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Reducing Expressions to Their Standard FormThe Standard Evaluation Procedure

2.5.3 Attributes

Definitions such as f[x_] = x^2 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 f.

In[1]:= SetAttributes[f, Flat]

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

In[2]:= f[f[a, b], c]


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 f.

In[3]:= f[x_, x_] := f[x]

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

In[4]:= f[a, a, a, b, b, b, c, c]


Manipulating attributes of symbols.

This shows the attributes assigned to f.

In[5]:= Attributes[f]


This removes the attributes assigned to f.

In[6]:= Attributes[f] = { }


The complete list of attributes for symbols in Mathematica.

Here are the attributes for the built-in function Plus.

In[7]:= Attributes[Plus]


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.

The built-in Log function is Listable.

In[8]:= Log[{5, 8, 11}]


This defines the function p to be listable.

In[9]:= SetAttributes[p, Listable]

Now p is automatically threaded over lists that appear as its arguments.

In[10]:= p[{a, b, c}, d]


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 Section 2.3.7. 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 Section 2.4.12 can be used as alternatives to SetAttributes and ClearAttributes to set and clear this attribute. As discussed in Section 2.4.12 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 g.

In[11]:= g[x_] = x + 1


This sets the Protected attribute for g.

In[12]:= Protect[g]


Now you cannot modify the definition of g.

In[13]:= g[x_] = x


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 g.

In[14]:= ?g

This sets the ReadProtected attribute for g.

In[15]:= SetAttributes[g, ReadProtected]

Now you can no longer read the definition of g.

In[16]:= ?g

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.

Clearing values and attributes.

This clears values and attributes of p which was given attribute Listable above.

In[17]:= ClearAll[p]

Now p is no longer listable.

In[18]:= p[{a, b, c}, d]


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 p were listable.

In[19]:= Thread[p[{a, b, c}, d]]


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.

Pure functions with attributes.

This pure function applies p to the whole list.

In[20]:= Function[{x}, p[x]] [{a, b, c}]


By adding the attribute Listable, the function gets distributed over the elements of the list before applying p.

In[21]:= Function[{x}, p[x], {Listable}] [{a, b, c}]


Reducing Expressions to Their Standard FormThe Standard Evaluation Procedure