Complex Patterns and Advanced Features
This Tutorial Is Intended for Advanced Users
Due to the complex inner workings of the Notation Package it is helpful to outline some of the more advanced features and structures of the Mathematica
front end and how they relate to the Notation Package. The following sections give a small overview of the functionality of tag boxes, the specific tags used by the Notation Package and the tag box option SyntaxForm
is a box structure just like RowBox
, or GridBox
. It is used to change the structure of an expression or indicate a grouping or interpretation of a subexpression at an underlying level. To illustrate tag boxes, consider the following input, which contains an embedded TagBox
input and output is made up of box structures at a low level. When Mathematica
receives input, these box structures are parsed into internal expressions, which can be thought of as full-form expressions. Internal evaluation then takes place, and finally the internal structures are transformed back into box structures for displaying in the Mathematica
front end. You can reveal how Mathematica
sees this input at a low level by choosing under the menu.
Here is the underlying representation of the expression above in terms of boxes, displayed using the menu item located under the menu.
The above expression contains a subexpression TagBox[SuperscriptBox["x", "2"], foo]]
. It is important to note that this box expression as normally viewed in Mathematica
looks visually like
even though it has an embedded TagBox
. Information contained in the tag is visually hidden from the user. When an expression containing a TagBox
is input into Mathematica,
the default interpretation of the subexpression surrounded by the TagBox
is to wrap the TagBox
name around the parsed subexpression, in this case to wrap foo
The embedded TagBox
has no special parsing behavior associated with it.
However, you can define your own rules for the way specific tag boxes are parsed. For instance, by using the low-level function MakeExpression
, you can change how Mathematica
will parse expressions containing tag boxes.
By defining a new rule for MakeExpression
, you can change how Mathematica
will parse expressions containing a TagBox
with the tag
The embedded TagBox
now has the special parsing behavior associated with it to just return the boxes.
Knowing that special behaviors can be set up for specific tags, you can now examine the tags defined by the Notation Package. The Notation Package defines three tags that have special behaviors: NotationTemplateTag
, and NotationMadeBoxesTag
. These are all string tags for two reasons. One, by using string tags you can avoid any potential problems to do with package contexts and redefinition of the symbols. Secondly, in Mathematica
if a TagBox
has a string tag and there is a named style on the stylesheet path with the same name as the string tag then the TagBox
will be displayed in that style. This lets you omit the BaseStyle
option from the TagBox
and consequently your box structures are smaller and more readable.
The Tag NotationTemplateTag
is a string tag used by the Notation Package to grab box structures before they can be parsed by Mathematica
. In fact NotationTemplateTag
acts rather like the tag
defined above. All Notation
, and InfixNotation
templates on the palette contain tag boxes with an embedded string tag: NotationTemplateTag
. The embedded TagBox
ensures that the Notation Package can obtain the correct parsing information and retain the proper styling and grouping information. This embedded tag is used to capture the box structure, and the captured structure is thus wrapped with a ParsedBoxWrapper
You can avoid using notation templates if you wrap raw box structures with a ParsedBoxWrapper
Complex Patterns and the Tag NotationPatternTag
For normal purposes it is usually sufficient that the patterns present in Notation
statements are simple patterns. However, it is sometimes necessary or desirable to use more complicated patterns in notations. For example, a notation might only be valid when a certain pattern is a number. To allow more complex patterns, you can embed a NotationPatternTag
tag box inside a notation statement. It is critical that any notation you define which uses a complex pattern has an embedded NotationPatternTag
, otherwise the pattern will be treated as a verbatim expression and not function as a pattern. Like NotationTemplateTag
, this should be a string tag. The palette has a button labeled that will embed a NotationPatternTag
around the selection, as well as tint the background of the selection to indicate that a complex pattern is present. (This tinting occurs as a result of the named style NotationPatternTag
, since this is a string tag.)
It must also be pointed out that the pattern matching on the external representation is performed on the box structures, so usually you will have to make small transformations to convert box structures into normal expressions. Pattern matching on the internal representation follows conventional pattern matching.
This defines a function analogous to NumericQ
that operates on box structures.
Now only input which matches the above pattern will be interpreted as a foo object.
Reciprocally only foo objects with numerical arguments will be formatted using the notation.
You should be careful to avoid unwanted evaluation through testing functions when parsing expressions (see "Parse without Evaluation Where Possible"
You can see that the patterns a_?StringNumericQ
do not appear literally since they were surrounded by a NotationPatternTag
in the notation statement.
The Tag NotationMadeBoxesTag
The tag NotationMadeBoxesTag
is intended for advanced users. It is also a string tag. It is used to indicate that box processing and formatting has already been done and that the Notation Package should not perform any processing. Typically you would use this tag for surrounding your own functions that return expressions that have already been turned into boxes or parsed into expressions. To illustrate the tag NotationMadeBoxesTag
, you can examine a notation statement that might be part of a number of statements used to create a notation for tensors.
You can see from the internal definition returned that there is no further processing of the expression
, i.e. it is not surrounded by a MakeBoxes[..., StandardForm]
Changing Precedences and the TagBox Option SyntaxForm
Using the option SyntaxForm
, you can change the precedence of an expression containing a TagBox
. A tag box containing a SyntaxForm
option will look like TagBox[boxStructure, tag, SyntaxForm->string]
, where string
is a string indicating the operator on which the precedence of the tag box is modeled. The following examples illustrate the SyntaxForm
An expression which uses the arrow
with standard precedences.
You can define a new notation for a composite arrow
surrounded by a TagBox
that has the SyntaxForm
option set to a low precedence.
This new composite arrow has a low precedence.
You can illustrate the underlying groupings of the expressions above in the following table.
A table illustrating the precedences and grouping of expressions with and without precedence-changing tag boxes.
option value can be any operator string valid in Mathematica,
that is, any operator contained in the UnicodeCharacters.tr file. The SyntaxForm
value can also include symbols before and after the operator to indicate whether the precedence is that of a prefix operator, an infix operator, or a postfix operator. Some typical values for the SyntaxForm
option are given in the table below.
|group as the operator times|
|group as a symbol|
|group as an infix plus operator|
|group as a for all operator|
|group as an integrate operator|
|group as a prefix union operator|
|group as white space|
Typical SyntaxForm values and their associated precedence behaviors.