Pattern matching

In computer science, pattern matching is the act of checking a given sequence of tokens for the presence of the constituents of some pattern. In contrast to pattern recognition, the match usually has to be exact: "either it will or will not be a match." The patterns generally have the form of either sequences or tree structures. Uses of pattern matching include outputting the locations (if any) of a pattern within a token sequence, to output some component of the matched pattern, and to substitute the matching pattern with some other token sequence (i.e., search and replace).

Sequence patterns (e.g., a text string) are often described using regular expressions and matched using techniques such as backtracking.

Tree patterns are used in some programming languages as a general tool to process data based on its structure, e.g. C#, F#, Haskell, ML, Python, Ruby, Rust, Scala, Swift and the symbolic mathematics language Mathematica have special syntax for expressing tree patterns and a language construct for conditional execution and value retrieval based on it.

Often it is possible to give alternative patterns that are tried one by one, which yields a powerful conditional programming construct. Pattern matching sometimes includes support for guards.

History

Early programming languages with pattern matching constructs include COMIT (1957), SNOBOL (1962), Refal (1968) with tree-based pattern matching, Prolog (1972), SASL (1976), NPL (1977), and KRC (1981).

Many text editors support pattern matching of various kinds: the QED editor supports regular expression search, and some versions of TECO support the OR operator in searches.

Computer algebra systems generally support pattern matching on algebraic expressions.

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 n 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 n * f (n-1) with n being the argument.

The 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. Algorithms for matching wildcards in simple string-matching situations have been developed in a number of recursive and non-recursive varieties.

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 a 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 Color that has a single data constructor ColorConstructor 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 Color 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 Color.

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 Color, we can use a simple tree pattern and write:

integerPart (ColorConstructor theInteger _) = theInteger

As well:

stringPart (ColorConstructor _ theString) = theString

The creations of these functions can be automated by Haskell's data record syntax.

This Ocaml example which defines a red-black tree and a function to re-balance it after element insertion shows how to match on a more complex structure generated by a recursive data type.

type color = Red , Black
type 'a tree = Empty , Tree of color * 'a tree * 'a * 'a tree

let rebalance t = match t with
, Tree (Black, Tree (Red, Tree (Red, a, x, b), y, c), z, d)
, Tree (Black, Tree (Red, a, x, Tree (Red, b, y, c)), z, d)
, Tree (Black, a, x, Tree (Red, Tree (Red, b, y, c), z, d))
, Tree (Black, a, x, Tree (Red, b, y, Tree (Red, c, z, d)))
-> Tree (Red, Tree (Black, a, x, b), y, Tree (Black, c, z, d))
, _ -> t (* the 'catch-all' case if no previous pattern matches *)

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 <- _1,_B_1,_A_2,_B_2

evaluates_to
_[A_1,_A_2

_Pattern_matching_in_Mathematica

In_Mathematica_

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