Patterns and Transformation Rules

Patterns

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

_any expression
x_any expression, given the name x
x:patterna pattern, given the name x
pattern?testa pattern that yields True when test is applied to its value
_hany expression with head h
x_hany 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 ___hsequences of expressions, each with head h
x__h and x___hsequences of expressions with head h, given the name x
PatternSequence[p1,p2,...]a sequence of patterns
x_:van expression with default value v
x_h:van 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
pattern1|pattern2|... a pattern which matches at least one of the
pattern/;conda 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 can stand for but not .

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

A pattern object such as stands for a sequence of expressions. So, for example, can stand for , with being Sequence[a, b, c]. If you use , say in the result of a transformation rule, the sequence will be spliced into the function in which appears. Thus would become .

When the pattern objects and appear as arguments of functions, they represent arguments which may be omitted. When the argument corresponding to is omitted, x is taken to have value v. When the argument corresponding to 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 when it appears as any argument of the function f
Default[f,n]default value for 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 can match an expression like with several different choices of , , and . The choices with and 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 or 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 and are equivalent
Flat and are equivalent
OneIdentity and x are equivalent

Attributes used in matching patterns.

Pattern objects like 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 when x corresponds to a sequence of just one argument.

Assignments

lhs=rhsimmediate assignment: rhs is evaluated at the time of assignment
lhs:=rhsdelayed 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]=rhsassignment is associated with f (downvalue)
t/:f[args]=rhsassignment is associated with t (upvalue)
f[g[args]]^=rhsassignment is associated with g (upvalue)

Assignments associated with different symbols.

In the case of an assignment like , 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 , 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 , , 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

    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[s1,s2,...]clear all values for the symbols , except for attributes, messages, and defaults
    ClearAll[s1,s2,...]clear all values for the , including attributes, messages, and defaults
    Remove[s1,s2,...]clear all values, and then remove the names of the

    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->rhsimmediate rule: rhs is evaluated when the rule is first given
    lhs:>rhsdelayed 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).

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