# Attributes

Definitions such as

specify

*values* for functions. Sometimes, however, you need to specify general properties of functions, without necessarily giving explicit values.

*Mathematica* provides a selection of

*attributes* that you can use to specify various properties of functions. For example, you can use the attribute

Flat to specify that a particular function is "flat", so that nested invocations are automatically flattened, and it behaves as if it were associative.

This assigns the attribute

Flat to the function

.

Now

behaves as a flat, or associative, function, so that nested invocations are automatically flattened.

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Attributes like

Flat can affect not only evaluation, but also operations such as pattern matching. If you give definitions or transformation rules for a function, you must be sure to have specified the attributes of the function first.

Here is a definition for the flat function

.

Because

is flat, the definition is automatically applied to every subsequence of arguments.

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Manipulating attributes of symbols.

This shows the attributes assigned to

.

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This removes the attributes assigned to

.

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Orderless | orderless, commutative function (arguments are sorted into standard order) |

Flat | flat, associative function (arguments are "flattened out") |

OneIdentity | , etc. are equivalent to a for pattern matching |

Listable | f is automatically "threaded" over lists that appear as arguments (e.g., becomes ) |

Constant | all derivatives of f are zero |

NumericFunction | f is assumed to have a numerical value when its arguments are numeric quantities |

Protected | values of f cannot be changed |

Locked | attributes of f cannot be changed |

ReadProtected | values of f cannot be read |

HoldFirst | the first argument of f is not evaluated |

HoldRest | all but the first argument of f are not evaluated |

HoldAll | none of the arguments of f are evaluated |

HoldAllComplete | the arguments of f are treated as completely inert |

NHoldFirst | the first argument of f is not affected by N |

NHoldRest | all but the first argument of f are not affected by N |

NHoldAll | none of the arguments of f are affected by N |

SequenceHold | Sequence objects appearing in the arguments of f are not flattened out |

Temporary | f is a local variable, removed when no longer used |

Stub | Needs is automatically called if f is ever explicitly input |

The complete list of attributes for symbols in *Mathematica*.

Here are the attributes for the built-in function

Plus.

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An important attribute assigned to built-in mathematical functions in

*Mathematica* is the attribute

Listable. This attribute specifies that a function should automatically be distributed or "threaded" over lists that appear as its arguments. This means that the function effectively gets applied separately to each element in any lists that appear as its arguments.

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This defines the function

to be listable.

Now

is automatically threaded over lists that appear as its arguments.

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Many of the attributes you can assign to functions in

*Mathematica* directly affect the evaluation of those functions. Some attributes, however, affect only other aspects of the treatment of functions. For example, the attribute

OneIdentity affects only pattern matching, as discussed in

"Flat and Orderless Functions". Similarly, the attribute

Constant is only relevant in differentiation, and operations that rely on differentiation.

The

Protected attribute affects assignments.

*Mathematica* does not allow you to make any definition associated with a symbol that carries this attribute. The functions

Protect and

Unprotect discussed in

"Modifying Built-in Functions" can be used as alternatives to

SetAttributes and

ClearAttributes to set and clear this attribute. As discussed in

"Modifying Built-in Functions", most built-in

*Mathematica* objects are initially protected so that you do not make definitions for them by mistake.

Here is a definition for the function

.

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Now you cannot modify the definition of

.

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You can usually see the definitions you have made for a particular symbol by typing

?f, or by using a variety of built-in

*Mathematica* functions. However, if you set the attribute

ReadProtected,

*Mathematica* will not allow you to look at the definition of a particular symbol. It will nevertheless continue to use the definitions in performing evaluation.

Although you cannot modify it, you can still look at the definition of

.

Now you can no longer read the definition of

.

Functions like

SetAttributes and

ClearAttributes usually allow you to modify the attributes of a symbol in any way. However, if you once set the

Locked attribute on a symbol, then

*Mathematica* will not allow you to modify the attributes of that symbol for the remainder of your

*Mathematica* session. Using the

Locked attribute in addition to

Protected or

ReadProtected, you can arrange for it to be impossible for users to modify or read definitions.

Clear[f] | remove values for f, but not attributes |

ClearAll[f] | remove both values and attributes of f |

Clearing values and attributes.

This clears values and attributes of

which was given attribute

Listable above.

Now

is no longer listable.

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By defining attributes for a function you specify properties that

*Mathematica* should assume whenever that function appears. Often, however, you want to assume the properties only in a particular instance. In such cases, you will be better off not to use attributes, but instead to call a particular function to implement the transformation associated with the attributes.

By explicitly calling

Thread, you can implement the transformation that would be done automatically if

were listable.

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Functions that perform transformations associated with some attributes.

Attributes in

*Mathematica* can only be permanently defined for single symbols. However,

*Mathematica* also allows you to set up pure functions which behave as if they carry attributes.

Function[vars,body,{attr_{1},...}] | a pure function with attributes , ... |

Pure functions with attributes.

This pure function applies

to the whole list.

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By adding the attribute

Listable, the function gets distributed over the elements of the list before applying

.

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