# Patterns and Transformation Rules

## Patterns

*Patterns* stand for classes of expressions. They contain

*pattern objects* which represent sets of possible expressions.

_ | any expression |

x_ | any expression, given the name *x* |

x:pattern | a pattern, given the name *x* |

pattern?test | a pattern that yields True when test is applied to its value |

_h | any expression with head *h* |

x_h | any expression with head *h*, given the name *x* |

__ | any sequence of one or more expressions |

___ | any sequence of zero or more expressions |

x__ and x___ | sequences of expressions, given the name *x* |

__h and ___h | sequences of expressions, each with head *h* |

x__h and x___h | sequences of expressions with head *h*, given the name *x* |

PatternSequence[p_{1},p_{2},...] | a sequence of patterns |

x_:v | an expression with default value *v* |

x_h:v | an expression with head *h* and default value *v* |

x_. | an expression with a globally defined default value |

Optional[x_h] | an expression that must have head *h*, and has a globally defined default value |

Except[c] | any expression except one that matches *c* |

Except[c,pattern] | any expression matching pattern, except one that matches *c* |

pattern.. | a pattern repeated one or more times |

pattern... | a pattern repeated zero or more times |

Repeated[pattern, spec] | a pattern repeated according to spec |

pattern_{1}|pattern_{2}|... | a pattern which matches at least one of the pattern_{i} |

pattern/;cond | a pattern for which cond evaluates to True |

HoldPattern[pattern] | a pattern not evaluated |

Verbatim[expr] | an expression to be matched verbatim |

OptionsPattern[] | a sequence of options |

Longest[pattern] | the longest sequence consistent with pattern |

Shortest[pattern] | the shortest sequence consistent with pattern |

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

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

Default[f] and so on.

Default[f] | default value for x_. when it appears as any argument of the function *f* |

Default[f,n] | default value for x_. when it appears as the *n*^{th} argument (negative *n* count from the end) |

Default[f,n,tot] | default value for the *n*^{th} argument when there are a total of tot arguments |

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

Shortest and

Longest.

Orderless | *f*[*x*, *y*] and *f*[*y*, *x*] are equivalent |

Flat | *f*[*f*[*x*], *y*] and *f*[*x*, *y*] are equivalent |

OneIdentity | *f*[*x*] and *x* are equivalent |

Attributes used in matching patterns.

Pattern objects like

x_ can represent any sequence of arguments in a function

*f* with attribute

Flat. The value of

*x* in this case is

*f* applied to the sequence of arguments. If

*f* has the attribute

OneIdentity, then

*e* is used instead of

*f*[*e*] when

*x* corresponds to a sequence of just one argument.

## Assignments

lhs=rhs | immediate assignment: rhs is evaluated at the time of assignment |

lhs:=rhs | delayed assignment: rhs is evaluated when the value of lhs is requested |

The two basic types of assignment in *Mathematica*.

Assignments in

*Mathematica* specify transformation rules for expressions. Every assignment that you make must be associated with a particular

*Mathematica* symbol.

f[args]=rhs | assignment is associated with *f* (downvalue) |

t/:f[args]=rhs | assignment is associated with *t* (upvalue) |

f[g[args]]^=rhs | assignment is associated with *g* (upvalue) |

Assignments associated with different symbols.

In the case of an assignment like

f[args]=rhs,

*Mathematica* 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,

*Mathematica* 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.

*Mathematica* 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

Attributes[f ] | attributes of *f* |

DefaultValues[f ] | default values for arguments of *f* |

DownValues[f ] | values for f[...], f[...][...], etc. |

FormatValues[f ] | print forms associated with *f* |

Messages[f ] | messages associated with *f* |

NValues[f ] | numerical values associated with *f* |

Options[f ] | defaults for options associated with *f* |

OwnValues[f ] | values for *f* itself |

UpValues[f ] | values for ...[..., f[...], ...] |

Types of values associated with symbols.

## Clearing and Removing Objects

expr=. | clear a value defined for expr |

f/:expr=. | clear a value associated with *f* defined for expr |

Clear[s_{1},s_{2},...] | clear all values for the symbols *s*_{i}, except for attributes, messages and defaults |

ClearAll[s_{1},s_{2},...] | clear all values for the *s*_{i}, including attributes, messages and defaults |

Remove[s_{1},s_{2},...] | clear all values, and then remove the names of the *s*_{i} |

Ways to clear and remove objects.

In

Clear,

ClearAll and

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.

Clear,

ClearAll and

Remove do nothing to symbols with the attribute

Protected.

## Transformation Rules

lhs->rhs | immediate rule: rhs is evaluated when the rule is first given |

lhs:>rhs | delayed rule: rhs is evaluated when the rule is used |

The two basic types of transformation rules in *Mathematica*.

Replacements for pattern variables that appear in transformation rules are effectively done using

ReplaceAll (the

/. operator).