# 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.

## 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 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 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. 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

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 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.

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 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.

Out[63]= | |

Reciprocally only foo objects with numerical arguments will be formatted using the notation.

Out[64]//TraditionalForm= |

| |

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 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.

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 option.

An expression which 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.

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.

| |

| 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.