# Control flow functions¶

## Evaluation control¶

MaxEvalDepth(n)

set the maximum evaluation depth

Use this command to set the maximum evaluation depth to n. The default value is 1000.

The point of having a maximum evaluation depth is to catch any infinite recursion. For example, after the definition f(x) := f(x), evaluating the expression f(x) would call f(x), which would call f(x), etc. The interpreter will halt if the maximum evaluation depth is reached. Also indirect recursion, e.g. the pair of definitions f(x) := g(x) and g(x) := f(x), will be caught.

An example of an infinite recursion, caught because the maximum evaluation depth is reached:

In> f(x) := f(x)
Out> True;
In> f(x)

Error on line 1 in file [CommandLine]
Max evaluation stack depth reached.
Please use MaxEvalDepth to increase the stack
size as needed.


However, a long calculation may cause the maximum evaluation depth to be reached without the presence of infinite recursion. The function MaxEvalDepth() is meant for these cases:

In> 10 # g(0) <-- 1;
Out> True;
In> 20 # g(n_IsPositiveInteger) <-- \
2 * g(n-1);
Out> True;
In> g(1001);
Error on line 1 in file [CommandLine]
Max evaluation stack depth reached.
Please use MaxEvalDepth to increase the stack
size as needed.
In> MaxEvalDepth(10000);
Out> True;
In> g(1001);
Out> 21430172143725346418968500981200036211228096234
1106721488750077674070210224987224498639675763139171
6255189345835106293650374290571384628087196915514939
7149607869135549648461970842149210124742283755908364
3060929499671638825347975351183310878921541258291423
92955373084335320859663305248773674411336138752;

Hold(expr)

keep expression unevaluated

The expression expr is returned unevaluated. This is useful to prevent the evaluation of a certain expression in a context in which evaluation normally takes place.

Example

In> Echo({ Hold(1+1), "=", 1+1 });
1+1 = 2
Out> True;

Eval(expr)

force evaluation of expression

This function explicitly requests an evaluation of the expression expr, and returns the result of this evaluation.

Example

In> a := x;
Out> x;
In> x := 5;
Out> 5;
In> a;
Out> x;
In> Eval(a);
Out> 5;


The variable a is bound to x, and x is bound to 5. Hence evaluating a will give x. Only when an extra evaluation of a is requested, the value 5 is returned. Note that the behavior would be different if we had exchanged the assignments. If the assignment a := x were given while x had the value 5, the variable a would also get the value 5 because the assignment operator :=() evaluates the right-hand side.

## Conditional execution¶

If(pred, then[, else])

branch point

This command implements a branch point. The predicate pred is evaluated, which should result in either True or False. In the first case, the expression then is evaluated and returned. If the predicate yields False, the expression else (if present) is evaluated and returned. If there is no else branch, the If() expression returns False.

The sign function is defined to be 1 if its argument is positive and -1 if its argument is negative. A possible implementation is:

In> mysign(x) := If (IsPositiveReal(x), 1, -1);
Out> True;
In> mysign(Pi);
Out> 1;
In> mysign(-2.5);
Out> -1;


Note that this will give incorrect results, if x cannot be numerically approximated:

In> mysign(a);
Out> -1;


Hence a better implementation would be:

In> mysign(_x)_IsNumber(N(x)) <-- If(IsPositiveReal(x), 1, -1);
Out> True;


## Loops¶

While(pred) expr

loop while a condition is met

Keep on evaluating expr while pred evaluates to True. More precisely, While() evaluates the predicate pred, which should evaluate to either True or False. If the result is True, the expression expr is evaluated and then the predicate pred is evaluated again. If it is still True, the expressions expr and pred are again evaluated and so on until pred evaluates to False. At that point, the loop terminates and While() returns True.

In particular, if pred immediately evaluates to False, the body is never executed. While() is the fundamental looping construct on which all other loop commands are based. It is equivalent to the while command in the programming language C.

Example

In> x := 0;
Out> 0;
In> While (x! < 10^6) [ Echo({x, x!}); x++; ];
0  1
1  1
2  2
3  6
4  24
5  120
6  720
7  5040
8  40320
9  362880
Out> True;


Until(pred) expr

loop until a condition is met

Keep on evaluating expr until pred becomes True. More precisely, Until() first evaluates the expression body. Then the predicate pred is evaluated, which should yield either True or False. In the latter case, the expressions expr and pred are again evaluated and this continues as long as “pred” is False. As soon as pred yields True, the loop terminates and Until() returns True.

The main difference with While() is that Until() always evaluates expr at least once, but While() may not evaluate it at all. Besides, the meaning of the predicate is reversed: While() stops if pred is False while Until() stops if pred is True. The command Until(pred) expr; is equivalent to pred; While(Not pred) body;. In fact, the implementation of Until() is based on the internal command While(). The Until() command can be compared to the do ... while construct in the programming language C.

Example

In> x := 0;
Out> 0;
In> Until (x! > 10^6) [ Echo({x, x!}); x++; ];
0  1
1  1
2  2
3  6
4  24
5  120
6  720
7  5040
8  40320
9  362880
Out> True;


For(init, pred, incr) expr

C-style for loop

This commands implements a C style for loop. First of all, the expression init is evaluated. Then the predicate pred is evaluated, which should return True or False. Next, the loop is executed as long as the predicate yields True. One traversal of the loop consists of the subsequent evaluations of expr, incr, and pred. Finally, True is returned.

This command is most often used in a form such as For(i=1, i<=10, i++) expr, which evaluates expr with i subsequently set to 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

The expression For(init, pred, incr) expr is equivalent to init; While(pred) [expr; incr;].

Example

In> For (i:=1, i<=10, i++) Echo({i, i!});
1  1
2  2
3  6
4  24
5  120
6  720
7  5040
8  40320
9  362880
10  3628800
Out> True;

ForEach(var, list) expr

loop over all entries in list

The expression expr is evaluated multiple times. The first time, var has the value of the first element of “list”, then it gets the value of the second element and so on. ForEach() returns True.

Example

In> ForEach(i,{2,3,5,7,11}) Echo({i, i!});
2  2
3  6
5  120
7  5040
11  39916800
Out> True;

Function() func(args)
Function(funcname, {args}) body

declare or define a function

This command can be used to define a new function with named arguments.

The number of arguments of the new function and their names are determined by the list args. If the ellipsis ... follows the last atom in args, a function with a variable number of arguments is declared (using RuleBaseListed()). Note that the ellipsis cannot be the only element of args and must be preceded by an atom.

A function with variable number of arguments can take more arguments than elements in args; in this case, it obtains its last argument as a list containing all extra arguments.

The short form of the Function() call merely declares a RuleBase() for the new function but does not define any function body. This is a convenient shorthand for RuleBase() and RuleBaseListed(), when definitions of the function are to be supplied by rules. If the new function has been already declared with the same number of arguments (with or without variable arguments), Function() returns false and does nothing.

The second, longer form of the Function() call declares a function and also defines a function body. It is equivalent to a single rule such as funcname(_arg1, _arg2) <-- body. The rule will be declared at precedence 1025. Any previous rules associated with funcname (with the same arity) will be discarded. More complicated functions (with more than one body) can be defined by adding more rules.

Example

This will declare a new function with two or more arguments, but define no rules for it. This is equivalent to RuleBase ("f1", {x, y, ...}):

In> Function() f1(x,y,...);
Out> True;
In> Function() f1(x,y);
Out> False;


This defines a function FirstOf which returns the first element of a list. Equivalent definitions would be FirstOf(_list) <-- list or FirstOf(list) := list:

In> Function("FirstOf", {list})  list;
Out> True;
In> FirstOf({a,b,c});
Out> a;


The following function will print all arguments to a string:

In> Function("PrintAll",{x, ...}) If(IsList(x), PrintList(x), ToString()Write(x));
Out> True;
In> PrintAll(1):
Out> " 1";
In> PrintAll(1,2,3);
Out> " 1 2 3";

Macro() func(args)
Macro(funcname, {args}) body

declare or define a macro

This does the same as Function(), but for macros. One can define a macro easily with this function, instead of having to use DefMacroRuleBase().

Example

The following example defines a looping function

In> Macro("myfor",{init,pred,inc,body}) [@init;While(@pred)[@body;@inc;];True;];
Out> True;
In> a:=10
Out> 10;


Here this new macro myfor is used to loop, using a variable a from the calling environment

In> myfor(i:=1,i<10,i++,Echo(a*i))
10
20
30
40
50
60
70
80
90
Out> True;
In> i
Out> 10;

Apply(fn, arglist)

apply a function to arguments

This function applies the function fn to the arguments in arglist and returns the result. The first parameter fn can either be a string containing the name of a function or a pure function. Pure functions, modeled after lambda-expressions, have the form {varlist,body}, where varlist is the list of formal parameters. Upon application, the formal parameters are assigned the values in arglist (the second parameter of Apply()) and the body is evaluated.

Another way to define a pure function is with the Lambda construct. Here, instead of passing in {varlist,body}, one can pass in Lambda(varlist,body). Lambda has the advantage that its arguments are not evaluated (using lists can have undesirable effects because lists are evaluated). Lambda can be used everywhere a pure function is expected, in principle, because the function Apply() is the only function dealing with pure functions. So all places where a pure function can be passed in will also accept Lambda.

An shorthand for Apply() is provided by the @() operator.

Example

In> Apply("+", {5,9});
Out> 14;
In> Apply({{x,y}, x-y^2}, {Cos(a), Sin(a)});
Out> Cos(a)-Sin(a)^2;
In>  Apply(Lambda({x,y}, x-y^2), {Cos(a), Sin(a)});
Out> Cos(a)-Sin(a)^2
In>  Lambda({x,y}, x-y^2) @ {Cos(a), Sin(a)}
Out> Cos(a)-Sin(a)^2

MapArgs(expr, fn)

apply a function to all top-level arguments

Every top-level argument in expr is substituted by the result of applying fn to this argument. Here fn can be either the name of a function or a pure function (see Apply() for more information on pure functions).

Example

In> MapArgs(f(x,y,z),"Sin");
Out> f(Sin(x),Sin(y),Sin(z));
In> MapArgs({3,4,5,6}, {{x},x^2});
Out> {9,16,25,36};

Subst(from, to) expr

perform a substitution

This function substitutes every occurrence of from in expr by to. This is a syntactical substitution: only places where from occurs as a subexpression are affected.

Example

In> Subst(x, Sin(y)) x^2+x+1;
Out> Sin(y)^2+Sin(y)+1;
In> Subst(a+b, x) a+b+c;
Out> x+c;
In> Subst(b+c, x) a+b+c;
Out> a+b+c;


The explanation for the last result is that the expression a+b+c is internally stored as (a+b)+c. Hence a+b is a subexpression, but b+c is not.

WithValue(var, val, expr)
WithValue(varlist, vallist, expr)

temporary assignment during an evaluation

First, the expression val is assigned to the variable var. Then, the expression expr is evaluated and returned. Finally, the assignment is reversed so that the variable var has the same value as it had before WithValue() was evaluated.

The second calling sequence assigns the first element in the list of values to the first element in the list of variables, the second value to the second variable, etc.

Example

In> WithValue(x, 3, x^2+y^2+1);
Out> y^2+10;
In> WithValue({x,y}, {3,2}, x^2+y^2+1);
Out> 14;


expression /: patterns

local simplification rules

Sometimes you have an expression, and you want to use specific simplification rules on it that are not done by default. This can be done with the /: and the /:: operators. Suppose we have the expression containing things such as Ln(a*b), and we want to change these into Ln(a)+Ln(b), the easiest way to do this is using the /: operator, as follows:

In> Sin(x)*Ln(a*b)
Out> Sin(x)*Ln(a*b);
In> % /: { Ln(_x*_y) <- Ln(x)+Ln(y) }
Out> Sin(x)*(Ln(a)+Ln(b));


A whole list of simplification rules can be built up in the list, and they will be applied to the expression on the left hand side of /:.

The forms the patterns can have are one of:

pattern <- replacement
{pattern,replacement}
{pattern,postpredicate,replacement}


Note that for these local rules, <- should be used instead of <-- which would be used in a global rule.

The /: operator traverses an expression much as Subst() does, that is, top down, trying to apply the rules from the beginning of the list of rules to the end of the list of rules. If the rules cannot be applied to an expression, it will try subexpressions of that expression and so on.

It might be necessary sometimes to use the /:: operator, which repeatedly applies the /: operator until the result doesn’t change any more. Caution is required, since rules can contradict each other, which could result in an infinite loop. To detect this situation, just use /: repeatedly on the expression. The repetitive nature should become apparent.

Example

In> Sin(u)*Ln(a*b) /: {Ln(_x*_y) <- Ln(x)+Ln(y)}
Out> Sin(u)*(Ln(a)+Ln(b));
In> Sin(u)*Ln(a*b) /:: { a <- 2, b <- 3 }
Out> Sin(u)*Ln(6);