Pattern matching

This article is about pattern matching infunctional programming.For other uses, seestring matchingandpattern recognition.

For the use of variable matching criteria in defining abstract patterns to match, seeregular expression.

Incomputer science,pattern matchingis the act of checking a given sequence oftokensfor the presence of the constituents of somepattern.In contrast topattern recognition,the match usually has to be exact: "either it will or will not be a match." The patterns generally have the form of eithersequencesortree 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 usingregular expressionsand matched using techniques such asbacktracking.

Tree patterns are used in someprogramming languagesas a general tool to process data based on its structure, e.g.C#,F#,Haskell,ML,Python,Ruby,Rust,Scala,Swiftand the symbolic mathematics languageMathematicahave specialsyntaxfor expressing tree patterns and alanguage constructforconditional executionand 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 forguards.

History

See also:Regular expression § History

Early programming languages with pattern matching constructs includeCOMIT(1957),SNOBOL(1962),Refal(1968) with tree-based pattern matching,Prolog(1972), St Andrews Static Language (SASL) (1976),NPL(1977), andKent Recursive Calculator(KRC) (1981).

Manytext editorssupport pattern matching of various kinds: theQED editorsupportsregular expressionsearch, and some versions ofTECOsupport the OR operator in searches.

Computer algebra systemsgenerally 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):

f0=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:

fn=n*f(n-1)

Here, the firstnis 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 leastHope), 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 returnsn * 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 formatching wildcardsin simple string-matching situations have been developed in a number ofrecursiveand 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 theabstract syntax treeof a programming language andalgebraic data types.

In Haskell, the following line defines an algebraic data typeColorthat has a single data constructorColorConstructorthat wraps an integer and a string.

dataColor=ColorConstructorIntegerString

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

When we want to writefunctionsto makeColoranabstract data type,we wish to write functions tointerfacewith 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 ofColor.

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

integerPart(ColorConstructortheInteger_)=theInteger

As well:

stringPart(ColorConstructor_theString)=theString

The creations of these functions can be automated by Haskell's datarecordsyntax.

ThisOCamlexample which defines ared–black treeand a function to re-balance it after element insertion shows how to match on a more complex structure generated by a recursive data type. The compiler verifies at compile-time that the list of cases is exhaustive and none are redundant.

typecolor=Red|Black
type'atree=Empty|Treeofcolor*'atree*'a*'atree

letrebalancet=matchtwith
|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 alist comprehensioncould be used for this kind of filtering:

[Ax|Ax<-[A1,B1,A2,B2]]

evaluates to

[A 1, A 2]

Pattern matching in Mathematica

InMathematica,the only structure that exists is thetree,which is populated by symbols. In theHaskellsyntax used thus far, this could be defined as

dataSymbolTree=SymbolString[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 instancea[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[1], A[2], or more generally A[x] wherexis any entity. In this case,Ais 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 asBlank[]for_andBlank[x]for_x.

The Mathematica functionCasesfilters 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 thestructureof 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 patterna[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 functionTracecan be used to monitor a computation, and return the elements that arise which match a pattern. For example, we can define theFibonacci sequenceas

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 patternfib[_]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, theMathematicafunctionCompilecan 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{{com[_], Integer}}instructsCompilethat expressions of the formcom[_]can be assumed to beintegersfor the purposes of compilation:

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

Mailboxes inErlangalso work this way.

TheCurry–Howard correspondencebetween proofs and programs relatesML-style pattern matching tocase analysisandproof 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 andfunctional programminglanguages in general, strings are represented as functionallistsof 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 thuselement: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 ofhead'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 forlist,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,guardscould be used to achieve the same matches:

[letter,digit]|isAlphaletter&&isDigitdigit

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

Main article:SNOBOL

SNOBOL (StriNg Oriented and symBOlic Language) is a computer programming language developed between 1962 and 1967 atAT&TBell LaboratoriesbyDavid J. Farber,Ralph E. Griswoldand Ivan P. Polonsky.

SNOBOL4 stands apart from most programming languages by having patterns as afirst-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 patternconcatenationandalternation.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 thehumanities.

Since SNOBOL's creation, newer languages such asAwkandPerlhave made string manipulation by means ofregular expressionsfashionable. SNOBOL4 patterns, however, subsumeBNFgrammars, which are equivalent tocontext-free grammarsand more powerful thanregular expressions.

See also

• Artificial Intelligence Markup Language(AIML) for an AI language based on matching patterns in speech
• AWKlanguage
• Coccinellepattern matches C source code
• Matching wildcards
• glob (programming)
• Pattern calculus
• Pattern recognitionfor fuzzy patterns
• PCREPerl Compatible Regular Expressions, a common modern implementation of string pattern matching ported to many languages
• REBOL parse dialectfor pattern matching used to implement language dialects
• Symbolic integration
• Tagged union
• Tom (pattern matching language)
• SNOBOLfor a programming language based on one kind of pattern matching
• Pattern language— metaphoric, drawn from architecture
• Graph matching

References

• The Mathematica Book, chapterSection 2.3: Patterns
• The Haskell 98 Report, chapter3.17 Pattern Matching.
• Python Reference Manual, chapter6.3 Assignment statements.
• ThePureProgramming Language, chapter4.3: Patterns

External links

• Views: An Extension to Haskell Pattern Matching
• Nikolaas N. Oosterhof, Philip K. F. Hölzenspies, and Jan Kuper.Application patterns.A presentation at Trends in Functional Programming, 2005
• JMatch:theJavalanguage extended with pattern matching
• ShowTrend:Online pattern matching for stock prices
• An incomplete history of the QED Text EditorbyDennis Ritchie- provides the history of regular expressions in computer programs
• The Implementation of Functional Programming Languages, pages 53–103Simon Peyton Jones, published by Prentice Hall, 1987.
• Nemerle, pattern matching.
• Erlang, pattern matching.
• Prop: a C++ based pattern matching language, 1999
• PatMat: a C++ pattern matching library based onSNOBOL/SPITBOL
• Temur Kutsia.Flat Matching.Journal of Symbolic Computation 43(12): 858–873. Describes in details flat matching in Mathematica.
• EasyPattern languagepattern matching language for non-programmers