Pattern matching
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In
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, pattern matching is the act of checking a given sequence of tokens for the presence of the constituents of some
pattern A pattern is a regularity in the world, in human-made design, or in abstraction, abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometry, geometric shapes and ...
. In contrast to
pattern recognition Pattern recognition is the automated recognition of patterns and regularities in data. It has applications in statistical data analysis, signal processing, image analysis, information retrieval, bioinformatics, data compression, computer ...
, 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 In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
or
tree structure A tree structure, tree diagram, or tree model is a way of representing the hierarchy, hierarchical nature of a structure in a graphical form. It is named a "tree structure" because the classic #Representing trees, representation resembles a tree ...
s. 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 expression A regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of Character (computing), characters that specifies a pattern matching, search pattern in String (computer science), text. Usually s ...
s and matched using techniques such as
backtracking Backtracking is a class of algorithms for finding solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it de ...
. Tree patterns are used in some
programming language A programming language is a system of notation for writing computer program, computer programs. Most programming languages are text-based formal languages, but they may also be visual programming language, graphical. They are a kind of computer ...
s as a general tool to process data based on its structure, e.g. C#, F#,
Haskell Haskell () is a General-purpose programming language, general-purpose, static typing, statically-typed, purely functional programming, purely functional programming language with type inference and lazy evaluation. Designed for teaching, resear ...
, ML, Python,
Ruby A ruby is a pinkish red to blood-red colored gemstone, a variety of the mineral corundum (aluminium oxide). Ruby is one of the most popular traditional jewelry gems and is very durable. Other varieties of gem-quality corundum are called sapp ...
,
Rust Rust is an iron oxide, a usually reddish-brown oxide formed by the reaction of iron and oxygen in the catalytic presence of water or air moisture. Rust consists of hydrous ferric oxides, hydrous iron(III) oxides (Fe2O3·nH2O) and iron(III) oxi ...
, Scala,
Swift Swift or SWIFT most commonly refers to: * SWIFT, an international organization facilitating transactions between banks ** SWIFT code * Swift (programming language) * Swift (bird), a family of birds It may also refer to: Organizations * SWIFT, a ...
and the symbolic mathematics language
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, Computer algebra, symbolic computation, data manipulation, network analysis, time series analysi ...
have special syntax for expressing tree patterns and a
language construct In computer programming, a language construct is a Syntax (programming languages), syntactically allowable part of a Computer program, program that may be formed from one or more lexical tokens in accordance with the rules of the programming lang ...
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 SNOBOL ("StriNg Oriented and symBOlic Language") is a series of programming languages developed between 1962 and 1967 at AT&T Corporation, AT&T Bell Laboratories by David J. Farber, Ralph E. Griswold and Ivan P. Polonsky, culminating in SNOBOL4 ...
(1962), Refal (1968) with tree-based pattern matching,
Prolog Prolog is a logic programming language associated with artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence dis ...
(1972), SASL (1976), NPL (1977), and KRC (1981). Many text editors support pattern matching of various kinds: the QED editor supports
regular expression A regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of Character (computing), characters that specifies a pattern matching, search pattern in String (computer science), text. Usually s ...
search, and some versions of TECO support the OR operator in searches.
Computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s 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 Hope is an Optimism, optimistic state of mind that is based on an wikt:expectation, expectation of positive outcomes with respect to events and circumstances in one's life or the world at large. As a verb, its definitions include: "expect with ...
), 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 In computer science, an abstract syntax tree (AST), or just syntax tree, is a Tree (data structure), tree representation of the abstract syntax, abstract syntactic structure of text (often source code) written in a formal language. Each node of ...
of a programming language and
algebraic data type In computer programming, especially functional programming and type theory, an algebraic data type (ADT) is a kind of Data type#Composite types, composite type, i.e., a type formed by combining other types. Two common classes of algebraic types ...
s. 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 In computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied scie ...
, 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 OCaml ( , formerly Objective Caml) is a general-purpose programming language, general-purpose, multi-paradigm programming language which extends the Caml dialect of ML (programming language), ML with object-oriented programming, object-oriented ...
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 A list comprehension is a Syntax of programming languages, syntactic construct available in some programming languages for creating a list based on existing list (computing), lists. It follows the form of the mathematical ''set-builder notation'' ( ...
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_ _ Wolfram_Mathematica_is_a_software_system_with_built-in_libraries_for_several_areas_of_technical_computing_that_allow_machine_learning,_statistics,_Computer_algebra,_symbolic_computation,_data_manipulation,_network_analysis,_time_series_analysi_...
,_the_only_structure_that_exists_is_the_Tree_(data_structure).html" "title="_1,_A_2.html" ;"title=" 1, B 1, A 2, B 2 evaluates to [A 1, A 2"> 1, B 1, A 2, B 2 evaluates to [A 1, A 2


Pattern matching in Mathematica

In
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, Computer algebra, symbolic computation, data manipulation, network analysis, time series analysi ...
, the only structure that exists is the Tree (data structure)">tree In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondar ...
, which is populated by symbols. In the
Haskell Haskell () is a General-purpose programming language, general-purpose, static typing, statically-typed, purely functional programming, purely functional programming language with type inference and lazy evaluation. Designed for teaching, resear ...
syntax used thus far, this could be defined as data SymbolTree = Symbol String [SymbolTree] 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 a[b,c] 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 A 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. Note that even blanks themselves are internally represented as Blank[] for _ and Blank[x] for _x. The Mathematica function Cases filters elements of the first argument that match the pattern in the second argument: Cases _a[_.html" ;"title=".html" ;"title=" a[_"> a[_">.html" ;"title=" a[_"> a[_ evaluates to Pattern matching applies to the ''structure'' of expressions. In the example below, Cases[ , a[b[_], _] ] returns because only these elements will match the pattern a[b[_],_] 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 Trace 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 In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
as fib 1=1 fib _= fib -1+ fib -2 Then, we can ask the question: Given fib what is the sequence of recursive Fibonacci calls? Trace _fib[_.html" ;"title="ib fib[_">ib fib[_ returns a structure that represents the occurrences of the pattern fib /code> in the computational structure:


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 Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, Computer algebra, symbolic computation, data manipulation, network analysis, time series analysi ...
function Compile 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 Compile that expressions of the form com /code> can be assumed to be
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
s for the purposes of compilation: com _:= Binomial i, iCompile _x^com[i_.html"_;"title=".html"_;"title="_x^com[i">_x^com[i_">.html"_;"title="_x^com[i">_x^com[i_ Mailboxes_in_Erlang_programming_language.html" ;"title="">_x^com[i_.html" ;"title=".html" ;"title=" x^com[i"> x^com[i ">.html" ;"title=" x^com[i"> x^com[i Mailboxes in Erlang programming language">Erlang also work this way. The Curry–Howard correspondence between proofs and programs relates ML (programming language), ML-style pattern matching to Proof by cases, case analysis and proof by exhaustion.


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 element:list. 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 head'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 list, so we can disregard it, and thus write the function: head (element:_) = element The equivalent Mathematica transformation is expressed as head lement, =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 etterCharacter, 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: etter, 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 build up the patterns themselves or analyze and transform the programs that contain them.


SNOBOL

SNOBOL (''StriNg Oriented and symBOlic Language'') is a computer programming language developed between 1962 and 1967 at
AT&T AT&T Inc. is an American multinational corporation, multinational telecommunications holding company headquartered at Whitacre Tower in Downtown Dallas, Texas. It is the world's List of largest companies by revenue, largest telecommunications ...
Bell Laboratories Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984), then AT&T Bell Laboratories (1984–1996) and Bell Labs Innovations (1996–2007), is an American industrial Research and development, research and scientific developm ...
by David J. Farber, Ralph E. Griswold and Ivan P. Polonsky. SNOBOL4 stands apart from most programming languages by having patterns as a first-class data type (''i.e.'' a data type whose values can be manipulated in all ways permitted to any other data type in the programming language) and by providing operators for pattern
concatenation In formal language, formal language theory and computer programming, string concatenation is the operation of joining character string (computer science), character strings wikt:end-to-end, end-to-end. For example, the concatenation of "sno ...
and alternation. Strings generated during execution can be treated as programs and executed. SNOBOL was quite widely taught in larger US universities in the late 1960s and early 1970s and was widely used in the 1970s and 1980s as a text manipulation language in the
humanities Humanities are List of academic disciplines, academic disciplines that study aspects of human society and culture. In the Renaissance, the term contrasted with Divinity (academic discipline), divinity and referred to what is now called classi ...
. Since SNOBOL's creation, newer languages such as Awk and
Perl Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offi ...
have made string manipulation by means of
regular expression A regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of Character (computing), characters that specifies a pattern matching, search pattern in String (computer science), text. Usually s ...
s fashionable. SNOBOL4 patterns, however, subsume BNF grammars, which are equivalent to
context-free grammar In formal language theory, a context-free grammar (CFG) is a formal grammar whose Production (computer science), production rules are of the form :A\ \to\ \alpha with A a ''single'' nonterminal symbol, and \alpha a string of Terminal and nonter ...
s and more powerful than
regular expression A regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of Character (computing), characters that specifies a pattern matching, search pattern in String (computer science), text. Usually s ...
s.Gimpel, J. F. 1973. A theory of discrete patterns and their implementation in SNOBOL4. Commun. ACM 16, 2 (Feb. 1973), 91–100. DOI=http://doi.acm.org/10.1145/361952.361960.


See also

* AIML for an AI language based on matching patterns in speech * AWK language * Coccinelle pattern matches C source code * Matching wildcards *
glob (programming) In computer programming, glob () patterns specify sets of filenames with wildcard characters. For example, the Unix Bash (Unix shell), Bash shell command mv *.txt textfiles/ moves (mv) all files with names ending in .txt from the current director ...
* Pattern calculus *
Pattern recognition Pattern recognition is the automated recognition of patterns and regularities in data. It has applications in statistical data analysis, signal processing, image analysis, information retrieval, bioinformatics, data compression, computer ...
for fuzzy patterns * PCRE Perl Compatible Regular Expressions, a common modern implementation of string pattern matching ported to many languages * REBOL parse dialect for pattern matching used to implement language dialects *
Symbolic integration In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or ''indefinite integral'', of a given function (mathematics), function ''f''(''x''), i.e. to find a differentiable function ''F''(''x'') such that :\fr ...
* Tagged union * Tom (pattern matching language) *
SNOBOL SNOBOL ("StriNg Oriented and symBOlic Language") is a series of programming languages developed between 1962 and 1967 at AT&T Corporation, AT&T Bell Laboratories by David J. Farber, Ralph E. Griswold and Ivan P. Polonsky, culminating in SNOBOL4 ...
for a programming language based on one kind of pattern matching * Pattern language — metaphoric, drawn from architecture * Graph matching


References

* The Mathematica Book, chapte
Section 2.3: Patterns
* The Haskell 98 Report, chapte

* Python Reference Manual, chapte

* The Pure (programming language), Pure Programming Language, chapte
4.3: Patterns


External links



* Nikolaas N. Oosterhof, Philip K. F. Hölzenspies, and Jan Kuper
Application patterns
A presentation at Trends in Functional Programming, 2005
JMatch
the Java (programming language), Java programming language extended with pattern matching
ShowTrend
Online pattern matching for stock prices

by Dennis Ritchie - provides the history of regular expressions in computer programs
The Implementation of Functional Programming Languages, pages 53–103
Simon Peyton Jones, published by Prentice Hall, 1987.
Nemerle, pattern matching



PatMat: a C++ pattern matching library based on
SNOBOL SNOBOL ("StriNg Oriented and symBOlic Language") is a series of programming languages developed between 1962 and 1967 at AT&T Corporation, AT&T Bell Laboratories by David J. Farber, Ralph E. Griswold and Ivan P. Polonsky, culminating in SNOBOL4 ...
/SPITBOL * Temur Kutsia
Flat Matching
Journal of Symbolic Computation 43(12): 858–873. Describes in details flat matching in Mathematica.

pattern matching language for non-programmers {{DEFAULTSORT:Pattern Matching Pattern matching, Conditional constructs Articles with example Haskell code Functional programming