NOTATION PACKAGE TUTORIAL

# 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.

The reader should be familiar with the concepts in "Textual Input and Output" and moreover understand the following tutorials: "The Representation of Textual Forms," "The Interpretation of Textual Forms," "Representing Textual Forms by Boxes," "String Representation of Boxes," "Converting between Strings, Boxes, and Expressions," and "Low-Level Input and Output Rules."

## Tag Boxes

A TagBox is a box structure just like RowBox, SubscriptBox, 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.

All Mathematica 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 Show Expression under the Cell menu.

Here is the underlying representation of the expression above in terms of boxes, displayed using the Show Expression menu item located under the Cell 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 around .

The embedded TagBox tag has no special parsing behavior associated with it.
 Out[55]=

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 tag now has the special parsing behavior associated with it to just return the boxes.
 Out[57]=

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, NotationPatternTag, and NotationMadeBoxesTag. These are all string tags for two reasons. First, by using string tags you can avoid any potential problems to do with package contexts and redefinition of the symbols. Second, 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

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, Symbolize, and InfixNotation templates on the Notation 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.

An expression containing an embedded NotationTemplateTag tag.
 Out[517]//FullForm=
You can avoid using notation templates if you wrap raw box structures with a ParsedBoxWrapper.
 Out[60]//FullForm=

## Complex Patterns and the Tag NotationPatternTag

For normal purposes it is usually sufficient that the patterns present in Notation and Symbolize 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 that 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 Notation palette has a button labeled InsertPatternWrapper 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.)

This creates a TraditionalForm notation that only functions with numbers.

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 that matches the above pattern will be interpreted as a object.
 Out[63]=
Reciprocally, only 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 examine the form of the rules created by using the option .

You can see that the patterns a_?StringNumericQ and a_?NumericQ 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.

An example notation for formatting tensors containing the tag NotationMadeBoxesTag.

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[, , SyntaxForm->], where string is a string indicating the operator on which the precedence of the tag box is modeled. The following examples illustrate the SyntaxForm option.

An expression that uses the arrow with standard precedences.
 Out[67]//FullForm=
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.
 Out[69]//FullForm=

You can illustrate the underlying groupings of the expressions above in the following table.

 visual form of expression grouping box form of expression a+bc a+bc

A table illustrating the precedences and grouping of expressions with and without precedence-changing tag boxes.

The SyntaxForm 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.

 SyntaxForm value precedence behavior 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.

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