true' nor to 'false' an error is signalled and a break loop (see "Break Loops") is entered. As usual you can leave the break loop with 'quit;'. If you enter 'return true;', execution of the 'if' statement continues as if the expression whose evaluation failed had evaluated to 'true'. Likewise, if you enter 'return false;', execution of the 'if' statement continues as if the expression whose evaluation failed had evaluated to 'false'. | gap> i := 10;; gap> if 0 < i then > s := 1; > elif i < 0 then > s := -1; > else > s := 0; > fi; gap> s; 1 # the sign of i | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{While}% \index{while}\index{while loop}\index{loop!while} 'while do od;' The 'while' loop executes the statement sequence while the condition evaluates to 'true'. First 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 are executed and the whole process begins again. The difference between the 'while' loop and the 'repeat until' loop (see "Repeat") is that the in the 'repeat until' loop are executed at least once, while the in the 'while' loop are not executed at all if is 'false' at the first iteration. If 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 , after which the next evaluation of may cause another error. | gap> i := 0;; s := 0;; gap> while s <= 200 do > i := i + 1; s := s + i^2; > od; gap> s; 204 # first sum of the first i squares larger than 200 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Repeat}% \index{repeat}\index{repeat loop}\index{loop!repeat}\index{until} 'repeat until ;' The 'repeat' loop executes the statement sequence until the condition evaluates to 'true'. First are executed. Then 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 . The difference between the 'while' loop (see "While") and the 'repeat until' loop is that the in the 'repeat until' loop are executed at least once, while the in the 'while' loop are not executed at all if is 'false' at the first iteration. If does not evaluate to 'true' or 'false' a error is signalled and a break loop (see "Break Loops") is entered. As usual you can leave the break loop with 'quit;'. If you enter 'return true;', execution continues with the next statement immediately following the 'repeat' loop. If you enter 'return false;', execution continues at , after which the next evaluation of may cause another error. | gap> i := 0;; s := 0;; gap> repeat > i := i + 1; s := s + i^2; > until s > 200; gap> s; 204 # first sum of the first i squares larger than 200 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{For}% \index{for}\index{for loop}\index{loop!for}\index{do}\index{od} 'for in do od;' The 'for' loop executes the statement sequence for every element of the list . The statement sequence is first executed with bound to the first element of the list , then with bound to the second element of and so on. must be a simple variable, it must not be a list element selection '[]' or a record component selection '.'. The execution of the 'for' loop is exactly equivalent to the 'while' loop | || := ||; || := 1; while || <= Length(||) do || := ||[||]; || || := || + 1; od; | with the exception that and are different variables for each 'for' loop that do not interfere with each other. The list is very often a range.\\ 'for in [..] do od;'\\ corresponds to the more common\\ 'for from to do 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; [ ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Functions}% \index{function}\index{end}\index{local}% \index{recursion}\index{recursive functions}% \index{environment}\index{body} |function (| [ \{|,| \} ] |) |[ |local | \{|, |\} |;| ]| || 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 and for each of the formal locals 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()-1' steps to compute 'fib()', while the iterative version of 'fib' needs only '-2' steps. Both are not optimal however, the library function 'Fibonacci' only needs on the order of 'Log()' steps. ' -> ' This is a shorthand for\\ 'function ( ) return ; end'.\\ must be a single identifier, i.e., it is not possible to write functions of several arguments this way. Also 'arg' is not treated specially, so it is also impossible to write functions that take a variable number of arguments this way. The following is an example of a typical use of such a function | gap> Sum( List( [1..100], x -> x^2 ) ); 338350 | When a function definition is evaluated inside another function , {\GAP} binds all the identifiers inside the function that are identifiers of an argument or a local of to the corresponding variable. This set of bindings is called the environment of the function . When 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Return}% \index{return} 'return;' In this form 'return' terminates the call of the innermost function that is currently executing, and control returns to the calling function. An error is signalled if no function is currently executing. No value is returned by the function. 'return ;' In this form 'return' terminates the call of the innermost function that is currently executing, and returns the value of the expression . 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 'C'. 'return ;' can be used to continue execution after an error. What happens with the value depends on the particular error. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{The Syntax in BNF}% \index{BNF} This section contains the definition of the {\GAP} syntax in Backus-Naur form. A BNF is a set of rules, whose left side is the name of a syntactical construct. Those names are enclosed in angle brackets and written in . 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 , or character sequences that must occur literally; they are written in 'typewriter style'. Furthermore each righthand side can contain the following metasymbols written in *boldface*. If the right hand side contains forms separated by a pipe symbol ($\|$) this means that one of the possible forms can occur. If a part of a form is enclosed in square brackets ([ ]) this means that this part is optional, i.e. might be present or missing. If part of the form is enclosed in curly braces (\{ \}) this means that the part may occur arbitrarily often, or possibly be missing. \newpage \begin{tabbing} \=\:= \= \kill \>\:= \>'a'$\|$...$\|$'z'$\|$'A'$\|$...$\|$'Z'$\|$'\_' \{'a'$\|$...$\|$'z'$\|$'A'$\|$...$\|$'Z'$\|$'0'$\|$...$\|$'9'$\|$'\_'\}\\ \>\:= \> \\ \>$\|$ \> '.' \\ \>$\|$ \> '.' '(' ')' \\ \>$\|$ \> '[' ']' \\ \>$\|$ \> '\{' '\}' \\ \>$\|$ \> '(' [ \{ ',' \} ] ')' \\ \>\:= \>'[' [ ] \{',' [ ] \} ']' \\ \>$\|$ \>'[' [, ] '..' ']' \\ \>\:= \>'rec(' [ '\:=' \{',' '\:=' \} ] ')' \\ \>\:= \>'(' \{',' \} ')' \{ '(' \{',' \} ')' \} \\ \>\:= \>'function (' [ \{',' \} ] ')' \\ \> \> [ 'local' \{',' \} ';' ] \\ \> \> \\ \> \>'end' \\ \>\:= \>|'| |'| \\ \>\:= \>'\"' \{ \} '\"' \\ \>\:= \>'0'$\|$'1'$\|$...$\|$'9' \{ '0'$\|$'1'$\|$...$\|$'9' \} \\ \>\:= \> \\ \>$\|$ \> \\ \>$\|$ \>'(' ')' \\ \>$\|$ \> \\ \>$\|$ \> \\ \>$\|$ \> \\ \>$\|$ \> \\ \>$\|$ \> \\ \>$\|$ \> \\ \>\:= \>\{'+'$\|$'-'\} [ '\^' \{'+'$\|$'-'\} ]\\ \>\:= \> \{ '\*'$\|$'/'$\|$'mod' \} \\ \>\:= \> \{ '+'$\|$'-' \} \\ \>\:= \>\{ 'not' \} \{ |=|$\|$|<>|$\|$|<|$\|$|>|$\|$|<=|$\|$|>=|$\|$|in| \} \\ \>\:= \> \{ 'and' \} \\ \>\:= \> \{ 'or' \} \\ \>\:= \> \\ \>$\|$ \> [ '->' ] \\ \>\:= \> \\ \>$\|$ \> '\:=' \\ \>$\|$ \>'if' 'then' \\ \> \>\{ 'elif' 'then' \= \} \\ \> \>[ 'else' \> ] 'fi'\\ \>$\|$ \>'for' 'in' 'do' 'od' \\ \>$\|$ \>'while' 'do' 'od' \\ \>$\|$ \>'repeat' 'until' \\ \>$\|$ \>'return' [ ] \\ \>$\|$ \>'quit' \\ \>\:= \>\{ ';' \} \\ \>$\|$ \>';' \end{tabbing} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E Emacs . . . . . . . . . . . . . . . . . . . . . local Emacs variables %% %% Local Variables: %% mode: outline %% outline-regexp: "\\\\Chapter\\|\\\\Section" %% fill-column: 73 %% eval: (hide-body) %% End: %%