File: coding.book.txt

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*BLURB
This document should get you started programming in Yacas. There
are some basic explanations and hands-on tutorials.
				Programming in Yacas

			Yacas under the hood

This part of the manual is a somewhat in-depth explanation of the
Yacas programming language and environment. It assumes that you have worked
through the introductory tutorial. You should consult the function reference about how to use the various Yacas functions mentioned here.

		The Yacas architecture

Yacas is designed as a small core engine that interprets a library
of scripts. The core engine provides the syntax parser and a number of
hard-wired functions, such as {Set()} or
{MathExp()} which cannot be redefined by the user. The script
library resides in the scripts directory "{scripts/}" and
contains higher-level definitions of functions and constants. The library
scripts are on equal footing with any code the user executes interactively
or any files the user loads.

Generally, all core functions have plain names and almost all are not
"bodied" or infix operators. The file {corefunctions.h} in the source
tree lists declarations of all kernel functions callable from Yacas; consult it for reference.
For many of the core functions, the script library already provides
convenient aliases. For instance, the addition operator "{+}" is defined in
the script {scripts/standard} while the actual addition of
numbers is performed through the built-in function
{MathAdd}.

		Startup, scripts and {.def} files

When Yacas is first started or restarted, it executes the script
{yacasinit.ys} in the scripts directory. This script may load some other
scripts. In order to start up quickly, Yacas does not execute all other
library scripts at first run or at restart. It only executes the file
{yacasinit.ys} and all {.def} files in the scripts. The {.def} files tell the system where it can find definitions
for various library functions. Library is divided into "packages" stored in "repository" directories. For example, the function {ArcTan} is defined in the {stdfuncs} package; the library file is {stdfuncs.rep/}{code.ys} and the {.def} file is {stdfuncs.rep}{/code.ys.def}. The function {ArcTan}
mentioned in the {.def} file, therefore Yacas will know to load the package
{stdfuncs} when the user invokes {ArcTan}. This way Yacas
knows where to look for any given function without actually loading the
file where the function is defined.

There is one exception to the strategy of delayed loading of the library
scripts. Namely, the syntax definitions of infix, prefix, postfix and bodied
functions, such as {Infix("*",4)} cannot be delayed (it is currently in the
file {stdopers.ys}). If it were delayed, the Yacas parser would encounter {1+2}
(typed by the user) and generate a syntax error before it has a chance to load
the definition of the operator "{+}".

		Object types

Yacas supports two basic kinds of objects: atoms and compounds. Atoms are (integer or real, arbitrary-precision) numbers such as {2.71828}, symbolic variables such as {A3} and character strings. Compounds include functions and expressions, e.g. {Cos(a-b)} and lists, e.g. {{1+a,2+b,3+c}}.

The type of an object is returned by the built-in function {Type}, for example:

	In> Type(a);
	Out> "";
	In> Type(F(x));
	Out> "F";
	In> Type(x+y);
	Out> "+";
	In> Type({1,2,3});
	Out> "List";

Internally, atoms are stored as strings and compounds as lists. (The Yacas
lexical analyzer is case-sensitive, so {List} and
{list} are different atoms.) The functions
{String()} and {Atom()} convert between atoms and
strings. A Yacas list {{1,2,3}} is internally a list
{(List 1 2 3)} which is the same as a function call
{List(1,2,3)} and for this reason the "type" of a list is the
string {"List"}. During evaluation, atoms can be interpreted as
numbers, or as variables that may be bound to some value, while compounds are
interpreted as function calls.

Note that atoms that result from an {Atom()} call may be
invalid and never evaluate to anything. For example,
{Atom(3X)} is an atom with string representation "3X" but
with no other properties.

Currently, no other lowest-level objects are provided by the core engine besides numbers, atoms, strings, and lists. There is, however, a possibility to link some externally compiled code that will provide additional types of objects.
Those will be available in Yacas as "generic objects."
For example, fixed-size arrays are implemented in this way.

		Yacas evaluation scheme

Evaluation of an object is performed either explicitly by the built-in 
command {Eval()} or implicitly when assigning variables or calling functions 
with the object as argument (except when a function does not evaluate that 
argument). Evaluation of an object can be explicitly inhibited using {Hold()}. 
To make a function not evaluate one of its arguments, a {HoldArg(funcname, argname)} 
must be declared for that function.

Internally, all expressions are either atoms or lists (perhaps nested). Use
{FullForm()} to see the internal form of an expression. A Yacas list expression
written as {{a, b}} is represented internally as {(List a b)}, equivalently to
a function call {List(a,b)}.

Evaluation of an atom goes as follows: if the atom is bound locally as a
variable, the object it is bound to is returned, otherwise, if it is bound as a
global variable then that is returned. Otherwise, the atom is returned
unevaluated.
Note that if an atom is bound to an expression, that expression is considered as final and is not evaluated again.

Internal lists of atoms are generally interpreted in the following way: the first
atom of the list is some command, and the atoms following in the list are
considered the arguments. The engine first tries to find out if it is a
built-in command (core function). In that case, the function is executed.
Otherwise, it could be a user-defined function (with a "rule database"),
and in that case the rules from the database are applied to it. If none of
the rules are applicable, or if no rules are defined for it, the object is
returned unevaluated.

Application of a rule to an expression transforms it into a different
expression to which other rules may be applicable. Transformation by matching
rules continues until no more rules are applicable, or until a "terminating"
rule is encountered. A "terminating" rule is one that returns {Hold()} or
{UnList()} of some expression. Calling these functions gives an unevaluated
expression because it terminates the process of evaluation itself.

The main properties of this scheme are the following. When objects are
assigned to variables, they generally are evaluated (except if you are
using the {Hold()} function) because assignment
{var := value} is really a function call to
{Set(var, value)} and this function evaluates its second
argument (but not its first argument). When referencing that variable
again, the object which is its value will not be re-evaluated. Also, the
default behavior of the engine is to return the original expression if it
could not be evaluated. This is a desired behavior if evaluation is used
for simplifying expressions.

One major design flaw in Yacas (one that other functional languages like
LISP also have) is that when some expression is re-evaluated in another
environment, the local variables contained in the expression to be
evaluated might have a different meaning. In this case it might be useful
to use the functions {LocalSymbols} and
{TemplateFunction}. Calling

	LocalSymbols(a,b)
	a*b;
results in "{a}" and "{b}" in the multiplication being substituted with
unique symbols that can not clash with other variables that may be used
elsewhere. Use {TemplateFunction} instead of
{Function} to define a function whose parameters should be
treated as unique symbols.

Consider the following example:

	In> f1(x):=Apply("+",{x,x});
	Out> True

The function {f1} simply adds its argument to itself. Now calling 
this function with some argument:

	In> f1(Sin(a))
	Out> 2*Sin(a)

yields the expected result. However, if we pass as an argument an 
expression containing the variable {x}, things go wrong:

	In> f1(Sin(x))
	Out> 2*Sin(Sin(x))

This happens because within the function, {x} is bound to {Sin(x)},
and since it is passed as an argument to {Apply} it will be re-evaluated,
resulting in {Sin(Sin(x))}. {TemplateFunction} solves this by making sure
the arguments can not collide like this (by using {LocalSymbols}:

	In> TemplateFunction("f2",{x}) Apply("+",{x,x});
	Out> True
	In> f2(Sin(a))
	Out> 2*Sin(a)
	In> f2(Sin(x))
	Out> 2*Sin(x)

In general one has to be careful when functions like {Apply}, {Map} 
or {Eval} (or derivatives) are used.


		Rules

<i>Rules</i> are special properties of functions that
are applied when the function object is being evaluated. A function object could
have just one rule bound to it; this is similar to a "subroutine" having a "function
body" in usual procedural languages. However, Yacas function objects can
also have several rules bound to them. This is analogous of having several
alternative "function bodies" that are executed under different
circumstances. This design is more suitable for symbolic manipulations.

A function is identified by its name as returned by {Type}
and the number of arguments, or "arity". The same name can be used with
different arities to define different functions: {f(x)} is
said to "have arity 1" and {f(x,y)} has arity 2. Each of
these functions may possess its own set of specific rules, which we shall
call a "rule database" of a function.

Each function should be first declared with the built-in command
{RuleBase} as follows:

	RuleBase("FunctionName",{argument list});

So, a new (and empty) rule database for {f(x,y)} could be
created by typing {RuleBase("f",{x,y})}. The names for the
arguments "x" and "y" here are arbitrary, but they will be globally stored
and must be later used in descriptions of particular rules for the function
{f}. After the new rulebase declaration, the evaluation engine of Yacas
will begin to really recognize {f} as a function, even though no function
body or equivalently no rules have been defined for it yet.

The shorthand operator {:=} for creating user functions that
we illustrated in the tutorial is actually defined in the scripts and it
makes the requisite call to the {RuleBase()} function.
After a {RuleBase()} call you can specify parsing
properties for the function; for example, you could make it an
infix or bodied operator.

Now we can add some rules to the rule database for a function. A rule
simply states that if a specific function object with a specific arity is
encountered in  an expression and if a certain predicate is true, then
Yacas should replace this function with some other expression. To tell Yacas about a new rule you can use the built-in {Rule} command. This command is what does the real work for the somewhat more aesthetically pleasing {... # ... <-- ...} construct we have seen in the tutorial. You do not have to call {RuleBase()} explicitly if you use that construct.

Here is the general syntax for a {Rule()} call:

	Rule("foo", arity, precedence, pred) body;
This specifies that for function {foo} with given
{arity} ({foo(a,b)} has arity 2), there is a
rule that if {pred} is true, then {body}
should be evaluated, and the original expression replaced by the result.
Predicate and body can use the symbolic names of arguments that were
declared in the {RuleBase} call.

All rules for a given function can be erased with a call to {Retract(funcname, arity)}. This is useful, for instance, when too many rules have been entered in the interactive mode. This call undefines the function and also invalidates the {RuleBase} declaration.

You can specify that function arguments are not evaluated before they are
bound to the parameter: {HoldArg("foo",a)} would then
declare that the a arguments in both {foo(a)} and {foo(a,b)} should not be evaluated before bound to {a}. Here
the argument name {a} should be the same as that used in the {RuleBase()} call when declaring these functions.
Inhibiting evaluation of certain arguments is useful for procedures
performing actions based partly on a variable in the expression, such as
integration, differentiation, looping, etc., and will be typically used for
functions that are algorithmic and procedural by
nature.

Rule-based programming normally makes heavy use of recursion and it is
important to control the order in which replacement rules are to be
applied. For this purpose, each rule is given a <i>precedence</i>.
Precedences go from low to high, so all rules with precedence 0 will be
tried before any rule with precedence 1.

You can assign several rules to one and the same function, as long as some of the predicates differ. If none of the predicates are true,
the function is returned with its arguments evaluated.

This scheme is slightly slower for ordinary functions that just have one
rule (with the predicate {True}), but it is a desired
behavior for symbolic manipulation. You can gradually build up your own functions, incrementally testing their properties.

		Examples of using rules

As a simple illustration, here are the actual {RuleBase()}
and {Rule()} calls needed to define the factorial function:

	In> RuleBase("f",{n});
	Out> True;
	In> Rule("f", 1, 10, n=0) 1;
	Out> True;
	In> Rule("f", 1, 20, IsInteger(n) \
	  And n>0) n*f(n-1);
	Out> True;

This definition is entirely equivalent to the one in the tutorial. {f(4)}
should now return 24, while {f(a)} should return just {f(a)} if {a} is not
bound to any value.

The {Rule} commands in this example specified two rules for function {f}
with arity 1: one rule with precedence 10 and predicate {n=0}, and another with precedence 20 and the predicate that returns {True} only
if {n} is a positive integer. Rules with lowest precedence get evaluated
first, so the rule with precedence 10 will be tried before the rule with
precedence 20. Note that the predicates and the body use the name "n"
declared by the {RuleBase()} call.

After declaring {RuleBase()} for a function, you could
tell the parser to treat this function as a postfix operator:

	In> Postfix("f");
	Out> True;
	In> 4 f;
	Out> 24;

There is already a function {Function} defined in the
standard scripts that allows you to construct simple functions. An example
would be

	Function ("FirstOf", {list})  list[1] ;

which simply returns the first element of a list. This could also have
been written as

	Function("FirstOf", {list})
	[
	  list[1] ;
	];

As mentioned before, the brackets {[ ]} are also used to combine multiple
operations to be performed one after the other. The result of the last
performed action is returned.

Finally, the function {FirstOf} could also have been
defined by typing

	FirstOf(list):=list[1] ;

		Structured programming and control flow

Some functions useful for control flow are already defined in Yacas's standard library. Let's look at a possible definition of a looping function {ForEach}. We shall here consider a somewhat simple-minded definition, while the actual {ForEach} as defined in the standard script "controlflow" is a little more sophisticated.

	Function("ForEach",{foreachitem,
	  foreachlist,foreachbody})
	[
	   Local(foreachi,foreachlen);
	   foreachlen:=Length(foreachlist);
	   foreachi:=0;
	   While (foreachi < foreachlen)
	   [
	     foreachi++;
	     MacroLocal(foreachitem);
	     MacroSet(foreachitem,
		   foreachlist[foreachi]);
	     Eval(foreachbody);
	   ];
	];
	
	Bodied("ForEach");
	UnFence("ForEach",3);
	HoldArg("ForEach",foreachitem);
	HoldArg("ForEach",foreachbody);

Functions like this should probably be defined in a separate file. You can
load such a file with the command {Load("file")}. This is an example of a macro-like function.  Let's first look at the last few
lines. There is a {Bodied(...)} call, which states that the syntax for the function {ForEach()} is {ForEach(item,{list}) body;}
-- that is, the last argument to the command {ForEach} should be outside its
brackets. {UnFence(...)} states that this function can use the local
variables of the calling function. This is necessary, since the body to be
evaluated for each item will probably use some local variables from that
surrounding.

Finally, {HoldArg("function",argument)} specifies that
the argument "{argument}" should not be evaluated before being
bound to that variable. This holds for {foreachitem} and
{foreachbody}, since {foreachitem} specifies
a variable to be set to that value, and {foreachbody} is the
expression that should be evaluated <i>after</i> that variable
is set.

Inside the body of the function definition there are calls to {Local(...)}.
{Local()} declares some local variable that will only be visible within a block
{[ ... ]}. The command {MacroLocal()} works almost the same. The difference is
that it evaluates its arguments before performing the action on it. This is
needed in this case, because the variable {foreachitem} is bound to a variable
to be used as the loop iterator, and it is <i>the variable it is bound to</i>
that we want to make local, not {foreachitem} itself. {MacroSet()} works
similarly: it does the same as {Set()} except that it also first evaluates the
first argument, thus setting the variable requested by the user of this
function. The {Macro}... functions in the built-in functions generally perform
the same action as their non-macro versions, apart from evaluating an argument
it would otherwise not evaluate.

To see the function in action, you could type:

	ForEach(i,{1,2,3}) [Write(i); NewLine();];
This should print 1, 2 and 3, each on a new line.

Note: the variable names "foreach..." have been chosen so they won't get
confused with normal variables you use. This is a major design flaw in this
language. Suppose there was a local variable {foreachitem}, defined in the
calling function, and used in {foreachbody}. These two would collide, and the
interpreter would use only the last defined version. In general, when writing a
function that calls {Eval()}, it is a good idea to use variable names that can
not collide with user's variables. This is generally the single largest cause
of bugs when writing programs in Yacas. This issue should be addressed in the
future.

		Additional syntactic sugar

The parser is extended slightly to allow for fancier constructs.

*	Lists, e.g. {{a,b}}. This then
is parsed into the internal notation {(List a b)} , but
will be printed again as {{a,b};}
*	Statement blocks such as
{[} statement1 {;} statement2{;];}. This
is parsed into a Lisp object {(Prog} {(}statement1 {)} {(}statement2 {))}, and printed out again in the proper form.
*	 Object argument accessors in the form of
{expr[ index ]}. These
are mapped internally to {Nth(expr,index)}. The value of {index}=0 returns the
operator of the object, {index}=1 the first argument, etc. So,
if {expr} is {foo(bar)}, then
{expr[0]} returns {foo}, and
{expr[1]} returns {bar}. Since lists of the form
{{...}} are essentially the same as
{List(...)}, the same accessors
can be used on lists.

*	Function blocks such as
	While (i < 10)
	  [
		Write(i);
		i:=i+1;
	  ];
The expression directly following the {While(...)} block is added as a last argument to the {While(...)} call. So {While(a)b;} is parsed to the internal form {(While a b).}

This scheme allows coding the algorithms in an almost C-like syntax.

Strings are generally represented with quotes around them, e.g.
"this is a string". Backslash {\} in a string will unconditionally add the
next character to the string, so a quote can be added with {\"} (a backslash-quote sequence).

		Using "Macro rules" (e.g. {NFunction})

The Yacas language allows to have rules whose definitions are generated at
runtime. In other words, it is possible to write rules (or "functions") that,
as a side-effect, will define other rules, and those other rules will depend on
some parts of the expression the original function was applied to.

This is accomplished using functions {MacroRuleBase}, {MacroRule}, {MacroRulePattern}. These functions evaluate their arguments (including the rule name, predicate and body) and define the rule that results from this evaluation.

Normal, "non-Macro" calls such as {Rule()} will not evaluate their arguments and this is a desired feature. For example, suppose we defined a new predicate like this,
	RuleBase("IsIntegerOrString, {x});
	Rule("IsIntegerOrString", 1, 1, True)
		IsInteger(x) And IsString(x);
If the {Rule()} call were to evaluate its arguments, then the "body" argument,
{IsInteger(x) And IsString(x)}, would be evaluated to {False} since {x} is an
atom, so we would have defined the predicate to be always {False}, which is not
at all what we meant to do. For this reason, the {Rule} calls do not evaluate
their arguments.

Consider however the following situation. Suppose we have a function
{f(arglist)} where {arglist} is its list of arguments, and suppose we want to
define a function {Nf(arglist)} with the same arguments which will evaluate
{f(arglist)} and return only when all arguments from {arglist} are numbers, and
return unevaluated {Nf(arglist)} otherwise. This can of course be done by a usual rule such as
	Rule("Nf", 3, 0, IsNumericList({x,y,z}))
	  <-- "f" @ {x,y,z};
Here {IsNumericList} is a predicate that checks whether all elements of a given
list are numbers. (We deliberately used a {Rule} call instead of an
easier-to-read {<--} operator to make it easier to compare with what follows.)

However, this will have to be done for every function {f} separately. We would
like to define a procedure that will define {Nf}, given <i>any</i> function
{f}. We would like to use it like this:
	NFunction("Nf", "f", {x,y,z});
After this function call we expect to be able to use the function {Nf}.

Here is how we could naively try to implement {NFunction} (and fail):

	NFunction(new'name, old'name, arg'list) := [
	  MacroRuleBase(new'name, arg'list);
	  MacroRule(new'name, Length(arg'list), 0,
	    IsNumericList(arg'list)
	    )
	  new'name @ arg'list;
	];

Now, this just does not do anything remotely right. {MacroRule} evaluates its 
arguments. Since {arg'list} is an atom and not a list of numbers at the time we are defining this,
{IsNumericList(arg'list)} will evaluate to {False} and the new
rule will be defined with a predicate that is always {False}, i.e. it will be
never applied.

The right way to figure this out is to realize that the {MacroRule} call
evaluates all its arguments and passes the results to a {Rule} call. So we need
to see exactly what {Rule()} call we need to produce and then we need to
prepare the arguments of {MacroRule} so that they evaluate to the right values.
The {Rule()} call we need is something like this:

	Rule("actual new name", <actual # of args>, 0,
	  IsNumericList({actual arg list})
	)  "actual new name" @ {actual arg list};

Note that we need to produce expressions such as {"new name" @ arg'list} and not <i>results</i> of evaluation of these expressions. We can produce these expressions by using {UnList()}, e.g.
	UnList({Atom("@"), "Sin", {x}})
produces
	"Sin" @ {x};
but not {Sin(x)}, and
	UnList({IsNumericList, {1,2,x}})
produces the expression
	IsNumericList({1,2,x});
which is not further evaluated.

Here is a second version of {NFunction()} that works:

	NFunction(new'name, old'name, arg'list) := [
	  MacroRuleBase(new'name, arg'list);
	  MacroRule(new'name, Length(arg'list), 0,
	    UnList({IsNumericList, arg'list})
	  )
	    UnList({Atom("@"), old'name, arg'list});
	];
Note that we used {Atom("@")} rather than just the bare atom {@} because {@} is
a prefix operator and prefix operator names as bare atoms do not parse (they
would be confused with applications of a prefix operator to what follows).

Finally, there is a more concise (but less general) way of defining
{NFunction()} for functions with known number of arguments, using the
backquoting mechanism. The backquote operation will first substitute variables
in an expression, without evaluating anything else, and then will evaluate the
resulting expression a second time. The code for functions of just one variable may look like this:

	N1Function(new'name, old'name) :=
		`( @new'name(x_IsNumber) <-- @old'name(x) );
This executes a little slower than the above version, because the backquote
needs to traverse the expression twice, but makes for much more readable code.


		Macro expansion

Yacas supports macro expansion (back-quoting). An expression can be
back-quoted by putting a {`} in front of it. Within the back-quoted
expression, all atoms that have a {@} in front of them get replaced
with the value of that atom (treated as a variable), and then the
resulting expression is evaluated:

	In> x:=y
	Out> y;
	In> `(@x:=2)
	Out> 2;
	In> x
	Out> y;
	In> y
	Out> 2;

This is useful in cases where within an expression one sub-expression
is not evaluated. For instance, transformation rules can be built
dynamically, before being declared. This is a particularly powerful
feature that allows a programmer to write programs that write programs.
The idea is borrowed from Lisp.

As the above example shows, there are similarities with the {Macro...}
functions, that serve the same purpose for specific expressions.
For example, for the above code, one could also have called {MacroSet}:

	In> MacroSet(x,3)
	Out> True;
	In> x
	Out> y;
	In> y
	Out> 3;

The difference is that {MacroSet}, and in general the {Macro...}
functions, are faster than their back-quoted counterparts.
This is because with back-quoting, first a new expression is 
built before it is evaluated. The advantages of back-quoting
are readability and flexibility (the number of {Macro...}
functions is limited, whereas back-quoting can be used anywhere).

When an {@} operator is placed in front of a function call, the
function call is replaced:

	In> plus:=Add
	Out> Add;
	In> `(@plus(1,2,3))
	Out> 6;

Application of pure functions is also possible (as of version 1.0.53)
by using macro expansion:

	In> pure:={{a,b},a+b}; 
	Out> {{a,b},a+b};
	In> ` @pure(2,3); 
	Out> 5;

Pure (nameless) functions are useful for declaring a temporary function,
that has functionality depending on the current environment it is in,
or as a way to call driver functions. In the case of drivers (interfaces
to specific functionality), a variable can be bound to a function to 
be evaluated to perform a specific task. That way several drivers can
be around, with one bound to the variables holding the functions that
will be called.

		Scope of variable bindings

When setting variables or retrieving variable values, variables are
automatically bound global by default. You can explicitly specify variables to
be local to a block such as a function body; this will make them invisible
outside the block. Blocks have the form {[} statement1{;} statement2{;} {]} and local variables are declared by the
{Local()} function.

When entering a block, a new stack frame is pushed for the local variables; it
means that the code inside a block doesn't see the local variables of the <i>caller</i> either!
You can tell the interpreter that a function should see local variables of the
calling environment; to do this, declare
	UnFence(funcname, arity)
on that function.


			Evaluation of expressions

When programming in some language, it helps to have a mental
model of what goes on behind the scenes when evaluating expressions,
or in this case simplifying expressions.

This section aims to explain how evaluation (and simplification)
of expressions works internally, in {Yacas}.

		The LISP heritage

	    Representation of expressions

Much of the inner workings is based on how LISP-like languages
are built up. When an expression is entered, or composed in some
fashion, it is converted into a prefix form much like you get
in LISP:

	a+b    ->    (+ a b)
	Sin(a) ->    (Sin a)
Here the sub-expression is changed into a list of so-called "atoms", 
where the first atom is a function name of the function to be 
invoked, and the atoms following are the arguments to be passed in
as parameters to that function. 

{Yacas} has the function {FullForm} to show the internal representation:

	In> FullForm(a+b)
	(+ a b )
	Out> a+b;
	In> FullForm(Sin(a))
	(Sin a )
	Out> Sin(a);
	In> FullForm(a+b+c)
	(+ (+ a b )c )
	Out> a+b+c;

The internal representation is very close to what {FullForm} shows
on screen. {a+b+c} would be {(+ (+ a b )c )} internally, or:

	()
	|
	|
	+  -> () -> c
	       |
	       |
	       + -> a -> b


	    Evaluation

An expression like described above is done in the following manner:
first the arguments are evaluated (if they need to be evaluated,
{Yacas} can be told to not evaluate certain parameters to functions),
and only then are these arguments passed in to the function for
evaluation. They are passed in by binding local variables to the
values, so these arguments are available as local values.

For instance, suppose we are evaluating {2*3+4}. This first gets
changed to the internal representation {(+ (* 2 3 )4 )}. Then,
during evaluation, the top expression refers to function "{+}".
Its arguments are {(* 2 3)} and {4}. First {(* 2 3)} gets evaluated.
This is a function call to the function "{*}" with arguments {2}
and {3}, which evaluate to themselves. Then the function "{*}" is
invoked with these arguments. The {Yacas} standard script library
has code that accepts numeric input and performs the multiplication
numerically, resulting in {6}. 

The second argument to the top-level "{+}" is {4}, which evaluates
to itself. 

Now, both arguments to the "{+}" function have been evaluated, and
the results are {6} and {4}. Now the "{+}" function is invoked.
This function also has code in the script library to actually
perform the addition when the arguments are numeric, so the result
is 10:

	In> FullForm(Hold(2*3+4))
	(+ (* 2 3 )4 )
	Out> 2*3+4;
	In> 2*3+4
	Out> 10;

Note that in {Yacas}, the script language does not define a "{+}" function
in the core. This and other functions are all implemented in the script library.
The feature "when the arguments to "{+}" are numeric, perform the numeric
addition" is considered to be a "policy" which should be configurable.
It should not be a part of the core language.

It is surprisingly difficult to keep in mind that evaluation is 
bottom up, and that arguments are evaluated before the function call
is evaluated. In some sense, you might feel that the evaluation of the 
arguments is part of evaluation of the function. It is not. Arguments
are evaluated before the function gets called.

Suppose we define the function {f}, which adds two numbers, and
traces itself, as:

	In> f(a,b):= \
	In> [\
	In> Local(result);\
	In> Echo("Enter f with arguments ",a,b);\
	In> result:=a+b;\
	In> Echo("Leave f with result ",result);\
	In> result;\
	In> ];
	Out> True;

Then the following interaction shows this principle:


	In> f(f(2,3),4)
	Enter f with arguments 2 3 
	Leave f with result 5 
	Enter f with arguments 5 4 
	Leave f with result 9 
	Out> 9;

The first Enter/Leave combination is for {f(2,3)}, and only then is
the outer call to {f} entered.

This has important consequences for the way {Yacas} simplifies
expressions: the expression trees are traversed bottom up, as 
the lowest parts of the expression trees are simplified first,
before being passed along up to the calling function. 

		{Yacas}-specific extensions for CAS implementations

{Yacas} has a few language features specifically designed for use
when implementing a CAS.

	    The transformation rules

Working with transformation rules is explained in the introduction
and tutorial book. This section mainly deals with how {Yacas}
works with transformation rules under the hood.

A transformation rule consists of two parts: a condition that
an expression should match, and a result to be substituted for
the expression if the condition holds. The most common way
to specify a condition is a pattern to be matched to an expression.

A pattern is again simply an expression, stored in internal format:

	In> FullForm(a_IsInteger+b_IsInteger*(_x))
	(+ (_ a IsInteger )(* (_ b IsInteger )(_ x )))
	Out> a _IsInteger+b _IsInteger*_x;

{Yacas} maintains structures of transformation rules, and tries
to match them to the expression being evaluated. It first tries
to match the structure of the pattern to the expression. In the above
case, it tries to match to {a+b*x}. If this matches, local variables
{a}, {b} and {x} are declared and assigned the sub-trees of the expression
being matched. Then the predicates are tried on each of them: in this
case, {IsInteger(a)} and {IsInteger(b)} should both return {True}.

Not shown in the above case, are post-predicates. They get evaluated
afterwards. This post-predicate must also evaluate to {True}.
If the structure of the expression matches the structure of the 
pattern, and all predicates evaluate to {True}, the pattern matches
and the transformation rule is applied, meaning the right hand side
is evaluated, with the local variables mentioned in the pattern
assigned. This evaluation means all transformation rules are re-applied
to the right-hand side of the expression.

Note that the arguments to a function are evaluated first, and only
then is the function itself called. So the arguments are evaluated,
and then the transformation rules applied on it. The main function
defines its parameters also, so these get assigned to local variables
also, before trying the patterns with their associated local variables.

Here is an example making the fact that the names in a pattern are
local variables more explicit:

	In> f1(_x,_a) <-- x+a
	Out> True;
	In> f2(_x,_a) <-- [Local(a); x+a;];
	Out> True;
	In> f1(1,2)
	Out> 3;
	In> f2(1,2)
	Out> a+1;

	    Using different rules in different cases

In a lot of cases, the algorithm to be invoked depends on the type
of the arguments. Or the result depends on the form of the input
expression. This results in the typical "case" or "switch" statement,
where the code to evaluate to determine the result depends on the 
form of the input expression, or the type of the arguments, or some other conditions.

{Yacas} allows to define several transformation rules
for one and the same function, if the rules are to be applied under different conditions.

Suppose the function
{f} is defined, a factorial function:


	10 # f(0) <-- 1;
	20 # f(n_IsPositiveInteger) <-- n*f(n-1);

Then interaction can look like:

	In> f(3)
	Out> 6;
	In> f(a)
	Out> f(a);

If the left hand side is matched by the expression being considered,
then the right hand side is evaluated. A subtle but important thing
to note is that this means that the whole body of transformation rules
is thus re-applied to the right-hand side of the {<--} operator.

Evaluation goes bottom-up, evaluating (simplifying) the lowest parts
of a tree first, but for a tree that matches a transformation rule,
the substitution essentially means return the result of evaluating the
right-hand side. Transformation rules are re-applied, on the right hand
side of the transformation rule, and the original expression can be thought
of as been substituted by the result of evaluating this right-hand side,
which is supposed to be a "simpler" expression, or a result closer to what
the user wants.

Internally, the function {f} is built up to resemble the following
pseudo-code:

	f(n)
	{
	   if (n = 1)
	     return 1;
	   else if (IsPositiveInteger(n))
	     return n*f(n-1);
	   else return f(n) unevaluated;
	}

The transformation rules are thus combined into one big 
statement that gets executed, with each transformation 
rule being a if-clause in the statement to be evaluated.
Transformation rules can be spread over different files,
and combined in functional groups. This adds to the readability.
The alternative is to write the full body of each function as
one big routine, which becomes harder to maintain as the function
becomes larger and larger, and hard or impossible to extend.

One nice feature is that functionality is easy to extend without
modifying the original source code:

	In> Ln(x*y)
	Out> Ln(x*y);
	In> Ln(_x*_y) <-- Ln(x) + Ln(y)
	Out> True;
	In> Ln(x*y)
	Out> Ln(x)+Ln(y);

This is generally not advisable, due to the fact that it alters
the behavior of the entire system. But it can be useful in some
instances. For instance, when introducing a new function {f(x)},
one can decide to define a derivative explicitly, and a way
to simplify it numerically:

	In> f(_x)_InNumericMode() <-- Exp(x)
	Out> True;
	In> (Deriv(_x)f(_y)) <-- f(y)*(Deriv(x)y);
	Out> True;
	In> f(2)
	Out> f(2);
	In> N(f(2))
	Out> 7.3890560989;
	In> Exp(2)
	Out> Exp(2);
	In> N(Exp(2))
	Out> 7.3890560989;
	In> D(x)f(a*x)
	Out> f(a*x)*a;


	    The "Evaluation is Simplification" hack

One of the ideas behind the {Yacas} scripting language is that evaluation
is used for simplifying expressions.
One consequence of this is that
objects can be returned unevaluated when they can not be simplified
further. This happens to variables that are not assigned, functions that
are not defined, or function invocations where the arguments passed
in as parameters are not actually handled by any code in the scripts.
An integral that can not be performed by {Yacas} should be returned 
unevaluated:

	In> 2+3
	Out> 5;
	In> a+b
	Out> a+b;
	In> Sin(a)
	Out> Sin(a);
	In> Sin(0)
	Out> 0;
	In> Integrate(x)Ln(x)
	Out> x*Ln(x)-x;
	In> Integrate(x)Ln(Sin(x))
	Out> Integrate(x)Ln(Sin(x));
	In> a!
	Out> a!;
	In> 3!
	Out> 6;

Other languages usually do not allow evaluation of unbound variables,
or undefined functions. In {Yacas}, these are interpreted as some yet
undefined global variables or functions, and returned unevaluated.

		Destructive operations

{Yacas} tries to keep as few copies of objects in memory as
possible. Thus when assigning the value of one variable to another,
a reference is copied, and both variables refer to the same memory,
physically. This is relevant for programming; for example, one
should use {FlatCopy} to actually make a new copy of an object.
Another feature relevant to reference semantics is "destructive 
operations"; these are functions that modify their arguments rather than work on a copy. Destructive operations on lists are generally recognized 
because their name starts with "Destructive",
e.g. {DestructiveDelete}. One other destructive
operation is assignment of a list element through {list[index] := ...}.

Some examples to illustrate destructive operations on lists:

	In> x1:={a,b,c}
	Out> {a,b,c};
A list {x1} is created.
	In> FullForm(x1)
	(List a b c )
	Out> {a,b,c};
	In> x2:=z:x1
	Out> {z,a,b,c};
A new list {x2} is {z} appended to {x1}. The {:} operation creates a copy of {x1} before appending, so {x1} is unchanged by this.
	In> FullForm(x2)
	(List z a b c )
	Out> {z,a,b,c};
	In> x2[1]:=y
	Out> True;
We have modified the first element of {x2}, but {x1} is still the same.
	In> x2
	Out> {y,a,b,c};
	In> x1
	Out> {a,b,c};
	In> x2[2]:=A
	Out> True;
We have modified the second element of {x2}, but {x1} is still the same.
	In> x2
	Out> {y,A,b,c};
	In> x1
	Out> {a,b,c};
	In> x2:=x1
	Out> {A,b,c};
Now {x2} and {x1} refer to the same list.
	In> x2[1]:=A
	Out> True;
We have modified the first element of {x2}, and {x1} is also modified.
	In> x2
	Out> {A,b,c};
	In> x1
	Out> {A,b,c};

A programmer should always be cautious when dealing with 
destructive operations. Sometimes it is not desirable to 
change the original expression.
The language deals with it this way because of performance considerations.
Operations can be made non-destructive by using {FlatCopy}:

	In> x1:={a,b,c}
	Out> {a,b,c};
	In> DestructiveReverse(x1)
	Out> {c,b,a};
	In> x1
	Out> {a};
	In> x1:={a,b,c}
	Out> {a,b,c};
	In> DestructiveReverse(FlatCopy(x1))
	Out> {c,b,a};
	In> x1
	Out> {a,b,c};

{FlatCopy} copies the elements of an expression only at the top level of nesting.
This means that if a list contains sub-lists, they are not copied, but
references to them are copied instead:

	In> dict1:={}
	Out> {};
	In> dict1["name"]:="John";
	Out> True;
	In> dict2:=FlatCopy(dict1)
	Out> {{"name","John"}};
	In> dict2["name"]:="Mark";
	Out> True;
	In> dict1
	Out> {{"name","Mark"}};

A workaround for this is to use {Subst} to copy the entire tree:

	In> dict1:={}
	Out> {};
	In> dict1["name"]:="John";
	Out> True;
	In> dict2:=Subst(a,a)(dict1)
	Out> {{"name","John"}};
	In> dict2["name"]:="Mark";
	Out> True;
	In> dict1
	Out> {{"name","John"}};
	In> dict2
	Out> {{"name","Mark"}};


			Coding style

		Introduction

This chapter intends to describe the coding style and conventions
applied in Yacas in order to make sure the engine always returns
the correct result. This is an attempt at fending off such errors
by combining rule-based programming with a clear coding style
which should make help avoid these mistakes.

		Interactions of rules and types

One unfortunate disadvantage of rule-based programming is that rules
can sometimes cooperate in unwanted ways.

One example of how rules can produce unwanted results is the rule {a*0 <-- 0}.
This would always seem to be true. However, when a is a vector, e.g.
{a:={b,c,d}}, then {a*0} should actually return {{0,0,0}}, that is, a zero
vector. The rule {a*0 <-- 0} actually changes the type of the expression from a
vector to an integer! This can have severe consequences when other functions
using this expressions as an argument expect a vector, or even worse, have a
definition of how to work on vectors, and a different one for working on
numbers.

When writing rules for an operator, it is assumed that the operator working on
arguments, e.g. {Cos} or {*}, will always have the same properties regardless
of the arguments. The Taylor series expansion of $Cos(a)$ is the same
regardless of whether $a$ is a real number, complex number or even a matrix.
Certain trigonometric identities should hold for the {Cos} function, regardless
of the type of its argument.

If a function is defined which does not adhere to these rules when applied
to another type, a different function name should be used, to avoid confusion.

By default, if a variable has not been bound yet, it is assumed to
be a number. If it is in fact a more complex object, e.g. a vector,
then you can declare it to be an "incomplete type" vector, using
{Object("IsVector",x)} instead of {x}. This  expression will evaluate to {x} if and
only if {x} is a vector at that moment of evaluation. Otherwise
it returns unevaluated, and thus stays an incomplete type.

So this means the type of a variable is numeric unless otherwise
stated by the user, using the "{Object}" command. No rules should
ever work on incomplete types. It is just meant for delayed
simplification.

The topic of implicit type of an object is important, since many rules
need to assume something about their argument types.

		Ordering of rules

The implementor of a rule set can specify the order in which rules should be
tried. This can be used to let the engine try more specific rules (those
involving more elements in the pattern) before trying less specific rules.
Ordering of rules can be also explicitly given by precedence numbers. The Yacas
engine will split the expression into subexpressions, and will try to apply all
matching rules to a given subexpression in order of precedence.

A rule with precedence 100 is defined by the syntax such as

	100 # f(_x + _y) <-- f(x) + f(y);

The problem mentioned above with a rule for vectors and scalars could be solved by making two rules:

*	1. $a*b$ (if $b$ is a vector and $a$ is a number) {<--} return vector of each component multiplied by $a$.
*	1. $a*0$ {<--} $0$

So vector multiplication would be tried first.

The ordering of the precedence of the rules in the standard math
scripts is currently:

*	50-60: Args are numbers: directly calculate. These are put in the beginning, so they are tried first. This is useful for quickly obtaining numeric results if all the arguments are numeric already, and symbolic transformations are not necessary.
*	100-199: tautologies. Transformations that do not change the type of the argument, and are always true.
*	200-399: type-specific transformations. Transformations for specific types of objects.
*	400-599: transformations on scalars (variables are assumed to be scalars). Meaning transformations that can potentially change the type of an argument.

		Writing new library functions

When you implement new library functions, you need to make your new code compatible and consistent with the rest of the library. Here are some relevant considerations.

	    To evaluate or not to evaluate

Currently, a general policy in the library is that functions do nothing
unless their arguments actually allow something to be evaluated. For
instance, if the function expects a variable name but instead gets a
list, or expects a list but instead gets a string, in most cases it
seems to be a good idea to do nothing and return unevaluated. The
unevaluated expression will propagate and will be easy to spot. Most
functions can accomplish this by using type-checking predicates such as
{IsInteger} in rules.

When dealing with numbers, Yacas tries to maintain exact answers as much as
possible and evaluate to floating-point only when explicitly told so (using
{N()}). The general evaluation strategy for numerical functions such as {Sin} or {Gamma}
is the following:

*	1. If {InNumericMode()} returns {True} and the arguments are numbers (perhaps complex
numbers), the function should evaluate its result in floating-point to current precision.
*	2. Otherwise, if the arguments are such that the result can be calculated exactly, it should be
evaluated and returned. E.g. {Sin(Pi/2)} returns {1}.
*	3. Otherwise the function should return unevaluated (but usually with its arguments evaluated).

Here are some examples of this behavior:

	In> Sin(3)
	Out> Sin(3);
	In> Gamma(8)
	Out> 5040;
	In> Gamma(-11/2)
	Out> (64*Sqrt(Pi))/10395;
	In> Gamma(8/7)
	Out> Gamma(8/7);
	In> N(Gamma(8/7))
	Out> 0.9354375629;
	In> N(Gamma(8/7+x))
	Out> Gamma(x+1.1428571428);
	In> Gamma(12/6+x)
	Out> Gamma(x+2);

To implement this behavior, {Gamma} and other mathematical functions usually
have two variants: the "symbolic" one and the "numerical" one. For instance,
there are {Sin} and {MathSin}, {Ln} and {Internal'LnNum}, {Gamma} and {Internal'GammaNum}. (Here
{MathSin} happens to be a core function but it is not essential.) The "numerical"
functions always evaluate to floating-point results. The "symbolic" function
serves as a front-end; it evaluates when the result can be expressed exactly,
or calls the "numerical" function if {InNumericMode()} returns {True}, and otherwise returns
unevaluated.

The "symbolic" function usually has multiple rules while the
"numerical" function is usually just one large block of
number-crunching code.


	    Using {N()} and {InNumericMode()} in scripts

As a rule, {N()} should be avoided in code that implements basic
numerical algorithms. This is because {N()} itself is implemented in
the library and it may need to use some of these algorithms.
Arbitrary-precision math can be handled by core functions such as
{MathDivide}, {MathSin} and so on, without using {N()}. For example, if
your code needs to evaluate $Sqrt(Pi)$ to many digits as an
intermediate result, it is better to write {MathSqrt(Internal'Pi())} than
{N(Sqrt(Pi))} because it makes for faster, more reliable code.

	    Using {Builtin'Precision'Set()}

The usual assumption is that numerical functions will evaluate
floating-point results to the currently set precision. For intermediate
calculations, a higher working precision is sometimes needed. In this
case, your function should set the precision back to the original value
at the end of the calculation and round off the result.

	    Using verbose mode

For routines using complicated algorithms, or when evaluation takes a
long time, it is usually helpful to print some diagnostic information,
so that the user can at least watch some progress. The current
convention is that if {InVerboseMode()} returns {True}, functions may 
print diagnostic information. (But do not print too much!). Verbose
mode is turned on by using the function {V(expression)}. The expression
is evaluated in verbose mode.

	    Procedural programming or rule-based programming?

Two considerations are relevant to this decision. First, whether to use
multiple rules with predicates or one rule with multiple {If()}s.
Consider the following sample code for the "double factorial"
function $n!! :=n*(n-2)*...$ written using predicates and rules:
	1# 0 !! <-- 1;
	1# 1 !! <-- 1;
	2# (n_IsEven) !! <-- 2^(n/2)*n!;
	3# (n_IsOdd) !! <-- n*(n-2)!!;
and an equivalent code with one rule:
	n!! := If(n=0 Or n=1, 1,
	  If(IsEven(n), 2^(n/2)*n!,
	  If(IsOdd(n), n*(n-2)!!, Hold(n!!)))
	);
(Note: This is not the way $n!!$ is implemented in the library.) The first version is a lot more clear. Yacas is very quick in rule matching and evaluation of predicates, so the first version is (marginally) faster. So it seems better to write a few rules with predicates than one rule with multiple {If()} statements.

The second question is whether to use recursion or loops. Recursion
makes code more elegant but it is slower and limited in depth.
Currently the default recursion depth of $1000$ is enough for most
casual calculations and yet catches infinite recursion errors
relatively quickly. Because of clearer code, it seems better to use
recursion in situations where the number of list elements will never
become large. In numerical applications, such as evaluation of Taylor series, recursion usually does not pay off.

		Reporting errors

Errors occurring because of invalid argument types should be reported only if absolutely necessary. (In the future there may be a static type checker implemented that will make explicit checking unnecessary.)

Errors of invalid values, e.g. a negative argument of real logarithm
function, or a malformed list, mean that a human has probably made a
mistake, so the errors need to be reported. "Internal errors", i.e.
program bugs, certainly need to be reported.

There are currently two facilities for reporting errors: a "hard" one and a "soft" one.

The "hard" error reporting facility is the function {Check}. For example, if {x}={-1}, then
	Check(x>0,"bad x");
will immediately halt the execution of a Yacas script and print the
error messsage. This is implemented as a C++ exception. A drawback of
this mechanism is that the Yacas stack unwinding is not performed by
the Yacas interpreter, so global variables such as {InNumericMode()},
{Verbose}, {Builtin'Precision'Set()} may keep the intermediate values they had been
assigned just before the error occurred. Also, sometimes it is better
for the program to be able to catch the error and continue.

*REM TODO: the above will hopefully be solved soon, as we can now trap exceptions in the scripts.

The "soft" error reporting is provided by the functions {Assert} and {IsError}, e.g.
	Assert("domain", x) x>0;
	If(IsError("domain"), ...);
The error will be reported but execution will continue normally until
some other function "handles" the error (prints the error message or
does something else appropriate for that error). Here the string
{"domain"} is the "error type" and {x} will be the information object
for this error. The error object can be any expression, but it is
probably a good idea to choose a short and descriptive string for the
error type.

The function {GetErrorTableau()} returns an associative list that
accumulates all reported error objects. When errors are "handled",
their objects should be removed from the list. The utility function
{DumpErrors()} is a simple error handler that prints all errors and
clears the list.
Other handlers are {GetError} and {ClearError}. These functions may be used to handle errors when it is safe to do so.

The "soft" error reporting facility is safer and more flexible than the
"hard" facility. However, the disadvantage is that errors are not
reported right away and pointless calculations may continue for a
while until an error is handled.

			Advanced example 1: parsing expressions ({CForm})

In this chapter we show how Yacas represents expressions and how one can build
functions that work on various types of expressions. Our specific example will
be {CForm()}, a standard library function that converts Yacas expressions into
C or C++ code. Although the input format of Yacas expressions is already very
close to C and perhaps could be converted to C by means of an external text
filter, it is instructive to understand how to use Yacas to parse its own
expressions and produce the corresponding C code. Here we shall only design the
core mechanism of {CForm()} and build a limited version that handles only
expressions using the four arithmetic actions.

		Recursive parsing of expression trees

As we have seen in the tutorial, Yacas represents all expressions as trees, or equivalently, as lists of lists. For example, the expression "{a+b+c+d+e}" is for Yacas a tree of depth 4 that could be visualized as

	  "+"
	 a  "+"
	   b  "+"
	     c  "+"
	       d   e
or as a nested list: {("+" a ("+" b ("+" c ("+" d e))))}. 

Complicated expressions are thus built from simple ones in a general and flexible way. If we want a function that acts on sums of any number of terms, we only need to define this function on a single atom and on a sum of two terms, and the Yacas engine will recursively perform the action on the entire tree.

So our first try is to define rules for transforming an atom and for transforming sums and products. The result of {CForm()} will always be a string. We can use recursion like this:

	In> 100 # CForm(a_IsAtom) <-- String(a);
	Out> True;
	In> 100 # CForm(_a + _b) <-- CForm(a) : \
	  " + " : CForm(b);
	Out> True;
	In> 100 # CForm(_a * _b) <-- CForm(a) : \
	  " * " : CForm(b);
	Out> True;

We used the string concatenation operator "{:}" and we
added spaces around the binary operators for clarity. All rules have the
same precedence 100 because there are no conflicts in rule ordering so far:
these rules apply in mutually exclusive cases. Let's try converting some
simple expressions now:

	In> CForm(a+b*c);
	Out> "a + b * c";
	In> CForm(a+b*c*d+e+1+f);
	Out> "a + b * c * d + e + 1 + f";

With only three rules, we were able to process even some complicated expressions. How did it work? We could illustrate the steps Yacas went through when simplifying {CForm(a+b*c)} roughly like this:

	CForm(a+b*c)
	    ... apply 2nd rule
	CForm(a) : " + " : CForm(b*c)
	    ... apply 1st rule and 3rd rule
	"a" : " + " : CForm(b) : " * " : CForm(c)
	    ... apply 1st rule
	"a" : " + " : "b" : " * " : "c"
	    ... concatenate strings
	"a + b * c"

		Handling precedence of infix operations

It seems that recursion will do all the work for us. The power of recursion is indeed great and extensive use of recursion is built into the design of Yacas. We might now add rules for more operators, for example, the unary addition, subtraction and division:

	100 # CForm(+ _a) <-- "+ " : CForm(a);
	100 # CForm(- _a) <-- "- " : CForm(a);
	100 # CForm(_a - _b) <-- CForm(a) : " - "
	  : CForm(b);
	100 # CForm(_a / _b) <-- CForm(a) : " / "
	  : CForm(b);

However, soon we find that we forgot about operator precedence. Our simple-minded {CForm()} gives wrong C code for expressions like this:

	In> CForm( (a+b) * c );
	Out> "a + b * c";

We need to get something like "(a+b)*c" in this case. How would we add a rule to insert parentheses around subexpressions? A simple way out would be to put parentheses around every subexpression, replacing our rules by something like this:

	100 # CForm(_a + _b) <-- "(" : CForm(a)
	  : " + " : CForm(b) : ")";
	100 # CForm(- _a) <-- "(- " : CForm(a)
	  : ")";
and so on. This will always produce correct C code, e.g. in our case "((a+b)*c)", but generally the output will be full of unnecessary parentheses. It is instructive to find a better solution.

We could improve the situation by inserting parentheses only if the higher-order expression requires them; for this to work, we need to make a call such as {CForm(a+b)} aware that the enveloping expression has a multiplication by {c} around the addition {a+b}. This can be implemented by passing an extra argument to {CForm()} that will indicate the precedence of the enveloping operation. A compound expression that uses an infix operator must be bracketed if the precedence of that infix operator is higher than the precedence of the enveloping infix operation.

We shall define an auxiliary function also named "CForm" but with a second argument, the precedence of the enveloping infix operation. If there is no enveloping operation, we shall set the precedence to a large number, e.g. 60000, to indicate that no parentheses should be inserted around the whole expression. The new "CForm(expr, precedence)" will handle two cases: either parentheses are necessary, or unnecessary. For clarity we shall implement these cases in two separate rules. The initial call to "CForm(expr)" will be delegated to "CForm(expr, precedence)".

The precedence values of infix operators such as "{+}" and "{*}" are defined in the Yacas library but may change in a future version. Therefore, we shall not hard-code these precedence values but instead use the function {OpPrecedence()} to determine them. The new rules for the "{+}" operation could look like this:

	PlusPrec := OpPrecedence("+");
	100 # CForm(_expr) <-- CForm(expr, 60000);
	100 # CForm(_a + _b, _prec)_(PlusPrec>prec)
	  <-- "(" : CForm(a, PlusPrec) : " + "
	  : CForm(b, PlusPrec) : ")";
	120 # CForm(_a + _b, _prec) <--
	    CForm(a, PlusPrec) : " + "
		: CForm(b, PlusPrec);
and so on. We omitted the predicate for the last rule because it has a later precedence than the preceding rule.

The way we wrote these rules is unnecessarily repetitive but straightforward and it illustrates the central ideas of expression processing in Yacas. The standard library implements {CForm()} essentially in this way. In addition the library implementation supports standard mathematical functions, arrays and so on, and is somewhat better organized to allow easier extensions and avoid repetition of code.


			Yacas programming pitfalls

No programming language is without programming pitfalls, and
{Yacas} has its fair share of pitfalls.



		All rules are global

All rules are global, and a consequence is that rules can clash or 
silently shadow each other, if the user defines two rules with the 
same patterns and predicates but different bodies.

For example:

	In> f(0) <-- 1
	Out> True;
	In> f(x_IsConstant) <-- Sin(x)/x
	Out> True;

This can happen in practice, if care is not taken. Here two 
transformation rules are defined which both have the same precedence 
(since their precedence was not explicitly set). In that case 
{Yacas} gets to decide which one to try first. 
Such problems can also occur where one transformation rule (possibly
defined in some other file) has a wrong precedence, and thus masks
another transformation rule. It is necessary to think of a scheme
for assigning precedences first. In many cases, the order in which transformation
rules are applied is important.

In the above example, because {Yacas} gets to decide which rule
to try first, it is possible that f(0) invokes the second rule,
which would then mask the first so the first rule is never called.
Indeed, in {Yacas} version 1.0.51, 

	In> f(0)
	Out> Undefined;

The order the rules are applied in is undefined if the precedences
are the same. The precedences should only be the same if order
does not matter. This is the case if, for instance, the two rules apply to different argument patters that could not possibly
mask each other.

The solution could have been either:

	In> 10 # f(0) <-- 1
	Out> True;
	In> 20 # f(x_IsConstant) <-- Sin(x)/x
	Out> True;
	In> f(0)
	Out> 1;
or
	In> f(0) <-- 1
	Out> True;
	In> f(x_IsConstant)_(x != 0) <-- Sin(x)/x
	Out> True;
	In> f(0)
	Out> 1;

So either the rules should have distinct precedences, or they should have
mutually exclusive predicates, so that they do not collide.

		Objects that look like functions

An expression that looks like a "function", for example {AbcDef(x,y)}, is in fact
either a call to a "core function" or  to a "user function", and there 
is a huge difference between the behaviors. Core functions immediately 
evaluate to something, while user functions are really just symbols to which
evaluation rules may or may not be applied. 

For example:

	In> a+b
	Out> a+b;
	In> 2+3
	Out> 5;
	In> MathAdd(a,b)
	In function "MathAdd" : 
	bad argument number 1 (counting from 1)
	The offending argument a evaluated to a
	CommandLine(1) : Invalid argument
	
	In> MathAdd(2,3)
	Out> 5;

The {+} operator will return the object unsimplified if the arguments
are not numeric. The {+} operator is defined in the standard scripts.
{MathAdd}, however, is a function defined in the "core" to
performs the numeric addition. It can only do this if the arguments
are numeric and it fails on symbolic arguments.  (The {+} operator calls {MathAdd} after it has verified that
the arguments passed to it are numeric.)

A core function such as {MathAdd} can never return unevaluated, but an operator such as "{+}" is a "user function"
which might or might not be evaluated to something.

A user function does not have to be defined before it is used. A consequence of this is that a typo in a function name or a variable name 
will always go unnoticed.
For example:

	In> f(x_IsInteger,y_IsInteger) <-- Mathadd(x,y)
	Out> True;
	In> f(1,2)
	Out> Mathadd(1,2);
Here we made a typo: we should have written {MathAdd}, but wrote
{Mathadd} instead. {Yacas} happily assumed that we mean a new and (so far) undefined "user function" {Mathadd} and returned the expression unevaluated.

In the above example it was easy to spot the error. But this feature becomes more dangerous when it this mistake
is made in a part of some procedure. A call that should have
been made to an internal function, if a typo was made, passes
silently without error and returns unevaluated.
The real problem occurs if we meant to call a function that has side-effects
and we  not use its return value. In this case we shall not immediately find
that the function was not evaluated, but instead we shall encounter a
mysterious bug later.

		Guessing when arguments are evaluated and when not

If your new function does not work as expected, there is a good chance
that it happened because you did not expect some expression which is an argument
to be passed to a function to be evaluated when it is in fact evaluated, or vice versa. 

For example:

	In> p:=Sin(x)
	Out> Sin(x);
	In> D(x)p
	Out> Cos(x);
	In> y:=x
	Out> x;
	In> D(y)p
	Out> 0;

Here the first argument to the differentiation function is not
evaluated, so {y} is not evaluated to {x}, and {D(y)p} is indeed 0.

	    The confusing effect of {HoldArg}

The problem of distinguishing evaluated and unevaluated objects becomes worse when we need to create a function that does not evaluate its arguments.

Since in {Yacas} evaluation starts from the bottom of the expression tree, all "user functions" will appear to evaluate their arguments by default. But sometimes it is convenient to prohibit evaluation of a particular argument (using {HoldArg} or {HoldArgNr}).

For example, suppose we need a function {A(x,y)} that, as a side-effect, assigns the variable {x} to the sum of {x} and {y}. This function will be called when {x} already has some value, so clearly the argument {x} in {A(x,y)} should be unevaluated. It is possible to make this argument unevaluated by putting {Hold()} on it and always calling {A(Hold(x), y)}, but this is not very convenient and easy to forget. It would be better to define {A} so that it always keeps its first argument unevaluated.

If we define a rule base for {A} and declare {HoldArg},
	Function() A(x,y);
	HoldArg("A", x);
then we shall encounter a difficulty when working with the argument {x} inside of a rule body for {A}. For instance, the simple-minded implementation
	A(_x, _y) <-- (x := x+y);
does not work:
	In> [ a:=1; b:=2; A(a,b);]
	Out> a+2;
In other words, the {x} inside the body of {A(x,y)} did not evaluate to {1}
when we called the function {:=}. Instead, it was left unevaluated as the atom
{x} on the left hand side of {:=}, since {:=} does not evaluate its left
argument. It however evaluates its right argument, so the {y} argument was
evaluated to {2} and the {x+y} became {a+2}.

The evaluation of {x} in the body of {A(x,y)} was prevented by the {HoldArg}
declaration. So in the body, {x} will just be the atom {x}, unless it is
evaluated again. If you pass {x} to other functions, they will just get
the atom {x}. Thus in our example, we passed {x} to the function {:=}, thinking that it will get {a}, but it got an unevaluated atom {x} on the left side and proceeded with that.

We need an explicit evaluation of {x} in this case. It can be performed using
{Eval}, or with backquoting, or by using a core function that evaluates its
argument. Here is some code that illustrates these three possibilities:
	A(_x, _y) <-- [ Local(z); z:=Eval(x); z:=z+y; ]
(using explicit evaluation) or
	A(_x, _y) <-- `(@x := @x + y);
(using backquoting) or
	A(_x, _y) <-- MacroSet(x, x+y);
(using a core function {MacroSet} that evaluates its first argument).

However, beware of a clash of names when using explicit evaluations (as
explained above). In other words, the function {A} as defined
above will not work correctly if we give it a variable also named {x}. The
{LocalSymbols} call should be used to get around this problem.

Another caveat is that when we call another function that does not evaluate its
argument, we need to substitute an explicitly evaluated {x} into it. A frequent
case is the following: suppose we have a function {B(x,y)} that  does not
evaluate {x}, and we need to write an interface function {B(x)} which will just
call {B(x,0)}. We should use an explicit evaluation of {x} to accomplish this, for example
	B(_x) <-- `B(@x,0);
or
	B(_x) <-- B @ {x, 0};
Otherwise {B(x,y)} will not get the correct value of its first parameter {x}.

	    Special behavior of {Hold}, {UnList} and {Eval}

When an expression is evaluated, all matching rules are applied to it repeatedly until no more rules match. Thus an expression is "completely" evaluated. There are, however, two cases when recursive application of rules is stopped at a certain point, leaving an expression not "completely" evaluated:

*	1. The expression which is the result of a call to a Yacas core function is
not evaluated further, even if some rules apply to it.
*	2. The expression is a variable that has a value assigned to it; for
example, the variable {x} might have the expression {y+1} as the value. That
value is not evaluated again, so even if {y} has been assigned another value,
say, {y=2} a Yacas expression such as {2*x+1} will evaluate to {2*(y+1)+1} and
not to {7}. Thus, a variable can have some unevaluated expression as its value
and the expression will not be re-evaluated when the variable is used.

The first possibility is mostly without consequence because  almost all core
functions return a simple atom that does not require further evaluation.
However, there are two core functions that can return a complicated expression:
{Hold} and {UnList}. Thus, these functions can produce arbitrarily complicated
Yacas expressions that will be left unevaluated.
For example, the result of 
	UnList({Sin, 0})
is the same as the result of
	Hold(Sin(0))
and is the unevaluated expression {Sin(0)} rather than {0}.

Typically you want to use {UnList} because you need to construct a
function call out of some objects that you have. But you need to call
{Eval(UnList(...))} to actually evaluate this function call. For example:

	In> UnList({Sin, 0})
	Out> Sin(0);
	In> Eval(UnList({Sin, 0}))
	Out> 0;

In effect, evaluation can be stopped with {Hold} or {UnList} and can be
explicitly restarted by using {Eval}. If several levels of un-evaluation are
used, such as {Hold(Hold(...))}, then the same number of {Eval} calls will be
needed to fully evaluate an expression.

	In> a:=Hold(Sin(0))
	Out> Sin(0);
	In> b:=Hold(a)
	Out> a;
	In> c:=Hold(b)
	Out> b;
	In> Eval(c)
	Out> a;
	In> Eval(Eval(c))
	Out> Sin(0);
	In> Eval(Eval(Eval(c)))
	Out> 0;

A function {FullEval} can be defined for "complete" evaluation of expressions, as follows:

	LocalSymbols(x,y)
	[
	  FullEval(_x) <-- FullEval(x,Eval(x));
	  10 # FullEval(_x,_x) <-- x;
	  20 # FullEval(_x,_y) <-- FullEval(y,Eval(y));
	];
Then the example above will be concluded with:
	In> FullEval(c);
	Out> 0;


	    Correctness of parameters to functions is not checked

Because {Yacas} does not enforce type checking of arguments,
it is possible to call functions with invalid arguments. The default
way functions in {Yacas} should deal with situations where an action can not
be performed, is to return the expression unevaluated. A function should 
know when it is failing to perform a task. The typical symptoms are 
errors that seem obscure, but just mean the function called should have
checked that it can perform the action on the object.

For example:

	In> 10 # f(0) <-- 1;
	Out> True;
	In> 20 # f(_n) <-- n*f(n-1);
	Out> True;
	In> f(3)
	Out> 6;
	In> f(1.3)
	CommandLine(1): Max evaluation stack depth reached.

Here, the function {f} is defined to be a factorial function,
but the function fails to check that its argument is a positive
integer, and thus exhausts the stack when called with a non-integer
argument.
A better way would be to write
	In> 20 # f(n_IsPositiveInteger) <-- n*f(n-1);
Then the function would have returned unevaluated when passed a non-integer or a symbolic expression.

		Evaluating variables in the wrong scope

There is a subtle problem that occurs when {Eval} is used
in a function, combined with local variables. The following
example perhaps illustrates it:

	In> f1(x):=[Local(a);a:=2;Eval(x);];
	Out> True;
	In> f1(3)
	Out> 3;
	In> f1(a)
	Out> 2;

Here the last call should have returned {a}, but it returned {2},
because {x} was assigned the value {a}, and {a} was assigned
locally the value of {2}, and {x} gets re-evaluated. This problem
occurs when the expression being evaluated contains variables which
are also local variables in the function body. The solution is to use
the {LocalSymbols} function for all local variables defined in the body.

The following illustrates this:

	In> f2(x):=LocalSymbols(a)[Local(a);a:=2;Eval(x);];
	Out> True;
	In> f1(3)
	Out> 3;
	In> f2(a)
	Out> a;

Here {f2} returns the correct result. {x} was assigned the value {a},
but the {a} within the function body is made distinctly different
from the one referred to by {x} (which, in a sense, refers to a global
{a}), by using {LocalSymbols}.

This problem generally occurs when defining functions that re-evaluate
one of its arguments, typically functions that perform a loop of some
sort, evaluating a body at each iteration.


			Debugging in Yacas

		Introduction

When writing a code segment, it is generally a good idea to separate the
problem into many small functions. Not only can you then reuse these
functions on other problems, but it makes debugging easier too.

For debugging a faulty function, in addition to the usual trial-and-error method and the "print everything" method, Yacas offers some trace facilities. You can try to trace applications of rules during evaluation of the function ({TraceRule()}, {TraceExp()}) or see the stack after an error has occurred ({TraceStack()}). 

There is also an interactive debugger, which shall be introduced
in this chapter.

Finally, you may want to run a debugging version of Yacas. This
version of the executable maintains more information about 
the operations it performs, and can report on this. 

This chapter will start with the interactive debugger, as it
is the easiest and most useful feature to use, and then proceed
to explain the trace and profiling facilities. Finally, the
internal workings of the debugger will be explained. It is highly
customizable (in fact, most of the debugging code is written in
Yacas itself), so for bugs that are really difficult to track
one can write custom code to track it.



		The trace facilities

The trace facilities are:

*	0. {TraceExp} : traces the full expression, showing all calls to user- or system-defined functions, their arguments, and the return values. For complex functions this can become a long list of function calls.
*	0. {TraceRule} : traces one single user-defined function (rule). It shows each invocation, the arguments passed in, and the returned values. This is useful for tracking the behavior of that function in the environment it is intended to be used in.
*	0. {TraceStack} : shows a few last function calls before an error has occurred.
*	0. {Profile} : report on statistics (number of times functions
were called, etc.). Useful for performance analysis.

The online manual pages (e.g. {?TraceStack}) have more information about the use of these functions.

An example invocation of {TraceRule} is

	In> TraceRule(x+y)2+3*5+4;

Which should then show something to the effect of

	  TrEnter(2+3*5+4);
	    TrEnter(2+3*5);
	       TrArg(2,2);
	          TrArg(3*5,15);
	       TrLeave(2+3*5,17);
	        TrArg(2+3*5,17);
	        TrArg(4,4);
	    TrLeave(2+3*5+4,21);
	Out> 21;



			Custom evaluation facilities

Yacas supports a special form of evaluation where hooks are
placed when evaluation enters or leaves an expression. 

This section will explain the way custom evaluation is supported
in {Yacas}, and will proceed to demonstrate how it can be used
by showing code to trace, interactively step through, profile,
and write custom debugging code.

Debugging, tracing and profiling has been implemented in the
debug.rep/ module, but a simplification of that code
will be presented here to show the basic concepts.

		The basic infrastructure for custom evaluation

The name of the function is {CustomEval}, and the calling sequence
is:

	CustomEval(enter,leave,error,expression);

Here, {expression} is the expression to be evaluated, {enter}
some expression that should be evaluated when entering an
expression, and {leave} an expression to be evaluated when leaving
evaluation of that expression.

The {error} expression is evaluated when an error occurred. If an error
occurs, this is caught high up, the {error} expression is called, and 
the debugger goes back to evaluating {enter} again so the situation can
be examined. When the debugger needs to stop, the {error} expression
is the place to call {CustomEval'Stop()} (see explanation below).

The {CustomEval} function can be used to write custom debugging
tools. Examples are: 

*	1. a trace facility following entering and leaving functions
*	1. interactive debugger for stepping through evaluation of an 
expression.
*	1. profiler functionality, by having the callback functions do the 
bookkeeping on counts of function calls for instance.

In addition, custom code can be written to for instance halt evaluation
and enter interactive mode as soon as some very specific situation
occurs, like "stop when function foo is called while the function bar
is also on the call stack and the value of the local variable x is
less than zero". 

As a first example, suppose we define the functions TraceEnter(),
TraceLeave() and {TraceExp()} as follows:

	TraceStart() := [indent := 0;];
	TraceEnter() :=
	[
	   indent++;
	   Space(2*indent);
	   Echo("Enter ",CustomEval'Expression());
	];
	TraceLeave() :=
	[
	   Space(2*indent);
	   Echo("Leave ",CustomEval'Result());
	   indent--;
	];
	Macro(TraceExp,{expression})
	[
	   TraceStart();
	   CustomEval(TraceEnter(),
	              TraceLeave(),
	              CustomEval'Stop(),@expression);
	];


allows us to have tracing in a very basic way. We can now call:

	In> TraceExp(2+3)
	  Enter 2+3 
	    Enter 2 
	    Leave 2 
	    Enter 3 
	    Leave 3 
	    Enter IsNumber(x) 
	      Enter x 
	      Leave 2 
	    Leave True 
	    Enter IsNumber(y) 
	      Enter y 
	      Leave 3 
	    Leave True 
	    Enter True 
	    Leave True 
	    Enter MathAdd(x,y) 
	      Enter x 
	      Leave 2 
	      Enter y 
	      Leave 3 
	    Leave 5 
	  Leave 5 
	Out> 5;

This example shows the use of {CustomEval'Expression} and
{CustomEval'Result}. These functions give some extra access
to interesting information while evaluating the expression.
The functions defined to allow access to information while
evaluating are:

*	1. {CustomEval'Expression()} - return expression currently
on the top call stack for evaluation.
*	1. {CustomEval'Result()} - when the {leave} argument is called this
function returns what the evaluation of the top expression will return.
*	1. {CustomEval'Locals()} - returns a list with the current local variables.
*	1. {CustomEval'Stop()} - stop debugging execution

		A simple interactive debugger

The following code allows for simple interactive debugging:


	DebugStart():=
	[
	   debugging:=True;
	   breakpoints:={};
	];
	DebugRun():= [debugging:=False;];
	DebugStep():=[debugging:=False;nextdebugging:=True;];
	DebugAddBreakpoint(fname_IsString) <-- 
	   [ breakpoints := fname:breakpoints;];
	BreakpointsClear() <-- [ breakpoints := {};];
	Macro(DebugEnter,{})
	[
	   Echo(">>> ",CustomEval'Expression());
	   If(debugging = False And
	      IsFunction(CustomEval'Expression()) And 
	      Contains(breakpoints,
	      Type(CustomEval'Expression())),   
	        debugging:=True);
	   nextdebugging:=False;
	   While(debugging)
	   [
	      debugRes:=
	        Eval(FromString(
	          ReadCmdLineString("Debug> "):";")
	          Read());
	      If(debugging,Echo("DebugOut> ",debugRes));
	   ];
	   debugging:=nextdebugging;
	];
	Macro(DebugLeave,{})
	[
	   Echo(CustomEval'Result(),
	        " <-- ",CustomEval'Expression());
	];
	Macro(Debug,{expression})
	[
	   DebugStart();
	   CustomEval(DebugEnter(),
	              DebugLeave(),
	              debugging:=True,@expression);
	];

This code allows for the following interaction:

	In> Debug(2+3)
	>>> 2+3 
	Debug> 

The console now shows the current expression being evaluated, and a debug
prompt for interactive debugging. We can enter {DebugStep()}, which steps to
the next expression to be evaluated:

	Debug> DebugStep();
	>>> 2 
	Debug> 

This shows that in order to evaluate {2+3} the interpreter first needs
to evaluate {2}. If we step further a few more times, we arrive at:

	>>> IsNumber(x) 
	Debug> 

Now we might be curious as to what the value for {x} is. We can dynamically
obtain the value for {x} by just typing it on the command line. 

	>>> IsNumber(x) 
	Debug> x
	DebugOut> 2 

{x} is set to {2}, so we expect {IsNumber} to return {True}. Stepping again:

	Debug> DebugStep();
	>>> x 
	Debug> DebugStep();
	2  <-- x 
	True  <-- IsNumber(x) 
	>>> IsNumber(y) 

So we see this is true. We can have a look at which local variables are currently
available by calling {CustomEval'Locals()}:

	Debug> CustomEval'Locals()
	DebugOut> {arg1,arg2,x,y,aLeftAssign,aRightAssign} 

And when bored, we can proceed with {DebugRun()} which will continue the
debugger until finished in this case (a more sophisticated debugger can
add breakpoints, so running would halt again at for instance a breakpoint).

	Debug> DebugRun()
	>>> y 
	3  <-- y 
	True  <-- IsNumber(y) 
	>>> True 
	True  <-- True 
	>>> MathAdd(x,y) 
	>>> x 
	2  <-- x 
	>>> y 
	3  <-- y 
	5  <-- MathAdd(x,y) 
	5  <-- 2+3 
	Out> 5;


The above bit of code also supports primitive breakpointing, in that
one can instruct the evaluator to stop when a function will be entered.
The debugger then stops just before the arguments to the function are 
evaluated. In the following example, we make the debugger stop when 
a call is made to the {MathAdd} function:

	In> Debug(2+3)
	>>> 2+3 
	Debug> DebugAddBreakpoint("MathAdd")
	DebugOut> {"MathAdd"} 
	Debug> DebugRun()
	>>> 2 
	2  <-- 2 
	>>> 3 
	3  <-- 3 
	>>> IsNumber(x) 
	>>> x 
	2  <-- x 
	True  <-- IsNumber(x) 
	>>> IsNumber(y) 
	>>> y 
	3  <-- y 
	True  <-- IsNumber(y) 
	>>> True 
	True  <-- True 
	>>> MathAdd(x,y) 
	Debug> 

The arguments to {MathAdd} can now be examined, or execution continued.

One great advantage of defining much of the debugger in script code can
be seen in the {DebugEnter} function, where the breakpoints are checked,
and execution halts when a breakpoint is reached. In this case the condition
for stopping evaluation is rather simple: when entering a specific function,
stop. However, nothing stops a programmer from writing a custom debugger
that could stop on any condition, halting at e very special case.



		Profiling

A simple profiler that counts the number of times each function is
called can be written such:

	ProfileStart():=
	[
	   profilefn:={};
	];
	10 # ProfileEnter()
	     _(IsFunction(CustomEval'Expression())) <-- 
	[
	   Local(fname);
	   fname:=Type(CustomEval'Expression());
	   If(profilefn[fname]=Empty,profilefn[fname]:=0);
	   profilefn[fname] := profilefn[fname]+1;
	];
	Macro(Profile,{expression})
	[
	   ProfileStart();
	   CustomEval(ProfileEnter(),True,
	              CustomEval'Stop(),@expression);
	   ForEach(item,profilefn)
	     Echo("Function ",item[1]," called ",
	          item[2]," times");
	];


which allows for the interaction:

	In> Profile(2+3)
	Function MathAdd called 1  times
	Function IsNumber called 2  times
	Function + called 1  times
	Out> True;






*INCLUDE YacasDebugger.chapt


			Advanced example 2: implementing a non-commutative algebra

We need to understand how to simplify expressions in Yacas, and the best way is to try writing our own algebraic expression handler. In this chapter we shall consider a simple implementation of a particular non-commutative algebra called the Heisenberg algebra. This algebra was introduced by Dirac to develop quantum field theory. We won't explain any physics here, but instead we shall to delve somewhat deeper into the workings of Yacas.

		The problem

Suppose we want to define special symbols $A(k)$ and $B(k)$ that we can multiply with each other or by a number, or add to each other, but not commute with each other, i.e. $A(k)*B(k) != B(k)*A(k)$. Here $k$ is merely a label to denote that $A(1)$ and $A(2)$ are two different objects. (In physics, these are called "creation" and "annihilation" operators for "bosonic quantum fields".) Yacas already assumes that the usual multiplication operator "{*}" is commutative. Rather than trying to redefine {*}, we shall introduce a special multiplication sign "{**}" that we shall use with the objects $A(k)$ and $B(k)$; between usual numbers this would be the same as normal multiplication. The symbols $A(k)$, $B(k)$ will never be evaluated to numbers, so an expression such as {2 ** A(k1) ** B(k2) ** A(k3)} is just going to remain like that. (In physics, commuting numbers are called "classical quantities" or "c-numbers" while non-commuting objects made up of A(k) and B(k) are called "quantum quantities" or "q-numbers".) There are certain commutation relations for these symbols: the $A$'s commute between themselves, $A(k)*A(l) = A(l)*A(k)$, and also the $B$'s, $B(k)*B(l) = B(l)*B(k)$. However, the $A$'s don't commute with the $B$'s: $A(k)*B(l) - B(l)*A(k) = delta(k-l)$. Here the "{delta}" is a "classical" function (called the "Dirac $delta$-function") but we aren't going to do anything about it, just leave it symbolic.

We would like to be able to manipulate such expressions, expanding brackets, collecting similar terms and so on, while taking care to always keep the non-commuting terms in the correct order. For example, we want Yacas to automatically simplify {2**B(k1)**3**A(k2)} to {6**B(k1)**A(k2)}. Our goal is not to implement a general package to tackle complicated non-commutative operations; we merely want to teach Yacas about these two kinds of "quantum objects" called {A(k)} and {B(k)}, and we shall define one function that a physicist would need to apply to these objects. This function applied to any given expression containing $A$'s and $B$'s will compute something called a "vacuum expectation value", or "VEV" for short, of that expression. This function has "classical", i.e. commuting, values and is defined as follows: VEV of a commuting number is just that number, e.g. $VEV(4) = 4$, $VEV(delta(k-l)) = delta(k-l)$; and $VEV(X*A(k)) = 0$, $VEV(B(k)*X) = 0$ where $X$ is any expression, commutative or not. It is straightforward to compute VEV of something that contains $A$'s and $B$'s: one just uses the commutation relations to move all $B$'s to the left of all $A$'s, and then applies the definition of VEV, simply throwing out any remaining q-numbers.

		First steps

The first thing that comes to mind when we start implementing this in Yacas is to write a rule such as

	10 # A(_k)**B(_l) <-- B(l)**A(k)
	  + delta(k-l);

However, this is not going to work right away. In fact this will
immediately give a syntax error because Yacas doesn't know yet about the
new multiplication {**}. Let's fix that: we shall define a
new infix operator with the same precedence as multiplication.

	RuleBase("**", {x,y});
	Infix("**", OpPrecedence("*"));

Now we can use this new multiplication operator in expressions, and it doesn't evaluate to anything -- exactly what we need. But we find that things don't quite work:

	In> A(_k)**B(_l) <-- B(l)**A(k)+delta(k-l);
	Out> True;
	In> A(x)**B(y)
	Out> B(l)**A(k)+delta(k-l);
Yacas doesn't grok that {delta(k)}, {A(k)} and {B(k)} are functions. This can be fixed by declaring

	RuleBase("A", {k});
	RuleBase("B", {k});
	RuleBase("delta", {k});
Now things work as intended:

	In> A(y)**B(z)*2
	Out> 2*(B(z)**A(y)+delta(y-z));

		Structure of expressions

Are we done yet? Let's try to calculate more things with our $A$'s and $B$'s:

	In> A(k)*2**B(l)
	Out> 2*A(k)**B(l);
	In> A(x)**A(y)**B(z)
	Out> A(x)**A(y)**B(z);
	In> (A(x)+B(x))**2**B(y)*3
	Out> 3*(A(x)+B(x))**2**B(y);
After we gave it slightly more complicated input, Yacas didn't fully
evaluate expressions containing the new {**} operation: it didn't
move constants {2} and {3} together, didn't expand brackets, and, somewhat
mysteriously, it didn't apply the rule in the first line above -- although
it seems like it should have. Before we hurry to fix these things, let's
think some more about how Yacas represents our new expressions. Let's start
with the first line above:

	In> FullForm( A(k)*2**B(l) )
	(** (* 2 (A k ))(B l ))
	Out> 2*A(k)**B(l);
What looks like {2*A(k)**B(l)} on the screen is really {(2*A(k)) ** B(l)} inside Yacas. In other words, the commutation rule didn't apply because there is no subexpression of the form {A(...)**B(...)} in this expression. It seems that we would need many rules to exhaust all ways in which the adjacent factors {A(k)} and {B(l)} might be divided between subexpressions. We run into this difficulty because Yacas represents all expressions as trees of functions and leaves the semantics to us. To Yacas, the "{*}" operator is fundamentally no different from any other function, so {(a*b)*c} and {a*(b*c)} are two basically different expressions. It would take a considerable amount of work to teach Yacas to recognize all such cases as identical. This is a design choice and it was made by the author of Yacas to achieve greater flexibility and extensibility.

A solution for this problem is not to write rules for all possible cases (there are infinitely many cases) but to systematically reduce expressions to a <i>canonical form</i>. "Experience has shown that" (a phrase used when we can't come up with specific arguments) symbolic manipulation of unevaluated trees is not efficient unless these trees are forced to a pattern that reflects their semantics.

We should choose a canonical form for all such expressions in a way that makes our calculations -- namely, the function {VEV()} -- easier. In our case, our expressions contain two kinds of ingredients: normal, commutative numbers and maybe a number of noncommuting symbols {A(k)} and {B(k)} multiplied together with the "{**}" operator. It will not be possible to divide anything by $A(k)$ or $B(k)$ -- such division is undefined.

A possible canonical form for expressions with A's and B's is the
following. All commutative numbers are moved to the left of the expression
and grouped together as one factor; all non-commutative products are
simplified to a single chain, all brackets expanded. A canonical expression
should not contain any extra brackets in its non-commutative part. For
example, (A(x)+B(x)*x)**B(y)*y**A(z) should be regrouped as a sum of two
terms, (y)**(A(x)**(B(y))**A(z)) and (x*y)**(B(x)**(B(y))**A(z)). Here we
wrote out all parentheses to show explicitly which operations are grouped.
(We have chosen the grouping of non-commutative factors to go from left to
right, however this does not seem to be an important choice.) On the screen
this will look simply {y ** A(x) ** B(y)} and
{x*y** B(x) ** B(y) ** A(z)} because we have defined the
precedence of the "**" operator to be the same as that of the normal
multiplication, so Yacas won't insert any more
parentheses.

This canonical form will allow Yacas to apply all the usual rules on the commutative factor while cleanly separating all non-commutative parts for special treatment. Note that a commutative factor such as {2*x} will be multiplied by a single non-commutative piece with "{**}".

The basic idea behind the "canonical form" is this: we should define our evaluation rules in such a way that any expression containing {A(k)} and {B(k)} will be always automatically reduced to the canonical form after one full evaluation. All functions on our new objects will assume that the object is already in the canonical form and should return objects in the same canonical form.

		Implementing the canonical form

Now that we have a design, let's look at some implementation
issues. We would like to write evaluation rules involving the new operator
"{**}" as well as the ordinary multiplications and
additions involving usual numbers, so that all "classical" numbers and all
"quantum" objects are grouped together separately. This should be
accomplished with rules that expand brackets, exchange the bracketing order
of expressions and move commuting factors to the left. For now, we shall not concern ourselves with divisions and subtractions.

First, we need to distinguish "classical" terms from "quantum" ones. For this, we shall define a predicate {IsQuantum()} recursively, as follows:

	*
	    /* Predicate IsQuantum(): will return
		  True if the expression contains A(k)
		  or B(k) and False otherwise */
	10 # IsQuantum(A(_x)) <-- True;
	10 # IsQuantum(B(_x)) <-- True;
	    /* Result of a binary operation may
		  be Quantum */
	20 # IsQuantum(_x + _y) <-- IsQuantum(x)
	  Or IsQuantum(y);
	20 # IsQuantum(+ _y) <-- IsQuantum(y);
	20 # IsQuantum(_x * _y) <-- IsQuantum(x)
	  Or IsQuantum(y);
	20 # IsQuantum(_x ** _y) <-- IsQuantum(x)
	  Or IsQuantum(y);
	    /* If none of the rules apply, the
		  object is not Quantum */
	30 # IsQuantum(_x) <-- False;

Now we shall construct rules that implement reduction to the canonical form. The rules will be given precedences, so that the reduction proceeds by clearly defined steps. All rules at a given precedence benefit from all simplifications at earlier precedences.

	  /* First, replace * by ** if one of the
	    factors is Quantum to guard against
		user error */
	10 # (_x * _y)_(IsQuantum(x) Or
	  IsQuantum(y)) <-- x ** y;
	    /* Replace ** by * if neither of the
		  factors is Quantum */
	10 # (_x ** _y)_(Not(IsQuantum(x) Or
	 IsQuantum(y))) <-- x * y;
	    /* Now we are guaranteed that ** is
		  used between Quantum values */
	    /* Expand all brackets involving
		  Quantum values */
	15 # (_x + _y) ** _z <-- x ** z + y ** z;
	15 # _z ** (_x + _y) <-- z ** x + z ** y;
	    /* Now we are guaranteed that there are
		  no brackets next to "**" */
	    /* Regroup the ** multiplications
		  toward the right */
	20 # (_x ** _y) ** _z <-- x ** (y ** z);
	    /* Move classical factors to the left:
		  first, inside brackets */
	30 # (x_IsQuantum ** _y)_(Not(IsQuantum(y)))
	  <-- y ** x;
	    /* Then, move across brackets:
		  y and z are already ordered
	      by the previous rule */
	    /* First, if we have Q ** (C ** Q) */
	35 # (x_IsQuantum ** (_y ** _z))
	  _(Not(IsQuantum(y))) <-- y ** (x ** z);
	    /* Second, if we have C ** (C ** Q) */
	35 # (_x ** (_y ** _z))_(Not(IsQuantum(x)
	  Or IsQuantum(y))) <-- (x*y) ** z;

After we execute this in Yacas, all expressions involving additions and multiplications are automatically reduced to the canonical form. Extending these rules to subtractions and divisions is straightforward.

		Implementing commutation relations

But we still haven't implemented the commutation relations. It is perhaps not necessary to have commutation rules automatically applied at each evaluation. We shall define the function {OrderBA()} that will bring all $B$'s to the left of all $A$'s by using the commutation relation. (In physics, this is called "normal-ordering".) Again, our definition will be recursive. We shall assign it a later precedence than our quantum evaluation rules, so that our objects will always be in canonical form. We need a few more rules to implement the commutation relation and to propagate the ordering operation down the expression tree:

	  /* Commutation relation */
	40 # OrderBA(A(_k) ** B(_l))
	  <-- B(l)**A(k) + delta(k-l);
	40 # OrderBA(A(_k) ** (B(_l) ** _x))
	  <-- OrderBA(OrderBA(A(k)**B(l)) ** x);
	    /* Ordering simple terms */
	40 # OrderBA(_x)_(Not(IsQuantum(x))) <-- x;
	40 # OrderBA(A(_k)) <-- A(k);
	40 # OrderBA(B(_k)) <-- B(k);
	    /* Sums of terms */
	40 # OrderBA(_x + _y) <-- OrderBA(x)
	  + OrderBA(y);
	    /* Product of a classical and
		  a quantum value */
	40 # OrderBA(_x ** _y)_(Not(IsQuantum(x)))
	  <-- x ** OrderBA(y);
	    /* B() ** X : B is already at left,
		  no need to order it */
	50 # OrderBA(B(_k) ** _x)<-- B(k)
	  ** OrderBA(x);
	    /* A() ** X : need to order X first */
	50 # OrderBA(A(_k) ** _x) <-- OrderBA(A(k)
	  ** OrderBA(x));

These rules seem to be enough for our purposes. Note that the commutation relation is implemented by the first two rules; the first one is used by the second one which applies when interchanging factors A and B separated by brackets. This inconvenience of having to define several rules for what seems to be "one thing to do" is a consequence of tree-like structure of expressions in Yacas. It is perhaps the price we have to pay for conceptual simplicity of the design.

		Avoiding infinite recursion

However, we quickly discover that our definitions don't work. Actually, we have run into a difficulty typical of rule-based programming:

	In> OrderBA(A(k)**A(l))
	Error on line 1 in file [CommandLine]
	Line error occurred on:
	>>>
	Max evaluation stack depth reached.
	Please use MaxEvalDepth to increase the
	  stack size as needed.
This error message means that we have created an infinite recursion. It is easy
to see that the last rule is at fault: it never stops applying itself when it
operates on a term containing only $A$'s and no $B$'s. When encountering a term
such as {A()**X}, the routine cannot determine whether {X} has already been
normal-ordered or not, and it unnecessarily keeps trying to normal-order it
again and again. We can circumvent this difficulty by using an auxiliary
ordering function that we shall call {OrderBAlate()}. This function will
operate only on terms of the form {A()**X} and only after {X} has been ordered.
It will not perform any extra simplifications but instead delegate all work to
{OrderBA()}.

	50 # OrderBA(A(_k) ** _x) <-- OrderBAlate(
	  A(k) ** OrderBA(x));
	55 # OrderBAlate(_x + _y) <-- OrderBAlate(
	  x) + OrderBAlate(y);
	55 # OrderBAlate(A(_k) ** B(_l)) <--
	  OrderBA(A(k)**B(l));
	55 # OrderBAlate(A(_k) ** (B(_l) ** _x))
	  <-- OrderBA(A(k)**(B(l)**x));
	60 # OrderBAlate(A(_k) ** _x) <-- A(k)**x;
	65 # OrderBAlate(_x) <-- OrderBA(x);
Now {OrderBA()} works as desired.


		Implementing VEV()

Now it is easy to define the function {VEV()}. This function should first execute the normal-ordering operation, so that all $B$'s move to the left of $A$'s. After an expression is normal-ordered, all of its "quantum" terms will either end with an $A(k)$ or begin with a $B(k)$, or both, and {VEV()} of those terms will return $0$. The value of {VEV()} of a non-quantum term is just that term. The implementation could look like this:

	100 # VEV(_x) <-- VEVOrd(OrderBA(x));
	    /* Everything is expanded now,
		  deal term by term */
	100 # VEVOrd(_x + _y) <-- VEVOrd(x)
	  + VEVOrd(y);
	    /* Now cancel all quantum terms */
	110 # VEVOrd(x_IsQuantum) <-- 0;
	    /* Classical terms are left */
	120 # VEVOrd(_x) <-- x;
To avoid infinite recursion in calling {OrderBA()}, we had to introduce an auxiliary function {VEVOrd()} that assumes its argument to be ordered.

Finally, we try some example calculations to test our rules:

	In> OrderBA(A(x)*B(y))
	Out> B(y)**A(x)+delta(x-y);
	In> OrderBA(A(x)*B(y)*B(z))
	Out> B(y)**B(z)**A(x)+delta(x-z)**B(y)
	  +delta(x-y)**B(z);
	In> VEV(A(k)*B(l))
	Out> delta(k-l);
	In> VEV(A(k)*B(l)*A(x)*B(y))
	Out> delta(k-l)*delta(x-y);
	In> VEV(A(k)*A(l)*B(x)*B(y))
	Out> delta(l-y)*delta(k-x)+delta(l-x)
	  *delta(k-y);
Things now work as expected. Yacas's {Simplify()} facilities can be used on the result of {VEV()} if it needs simplification.




*REM GNU Free Documentation License
*INCLUDE FDL.chapt