Patterns and Transformation Rules

Patterns

Patterns stand for classes of expressions. They contain pattern objects that represent sets of possible expressions.

(RefLink[Pattern, paclet:ref/Pattern]) objects.

When several pattern objects with the same name occur in a single pattern, all the objects must stand for the same expression. Thus f[x_,x_] can stand for f[2,2] but not f[2,3].

In a pattern object such as _h, the head h can be any expression, but cannot itself be a pattern.

A pattern object such as x__ stands for a sequence of expressions. So, for example, f[x__] can stand for f[a,b,c], with x being (RefLink[Sequence, paclet:ref/Sequence])[a,b,c]. If you use x, say in the result of a transformation rule, the sequence will be spliced into the function in which x appears. Thus g[u,x,u] would become g[u,a,b,c,u].

When the pattern objects x_:v and x_. appear as arguments of functions, they represent arguments which may be omitted. When the argument corresponding to x_:v is omitted, x is taken to have value v. When the argument corresponding to x_. is omitted, x is taken to have a default value that is associated with the function in which it appears. You can specify this default value by making assignments for (RefLink[Default, paclet:ref/Default])[f] and so on.

(RefLink[Default, paclet:ref/Default]) values.

A pattern like f[x__,y__,z__] can match an expression like f[a,b,c,d,e] with several different choices of x, y, and z. The choices with x and y of minimum length are tried first. In general, when there are multiple __ or ___ in a single function, the case that is tried first takes all the __ and ___ to stand for sequences of minimum length, except the last one, which stands for "the rest" of the arguments.

When x_:v or x_. are present, the case that is tried first is the one in which none of them correspond to omitted arguments. Cases in which later arguments are dropped are tried next.

The order in which the different cases are tried can be changed using (RefLink[Shortest, paclet:ref/Shortest]) and (RefLink[Longest, paclet:ref/Longest]).

Attributes used in matching patterns.

Pattern objects like x_ can represent any sequence of arguments in a function f with attribute (RefLink[Flat, paclet:ref/Flat]). The value of x in this case is f applied to the sequence of arguments. If f has the attribute (RefLink[OneIdentity, paclet:ref/OneIdentity]), then e is used instead of f[e] when x corresponds to a sequence of just one argument.

Assignments

The two basic types of assignment in the Wolfram Language.

Assignments in the Wolfram Language specify transformation rules for expressions. Every assignment that you make must be associated with a particular Wolfram Language symbol.

Assignments associated with different symbols.

In the case of an assignment like f[args]=rhs, the Wolfram Language looks at f, then the head of f, then the head of that, and so on, until it finds a symbol with which to associate the assignment.

When you make an assignment like lhs^=rhs, the Wolfram Language will set up transformation rules associated with each distinct symbol that occurs either as an argument of lhs, or as the head of an argument of lhs.

The transformation rules associated with a particular symbol s are always stored in a definite order, and are tested in that order when they are used. Each time you make an assignment, the corresponding transformation rule is inserted at the end of the list of transformation rules associated with s, except in the following cases:

The left-hand side of the transformation rule is identical to a transformation rule that has already been stored, and any /; conditions on the right-hand side are also identical. In this case, the new transformation rule is inserted in place of the old one.

The Wolfram Language determines that the new transformation rule is more specific than a rule already present, and would never be used if it were placed after this rule. In this case, the new rule is placed before the old one. Note that in many cases it is not possible to determine whether one rule is more specific than another; in such cases, the new rule is always inserted at the end.

Types of Values

Types of values associated with symbols.

Clearing and Removing Objects

Ways to clear and remove objects.

In (RefLink[Clear, paclet:ref/Clear]), (RefLink[ClearAll, paclet:ref/ClearAll]), and (RefLink[Remove, paclet:ref/Remove]), each argument can be either a symbol or the name of a symbol as a string. String arguments can contain the metacharacters * and @ to specify action on all symbols whose names match the pattern.

(RefLink[Clear, paclet:ref/Clear]), (RefLink[ClearAll, paclet:ref/ClearAll]), and (RefLink[Remove, paclet:ref/Remove]) do nothing to symbols with the attribute (RefLink[Protected, paclet:ref/Protected]).

Transformation Rules

The two basic types of transformation rules in the Wolfram Language.

Replacements for pattern variables that appear in transformation rules are effectively done using (RefLink[ReplaceAll, paclet:ref/ReplaceAll]) (the /. operator).