Pattern matching

Pattern matching is the act of checking for the presence of the constituents of a given pattern. In contrast to pattern recognition, the pattern is rigidly specified. Such a pattern concerns conventionally either sequences or tree structures. Pattern matching is used to check that things have the desired structure, to find relevant structure, to retrieve the aligning parts, and to substitute the matching part with something else. sequence (or specifically text string) patterns are often described using regular expressions (i.e. backtracking) and matched using respective algorithms. Sequences can also be seen as trees branching for each element into the respective element and the rest of the sequence, or as trees that immediately branch into all elements.

Tree patterns can be used in programming languages as a general tool to process data based on its structure. Some functional programming languages such as Haskell, ML and the symbolic mathematics language Mathematica have a special syntax for expressing tree patterns and a language construct for conditional execution and value retrieval based on it. Because of simplicity and efficiency reasons these tree patterns lack some features that are available in regular expressions. Depending on the languages, pattern matching can be used for function arguments, in case expressions, whenever new variables are bound, or in very limited situations such as only for sequences in assignment in Python. Often it is possible to give alternative patterns that are tried one by one. Pattern matching can benefit from guards.

Term rewriting languages rely on pattern matching for the fundamental way a program evaluates into a result. Pattern matching benefits most when the underlying datastructures are as simple and flexible as possible. This is especially the case in languages with a strong symbolic flavor. In symbolic programming languages, patterns are the same kind of datatype as everything else, and can therefore be fed in as arguments to functions.

Primitive patterns
The simplest pattern in pattern matching is an explicit value or a variable. For an example, consider a simple function definition in Haskell syntax (function parameters are not in parentheses but are separated by spaces, = is not assignment but definition):

f 0 = 1

Here, 0 is a single value pattern. Now, whenever f is given 0 as argument the pattern matches and the function returns 1. With any other argument, the matching and thus the function fail. As the syntax supports alternative patterns in function definitions, we can continue the definition extending it to take more generic arguments:

f n = n * f (n-1)

Here, the first  is a single variable pattern, which will match absolutely any argument and bind it to name n to be used in the rest of the definition. In Haskell (unlike at least Hope), patterns are tried in order so the first definition still applies in the very specific case of the input being 0, while for any other argument the function returns  with n being the argument.

Wildcard pattern (often written as ) is also simple: like a variable name, it matches any value, but does not bind the value to any name.

Tree patterns
More complex patterns can be built from the primitive ones of the previous section, usually in the same way as values are built by combining other values. The difference then is that with variable and wildcard parts, a pattern doesn't build into single value, but matches a group of values that are the combination of the concrete elements and the elements that are allowed to vary within the structure of the pattern.

A tree pattern describes a part of a tree by starting with a node and specifying some branches and nodes and leaving some unspecified with a variable or wildcard pattern. It may help to think of the abstract syntax tree of a programming language and algebraic data types.

In Haskell, the following line defines an algebraic data type  that has a single data constructor   that wraps an integer and a string. data Color = ColorConstructor Integer String

The constructor is a node in a tree and the integer and string are leaves in branches.

When we want to write functions to make  an abstract data type, we wish to write functions to interface with the data type, and thus we want to extract some data from the data type, for example, just the string or just the integer part of.

If we pass a variable that is of type Color, how can we get the data out of this variable? For example, for a function to get the integer part of, we can use a simple tree pattern and write: integerPart (ColorConstructor theInteger _) = theInteger

Or conversely: stringPart (ColorConstructor _ theString) = theString

Filtering data with patterns
Pattern matching can be used to filter data of a certain structure. For instance, in Haskell a list comprehension could be used for this kind of filtering:

[A x | A x <- [A 1, B 1, A 2, B 2]]

evaluates to [A 1, A 2]

Pattern matching in Mathematica
In Mathematica, the only structure that exists is the tree, which is populated by symbols. In the Haskell syntax used thus far, this could be defined as data SymbolTree = Symbol String [Symbol] An example tree could then look like Symbol "a" [Symbol "b" [], Symbol "c" []]

In the traditional, more suitable syntax, the symbols are written as they are and the levels of the tree are represented using [], so that for instance  is a tree with a as the parent, and b and c as the children.

A pattern in Mathematica involves putting "_" at positions in that tree. For instance, the pattern

A[_]

Will match elements such as A[1], A[2], or more generally A[x] where x is any entity. In this case, A is the concrete element, while _ denotes the piece of tree that can be varied. A symbol prepended to _ binds the match to that variable name while a symbol appended to _ restricts the matches to nodes of that symbol.

The Mathematica function  filters elements of the first argument that match the pattern in the second argument:

Cases[{a[1], b[1], a[2], b[2]}, a[_] ]

evaluates to

{a[1], a[2]}

Pattern matching applies to the structure of expressions. In the example below,

Cases[{a[b], a[b,c], a[b[c], d], a[b[c], d[e]], a[b[c], d, e]}, a[b[_],_]]

returns

{a[b[c],d], a[b[c],d[e]]}

because only these elements will match the pattern  above.

In Mathematica, it is also possible to extract structures as they are created in the course of computation, regardless of how or where they appear. The function  can be used to monitor a computation, and return the elements that arise which match a pattern. For example, we can define the Fibonacci sequence as

fib[0|1]:=1 fib[n_]:= fib[n-1] + fib[n-2]

Then, we can ask the question: Given fib[3], what is the sequence of recursive Fibonacci calls?

Trace[fib[3], fib[_]]

returns a structure that represents the occurrences of the pattern fib[_] in the computational structure:

{fib[3],{fib[2],{fib[1]},{fib[0]}},{fib[1]}}

Declarative programming
In symbolic programming languages, it is easy to have patterns as arguments to functions or as elements of data structures. A consequence of this is the ability to use patterns to declaratively make statements about pieces of data and to flexibly instruct functions how to operate.

For instance, the Mathematica function  can be used to make more efficient versions of the code. In the following example the details do not particularly matter; what matters is that the subexpression instructs   that expressions of the form com[_] can be assumed to be integers for the purposes of compilation:

com[i_] := Binomial[2i, i] Compile[{x, {i, _Integer}}, x^com[i], ]

Mailboxes in Erlang also work this way.

Pattern matching and strings
By far the most common form of pattern matching involves strings of characters. In many programming languages, a particular syntax of strings is used to represent regular expressions, which are patterns describing string characters.

However, it is possible to perform some string pattern matching within the same framework that has been discussed throughout this article.

Tree patterns for strings
In Mathematica, strings are represented as trees of root StringExpression and all the characters in order as children of the root. Thus, to match "any amount of trailing characters", a new wildcard ___ is needed in contrast to _ that would match only a single character.

In Haskell and functional programming languages in general, strings are represented as functional lists of characters. A functional list is defined as an empty list, or an element constructed on an existing list. In Haskell syntax: []     -- an empty list x:xs   -- an element x constructed on a list xs

The structure for a list with some elements is thus. When pattern matching, we assert that a certain piece of data is equal to a certain pattern. For example, in the function: head (element:list) = element

we assert that the first element of 's argument is called element, and the function returns this. We know that this is the first element because of the way lists are defined, a single element constructed onto a list. This single element must be the first. The empty list would not match the pattern at all, as an empty list does not have a head (the first element that is constructed).

In the example, we have no use for, so we can disregard it, and thus write the function: head (element:_) = element

The equivalent Mathematica transformation is expressed as

head[element_, ___]:=element

Example string patterns
In Mathematica, for instance,

StringExpression["a", _]

will match a string that has two characters and begins with "a".

The same pattern in Haskell:

['a', _]

Symbolic entities can be introduced to represent many different classes of relevant features of a string. For instance,

StringExpression[LetterCharacter, DigitCharacter]

will match a string that consists of a letter first, and then a number.

In Haskell, guards could be used to achieve the same matches:

[letter, digit] | isAlpha letter && isDigit digit

The main advantage of symbolic string manipulation is that it can be completely integrated with the rest of the programming language, rather than being a separate, special purpose subunit. The entire power of the language can be leveraged to built up the patterns themselves or analyze and transform the programs that contain them.

History
The first computer programs to use pattern matching were text editors. At Bell Labs, Ken Thompson extended the seeking and replacing features of the QED editor to accept regular expressions. Early programming languages with pattern matching constructs include SNOBOL from 1962, NPL from 1977, and KRC from 1981.