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| | I like Gaming. <br>I to learn English in my free time.<br><br>Here is my blog; [http://www.latestsms.in/hindi-sms-messages.htm Good Morning SMS] |
| In [[mathematics]], [[computer science]], and [[logic]], '''rewriting''' covers a wide range of (potentially [[Deterministic computation|non-deterministic]]) methods of replacing subterms of a [[Well-formed formula|formula]] with other terms. What is considered are '''rewriting systems''' (also known as '''rewrite systems''' or '''reduction systems'''). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects.
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| Rewriting can be [[Non-deterministic algorithm|non-deterministic]]. One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an [[algorithm]] for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems can be viewed as [[computer program]]s, and several [[declarative programming language]]s are based on term rewriting.
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| == Intuitive examples ==
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| <!-- Commented out because the example lack the rewriting rules are simply confusing.
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| === Arithmetic ===
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| Consider the rules of regular arithmetic. We can think of these rules as forming a rewriting system, with rules including:
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| : <math> \mathrm{plus}(a, b) \rightarrow a + b</math>
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| : <math> \mathrm{times}(a, b) \rightarrow a \times b</math>
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| where ''a''+''b'' denotes the sum of ''a'' and ''b'', and ''a'' × ''b'' denotes the product of ''a'' and ''b''.
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| Suppose we are given the expression
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| : <math> \mathrm{times}(\mathrm{plus}(11,9), \mathrm{plus}(2,4))</math>
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| We can rewrite this expression in two ways, simplifying either the first bracket or the second. Simplifying the first bracket, we have
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| : <math> \mathrm{times}(20, \mathrm{plus}(2, 4)) = \mathrm{times}(20, 6) = 120</math>
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| Simplifying the second gives
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| : <math> \mathrm{times}(\mathrm{plus}(11,9), 6) = \mathrm{times}(20, 6) = 120.</math>
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| === Basic algebra ===
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| Rewrite systems provide a convenient method of [[automating theorem proving]]. If we begin with a set of [[equational logic|equational]] hypotheses, then these may be used to formulate a set of rewrite rules. An example from school [[algebra]] goes under the heading ''collect [[like terms]]'' in an equation. There will usually be several ways to proceed, in collecting up and simplifying an equation
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| :''P''(''x'') = ''Q''(''x'')
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| in which ''P'' and ''Q'' are [[polynomial]]s. After some application of the conventional rules of algebra we may end with an equation
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| :''R''(''x'') = 0.
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| This is something like a [[normal form (term rewriting)|normal form]], though we may well have different signs (at least) for ''R'' found by different routes. If we insist that ''R'' is [[monic polynomial|monic]] there is actually a normal form (as is usually tacitly assumed) in which ''R''(''x'') is written in terms of decreasing powers of ''x''.
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| -->
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| === Logic ===
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| In [[logic]], the procedure for obtaining the [[conjunctive normal form]] (CNF) of a formula can be conveniently written as a rewriting system. The rules of such a system would be:
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| :<math>\neg\neg A \to A</math> ([[double negative elimination]])
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| :<math>\neg(A \land B) \to \neg A \lor \neg B</math> ([[De Morgan's laws]])
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| :<math>\neg(A \lor B) \to \neg A \land\neg B</math>
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| :<math> (A \land B) \lor C \to (A \lor C) \land (B \lor C)</math> ([[Distributivity]])
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| :<math> A \lor (B \land C) \to (A \lor B) \land (A \lor C)</math>,
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| where the symbol (<math>\to</math>) indicates that an expression matching the left hand side of the rule can be rewritten to one formed by the right hand side. In this system, we can perform a rewrite from left to right only when the logical interpretation of the left hand side is [[Logical equivalence|equivalent]] to that of the right.
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| == Abstract rewriting systems ==
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| {{Main|abstract rewriting system}}
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| {{Merge to|abstract rewriting system|date=February 2011}}
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| From the above examples, it's clear that we can think of rewriting systems in an abstract manner. We need to specify a set of objects and the rules that can be applied to transform them. The most general (unidimensional) setting of this notion is called an '''abstract reduction system''', (abbreviated '''ARS'''), although more recently authors use '''abstract rewriting system''' as well.<ref>Bezem et al, p. 7,</ref> (The preference for the word "reduction" here instead of "rewriting" constitutes a departure from the uniform use of "rewriting" in the names of systems that are particularizations of ARS. Because the word "reduction" does not appear in the names of more specialized systems, in older texts '''reduction system''' is a synonym for ARS).<ref name="Book and Otto, p. 10">Book and Otto, p. 10</ref>
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| An ARS is simply a set ''A'', whose elements are usually called objects, together with a [[binary relation]] on ''A'', traditionally denoted by →, and called the '''reduction relation''', '''rewrite relation'''<ref>Bezem et al, p. 7</ref> or just '''reduction'''.<ref name="Book and Otto, p. 10"/> This (entrenched) terminology using "reduction" is a little misleading, because the relation is not necessarily reducing some measure of the objects; this will become more apparent when we discuss string rewriting systems further in this article.
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| '''Example 1'''. Suppose the set of objects is ''T'' = {''a'', ''b'', ''c''} and the binary relation is given by the rules ''a'' → ''b'', ''b'' → ''a'', ''a'' → ''c'', and ''b'' → ''c''. Observe that these rules can be applied to both ''a'' and ''b'' in any fashion to get the term ''c''. Such a property is clearly an important one. Note also, that ''c'' is, in a sense, a "simplest" term in the system, since nothing can be applied to ''c'' to transform it any further. This example leads us to define some important notions in the general setting of an ARS. First we need some basic notions and notations.<ref>Baader and Nipkow, pp. 8-9</ref>
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| * <math>\stackrel{*}{\rightarrow}</math> is the [[transitive closure]] of <math>\rightarrow \cup =</math>, where = is the [[identity relation]], i.e. <math>\stackrel{*}{\rightarrow}</math> is the smallest [[preorder]] ([[reflexive relation|reflexive]] and [[transitive relation|transitive]] relation) containing <math>\rightarrow</math>. It is also called the [[reflexive transitive closure]] of <math>\rightarrow</math>.
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| * <math>\leftrightarrow</math> is <math>\rightarrow \cup \rightarrow^{-1}</math>, that is the union of the relation → with its [[inverse relation]], also known as the [[symmetric closure]] of <math>\rightarrow</math>.
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| * <math>\stackrel{*}{\leftrightarrow}</math> is the [[transitive closure]] of <math>\leftrightarrow \cup =</math>, that is <math>\stackrel{*}{\leftrightarrow}</math> is the smallest [[equivalence relation]] containing <math>\rightarrow</math>. It is also known as the [[reflexive transitive symmetric closure]] of <math>\rightarrow</math>.
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| === Normal forms, joinability and the word problem ===
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| An object ''x'' in ''A'' is called '''reducible''' if there exists some other ''y'' in ''A'' such that <math>x \rightarrow y</math>; otherwise it is called '''irreducible''' or a '''normal form'''. An object ''y'' is called a normal form of ''x'' if <math>x \stackrel{*}{\rightarrow} y</math>, and ''y'' is irreducible. If ''x'' has a ''unique'' normal form, then this is usually denoted with <math>x\downarrow</math>. In example 1 above, ''c'' is a normal form, and <math>c = a\downarrow = b\downarrow</math>. If every object has at least one normal form, the ARS is called '''normalizing'''.
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| A related, but weaker notion than the existence of normal forms is that of two objects being '''joinable''': ''x'' and ''y'' are said to be joinable if there exists some ''z'' with the property that <math>x \stackrel{*}{\rightarrow} z \stackrel{*}{\leftarrow} y</math>. From this definition, it's apparent one may define the joinability relation as <math>\stackrel{*}{\rightarrow} \circ \stackrel{*}{\leftarrow}</math>, where <math>\circ</math> is the [[composition of relations]]. Joinability is usually denoted, somewhat confusingly, also with <math>\downarrow</math>, but in this notation the down arrow is a binary relation, i.e. we write <math>x\mathbin\downarrow y</math> if ''x'' and ''y'' are joinable.
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| One of the important problems that may be formulated in an ARS is the '''word problem''': given ''x'' and ''y'' are they equivalent under <math>\stackrel{*}{\leftrightarrow}</math>? This is a very general setting for formulating the [[Word problem (mathematics)|word problem for the presentation of an algebraic structure]]. For instance, the [[word problem for groups]] is a particular case of an ARS word problem. Central to an "easy" solution for the word problem is the existence of unique normal forms: in this case if two objects have the same normal form, then they are equivalent under <math>\stackrel{*}{\leftrightarrow}</math>. The word problem for an ARS is [[undecidable problem|undecidable]] in general.
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| === The Church-Rosser property and confluence ===
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| {{main|Confluence (abstract rewriting)}}
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| An ARS is said to possess the '''Church-Rosser property''' if and only if <math>x\stackrel{*}{\leftrightarrow}y</math> implies <math>x\mathbin\downarrow y</math>. In words, the Church-Rosser property means that any two equivalent objects are joinable. [[Alonzo Church]] and [[J. Barkley Rosser]] proved in 1936 that [[lambda calculus]] has this property;<ref>Alonzo Church and J. Barkley Rosser. Some properties of conversion. Trans.
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| AMS, 39:472-482, 1936</ref> hence the name of the property.<ref>Baader and Nipkow, p. 9</ref> (The fact that lambda calculus has this property is also known as the [[Church-Rosser theorem]].) In an ARS with the Church-Rosser property the word problem may be reduced to the search for a common successor. In a Church-Rosser system, an object has ''at most one'' normal form; that is the normal form of an object is unique if it exists, but it may well not exist.
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| Several different properties are equivalent to the Church-Rosser property, but may be simpler to check in some particular setting. In particular, ''confluence'' is equivalent to Church-Rosser. <!---seemingly wrong, as the confluence def. quatifies over all objs: The notion of confluence can be defined for individual elements, something that's not possible for Church-Rosser.--> An ARS <math>(A,\rightarrow)</math> is said:
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| * '''confluent''' if for all ''w'', ''x'', and ''y'' in ''A'', <math>x \stackrel{*}{\leftarrow} w \stackrel{*}{\rightarrow} y</math> implies <math>x\mathbin\downarrow y</math>. Roughly speaking, confluence says that no matter how two paths diverge from a common ancestor (''w''), the paths are joining at ''some'' common successor. This notion may be refined as property of a particular object ''w'', and the system called confluent if all its elements are confluent.
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| * '''locally confluent''' if for all ''w'', ''x'', and ''y'' in ''A'', <math>x \leftarrow w \rightarrow y</math> implies <math>x\mathbin\downarrow y</math>. This property is sometimes called '''weak confluence'''.
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| '''Theorem.''' For an ARS the following conditions are equivalent: (i) it has the Church-Rosser property, (ii) it is confluent.<ref>Baader and Nipkow, p. 11</ref>
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| '''Corollary'''.<ref>Baader and Nipkow, p. 12</ref> In a confluent ARS if <math>x \stackrel{*}{\leftrightarrow} y</math> then
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| * If both ''x'' and ''y'' are normal forms, then ''x'' = ''y''.
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| * If ''y'' is a normal form, then <math>x \stackrel{*}{\rightarrow} y</math>
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| Because of these equivalences, a fair bit of variation in definitions is encountered in the literature. For instance, in Bezem ''et al.'' 2003 the Church-Rosser property and confluence are defined to be synonymous and identical to the definition of confluence presented here; Church-Rosser as defined here remains unnamed, but is given as an equivalent property; this departure from other texts is deliberate.<ref>Bezem et al, p.11</ref> Because of the above corollary, in a confluent ARS one may define a normal form ''y'' of ''x'' as an irreducible ''y'' with the property that <math>x \stackrel{*}{\leftrightarrow} y</math>. This definition, found in Book and Otto, is equivalent to common one given here in a confluent system, but it is more inclusive <ref group=note>i.e. it considers more objects as a normal for of ''x'' than our definition</ref> more in a non-confluent ARS.
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| Local confluence on the other hand is not equivalent with the other notions of confluence given in this section, but it is strictly weaker than confluence.
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| The relation <math>a \rightarrow b, \; b \rightarrow a, \; a \rightarrow c,\; b \rightarrow d</math> is locally confluent, but not confluent, as <math>c</math> and <math>d</math> are equivalent, but not joinable.<ref>{{cite journal|author=M.H.A. Neumann|title=On Theories with a Combinatorial Definition of ``Equivalence''|journal=Annals of Mathematics|volume=42|number=2|pages=223–243|year=1942}}</ref>
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| === Termination and convergence ===
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| An abstract rewriting system is said to be '''terminating''' or '''noetherian''' if there is no infinite chain <math>x_0 \rightarrow x_1 \rightarrow x_2 \rightarrow \cdots</math>. In a terminating ARS, every object has at least one normal form, thus it is normalizing. The converse is not true. In example 1 for instance, there is an infinite rewriting chain, namely <math>a \rightarrow b \rightarrow a \rightarrow b \rightarrow \cdots</math>, even though the system is normalizing. A confluent and terminating ARS is called '''convergent'''. In a convergent ARS, every object has a unique normal form. <!---hard to understand (what is sufficient for what?): But it is sufficient for the system to be confluent and normalizing for a unique normal to exist for every element, as seen in example 1.-->
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| '''Theorem''' ([[Newman's Lemma]]): A terminating ARS is confluent if and only if it is locally confluent.
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| == String rewriting systems ==
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| {{Main|String rewriting system}}
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| A '''string rewriting system''' (SRS), also known as '''semi-Thue system''', exploits the [[free monoid]] structure of the [[String (computer science)|strings]] (words) over an [[Alphabet (computer science)|alphabet]] to extend a rewriting relation, ''R'' to ''all'' strings in the alphabet that contain left- and respectively right-hand sides of some rules as [[substring]]s. Formally a semi-Thue systems is a [[tuple]] <math>(\Sigma, R)</math> where <math>\Sigma</math> is a (usually finite) alphabet, and ''R'' is a binary relation between some (fixed) strings in the alphabet, called '''rewrite rules'''. The '''one-step rewriting relation''' relation <math>\rightarrow_R</math> induced by ''R'' on <math>\Sigma^*</math> is defined as: for any strings ''s'', and ''t'' in <math>\Sigma^*</math> <math>s \rightarrow_R t</math> if and only if there exist ''x'', ''y'', ''u'', ''v'' in <math>\Sigma^*</math> such that ''s'' = ''xuy'', ''t'' = ''xvy'', and ''u'' ''R'' ''v''. Since <math>\rightarrow_R</math> is a relation on <math>\Sigma^*</math>, the pair <math>(\Sigma^*, \rightarrow_R)</math> fits the definition of an abstract rewriting system. Obviously ''R'' is subset of <math>\rightarrow_R</math>. If the relation <math>R</math> is [[symmetric relation|symmetric]], then the system is called a '''Thue system'''.
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| In a SRS, the '''reduction relation''' <math>\stackrel{*}{\rightarrow}_R</math> is compatible with the monoid operation, meaning that <math>x\stackrel{*}{\rightarrow}_R y</math> implies <math>uxv\stackrel{*}{\rightarrow}_R uyv</math> for all strings x, y, u, v in <math>\Sigma^*</math>. Similarly, the reflexive transitive symmetric closure of <math>\rightarrow_R</math>, denoted <math>\stackrel{*}{\leftrightarrow}_R</math>, is a [[Congruence relation|congruence]], meaning it is an [[equivalence relation]] (by definition) and it is also compatible with string concatenation. The relation <math>\stackrel{*}{\leftrightarrow}_R</math> is called the '''Thue congruence''' generated by ''R''. In a Thue system, i.e. if R is symmetric, the rewrite relation <math>\stackrel{*}{\rightarrow}_R</math> coincides with the Thue congruence <math>\stackrel{*}{\leftrightarrow}_R</math>.
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| The notion of a semi-Thue system essentially coincides with the [[presentation of a monoid]]. Since <math>\stackrel{*}{\leftrightarrow}_R</math> is a congruence, we can define the '''factor monoid''' <math>\mathcal{M}_R = \Sigma^*/\stackrel{*}{\leftrightarrow}_R</math> of the free monoid <math>\Sigma^*</math> by the Thue congruence in the [[factor monoid|usual manner]]. If a monoid <math>\mathcal{M}</math> is [[isomorphic]] with <math>\mathcal{M}_R</math>, then the semi-Thue system <math>(\Sigma, R)</math> is called a [[monoid presentation]] of <math>\mathcal{M}</math>.
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| We immediately get some very useful connections with other areas of algebra. For example, the alphabet {''a'', ''b''} with the rules { ''ab'' → ε, ''ba'' → ε }, where ε is the [[empty string]], is a presentation of the [[free group]] on one generator. If instead the rules are just { ''ab'' → ε }, then we obtain a presentation of the [[bicyclic monoid]]. Thus semi-Thue systems constitute a natural framework for solving the [[word problem (mathematics)|word problem]] for monoids and groups. In fact, every monoid has a presentation of the form <math>(\Sigma, R)</math>, i.e. it may be always be presented by a semi-Thue system, possibly over an infinite alphabet.
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| The word problem for a semi-Thue system is undecidable in general; this result is sometimes known as the '''Post-Markov theorem'''.<ref>Martin Davis et al. 1994, p. 178</ref>
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| == Term rewriting systems ==
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| {{Expand section|date=October 2009}}
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| [[File:Triangle diagram of rewrite rule application svg.svg|thumb|'''Pic.1:''' Schematic triangle diagram of application of a rewrite rule <math>l \longrightarrow r</math> at position <math>p</math> in a term, with matching substitution <math>\sigma</math>]]
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| [[File:Example term for position, path, depth, match svg.svg|thumb|'''Pic.2:''' Rule lhs term <math>x*(y*z)</math> matching in term <math>\frac{a*((a+1)*(a+2))}{1*(2*3)}</math>]]
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| A '''term rewriting system''' (TRS) is a rewriting system where the objects are ''[[term (logic)|terms]]'', or expressions with nested sub-expressions. For example, the system shown under Logic above is a term rewriting system. The terms in this system are composed of binary operators <math>(\vee)</math> and <math>(\wedge)</math> and the unary operator <math>(\neg)</math>. Also present in the rules are ''variables'',
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| <!---nonsense: which are part of the rules themselves rather than the term;--->
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| these each represent any possible term (though a single variable always represents the same term throughout a single rule).
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| <!---nonsense: The term structure in such a system is usually presented using a [[Formal grammar|grammar]].--->
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| In contrast to string rewriting systems, whose objects are flat sequences of symbols, the objects a term rewriting system works on, i.e. the terms, form a [[term algebra]]. A term can be visualized as a tree of symbols, the set of admitted symbols being fixed by a given [[signature (logic)|signature]].
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| ===Formal definition===
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| A '''term rewriting rule''' is a pair of terms, commonly written as <math>l \longrightarrow r</math>, to indicate that the left hand side <math>l</math> can be replaced by the right hand side <math>r</math>. A '''term rewriting system''' is a set <math>R</math> of such rules. A rule <math>l \longrightarrow r</math> can be '''applied''' to a term <math>s</math> if the left term <math>l</math> [[Pattern matching|matches]] some [[Term_(logic)#Operations_with_terms|subterm]] of <math>s</math>, that is, if <math>s \mid_p = l \sigma</math>
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| <ref group=note>here, <math>s \mid_p</math> denotes the subterm of <math>s</math> rooted at position <math>p</math>,
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| while <math>l \sigma</math> denotes the result of applying the [[substitution (logic)|substitution]] <math>\sigma</math> to the term <math>l</math></ref>
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| for some [[Term_(logic)#Operations_with_terms|position]] <math>p</math> in <math>s</math> and some [[substitution (logic)|substitution]] <math>\sigma</math>. The result term <math>t</math> of this rule application is then obtained as <math>t = s[r \sigma]_p</math>;
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| <ref group=note>here, <math>s[r \sigma]_p</math> denotes the result of [[Term_(logic)#Operations_with_terms|replacing the subterm]] at position <math>p</math> in <math>s</math> by the term <math>r \sigma</math></ref>
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| see picture 1. In this case, <math>s</math> is said to be '''rewritten in one step''', or '''rewritten directly''', to <math>t</math> by the system <math>R</math>, formally denoted as <math>s \longrightarrow_R t</math>, or as <math>s \stackrel{R}{\longrightarrow} t</math> by some authors.
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| If a term <math>t_1</math> can be rewritten in several steps into a term <math>t_n</math>, that is, if <math>t_1 \longrightarrow_R t_2 \longrightarrow_R \ldots \longrightarrow_R t_n</math>, the term <math>t_1</math> is said to be '''rewritten''' to <math>t_n</math>, formally denoted as <math>t_1 \longrightarrow_R^+ t_n</math>.
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| In other words, the relation <math>\longrightarrow_R^+</math> is the [[transitive closure]] of the relation <math>\longrightarrow_R</math>;
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| often, also the notation <math>\longrightarrow_R^*</math> is used to denote the [[Closure_(mathematics)#P_closures_of_binary_relations|reflexive-transitive closure]] of <math>\longrightarrow_R</math>, that is,
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| <math>s \longrightarrow_R^* t</math> if <math>s = t</math> or <math>s \longrightarrow_R^+ t</math>.
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| <ref>{{cite book| author=N. Dershowitz, J.-P. Jouannaud| title=Rewrite Systems| year=1990| volume=B| pages=243–320| publisher=Elsevier| editor=Jan van Leeuwen| series=Handbook of Theoretical Computer Science}}; here: Sect.2.3</ref>
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| A term rewriting given by a set <math>R</math> of rules can be viewed as an abstract rewriting system as defined [[#Abstract rewriting systems|above]], with terms as its objects and <math>\longrightarrow_R</math> as its rewrite relation.
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| For example, <math>x*(y*z) \longrightarrow (x*y)*z</math> is a rewrite rule, commonly used to establish a normal form with respect to the associativity of <math>*</math>.
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| That rule can be applied at the numerator in the term <math>\frac{a*((a+1)*(a+2))}{1*(2*3)}</math> with the matching substitution <math>\{ x \mapsto a, \; y \mapsto a+1, \; z \mapsto a+2 \}</math>, see picture 2.
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| <ref group=note>since applying that substitution to the rule's left hand side <math>x*(y*z)</math> yields the numerator <math>a*((a+1)*(a+2))</math></ref>
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| Applying that substitution to the rule's right hand side yields the term <math>(a*(a+1))*(a+2)</math>, and replacing the numerator by that term yields <math>\frac{(a*(a+1))*(a+2)}{1*(2*3)}</math>, which is the result term of applying the rewrite rule.
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| Altogether, applying the rewrite rule has achieved what is called "''applying the associativity law for <math>*</math> to <math>\frac{a*((a+1)*(a+2))}{1*(2*3)}</math>''" in elementary algebra.
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| Alternatively, the rule could have been applied again to the denominator of the original term, yielding <math>\frac{a*((a+1)*(a+2))}{(1*2)*3}</math>.
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| ===Termination===
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| Beyond section [[#Termination and convergence]], additional subtleties are to be considered for ''term'' rewriting systems.
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| Termination even of a system consisting of one rule with a [[Term_(logic)#Ground_and_linear_terms|linear]] left hand side is undecidable.<ref>{{cite book| author=M. Dauchet| chapter=Simulation of Turing Machines by a Left-Linear Rewrite Rule| title=Proc. 3rd RTA| year=1989|volume=355| pages=109–120| publisher=Springer LNCS|series=LNCS}}</ref>
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| Termination is also undecidable for systems using only unary function symbols; however, it is decidable for finite [[Term_(logic)#Ground_and_linear_terms|ground]] systems.
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| <ref>{{cite techreport| author=Gerard Huet, D.S. Lankford| title=On the Uniform Halting Problem for Term Rewriting Systems|date=Mar 1978| number=283| pages=8| institution=IRIA| url=http://www.ens-lyon.fr/LIP/REWRITING/TERMINATION/Huet_Lankford.pdf| accessdate=16 June 2013}}</ref>
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| The following examples of terminating term rewrite systems are due to Toyama:<ref>{{cite journal| author=Y. Toyama| title=Counterexamples to Termination for the Direct Sum of Term Rewriting Systems| journal=Inform. Process. Lett.| year=1987|volume=25| number=| pages=141–143| url=http://www.nue.riec.tohoku.ac.jp/user/toyama/research/paper/journal/counterexamples.pdf}}</ref> <math>f(0,1,x) \rightarrow f(x,x,x)</math> and <math>g(x,y) \rightarrow x, \; g(x,y) \rightarrow y</math>. Their union is a non-terminating system, since <math>f(g(0,1),g(0,1),g(0,1)) \rightarrow f(0,g(0,1),g(0,1)) \rightarrow f(0,1,g(0,1)) \rightarrow f(g(0,1),g(0,1),g(0,1)) \rightarrow \ldots</math>. This result disproves a conjecture of Dershowitz,<ref>{{cite book| author=N. Dershowitz| chapter=Termination| title=Proc. RTA| year=1985| volume=220| pages=180–224| publisher=Springer| editor=Jean-Pierre Jouannaud| series=LNCS| url=http://www.cs.tau.ac.il/~nachum/papers/LNCS/Termination.pdf}}; here: p.210</ref> who claimed that the union of two terminating term rewrite systems <math>R_1</math> and <math>R_2</math> is again terminating if all left hand sides of <math>R_1</math> and right hand sides of <math>R_2</math> are [[Term_(logic)#Ground_and_linear_terms|linear]], and there are no "''overlaps''" between left hand sides of <math>R_1</math> and right hand sides of <math>R_2</math>. All these properties are satisfied by Toyama's examples.
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| === Graph rewriting systems ===
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| A generalization of term rewrite systems are [[graph rewriting|graph rewrite systems]], operating on [[graph (graph theory)|graphs]] instead of ([[ground term|ground]]-) [[term (mathematics)|terms]] / their corresponding [[tree (graph theory)|tree]] representation.
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| == Trace rewriting systems ==
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| [[Trace theory]] provides a means for discussing multiprocessing in more formal terms, such as via the [[trace monoid]] and the [[history monoid]]. Rewriting can be performed in trace systems as well.
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| {{Expand section|date=October 2009}}
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| == Philosophy ==
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| Rewriting systems can be seen as programs that infer end-effects from a list of cause-effect relationships. In this way, rewriting systems can be considered to be automated [[causality]] provers.
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| ==See also==
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| * [[Rewrite rule]]<ref group=note>About the "rewrite rule" notion in [[linguistics]], corresponding to a [[production rule (formal languages)|production rule]] of a [[context-free grammar]]. The [[Formal grammar#The semantics of grammars|derivation relation]] of such a grammar constitutes an abstract rewriting system in the [[#Abstract rewriting systems|above]] sense.</ref>
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| * [[Critical pair (logic)]]
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| * [[Knuth-Bendix completion algorithm]]
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| * [[L-system]]s specify rewriting that is done in parallel.
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| * [[Regulated rewriting]]
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| * [[Rho calculus]]
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| == Notes ==
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| {{reflist|group=note}}
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| == References ==
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| {{reflist|3}}
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| == Further reading ==
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| *{{cite book|first1=Franz|last1=Baader|authorlink1=Franz Baader|first2=Tobias|last2=Nipkow|authorlink2=Tobias Nipkow|title=Term rewriting and all that|publisher=Cambridge University Press |isbn=0-521-77920-0|year=1999
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| |url=http://books.google.com/books?id=N7BvXVUCQk8C&dq}} 316 pages. A textbook suitable for undergraduates.
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| * [[Marc Bezem]], [[Jan Willem Klop]], [[Roel de Vrijer]] ("Terese"), ''[http://books.google.com/books?id=7QQ5u-4tRUkC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Term Rewriting Systems]'', Cambridge University Press, 2003, ISBN 0-521-39115-6. This is the most recent comprehensive monograph. It uses however a fair deal of non-yet-standard notations and definitions. For instance the Church-Rosser property is defined to be identical with confluence.
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| * [[Ronald V. Book]] and Friedrich Otto, ''String-Rewriting Systems'', Springer (1993).
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| * Nachum Dershowitz and Jean-Pierre Jouannaud [http://citeseer.ist.psu.edu/dershowitz90rewrite.html ''Rewrite Systems''], Chapter 6 in [[Jan van Leeuwen]] (Ed.), ''Handbook of Theoretical Computer Science, Volume B: Formal Models and Semantics.'', Elsevier and MIT Press, 1990, ISBN 0-444-88074-7, pp. 243–320. The [[preprint]] of this chapter is freely available from the authors, but it misses the figures.
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| * [[Martin Davis]], [[Ron Sigal]], [[Elaine J. Weyuker]], (1994) ''Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science – 2nd edition'', Academic Press, ISBN 0-12-206382-1.
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| * Sinyor, J.; [http://downloads.hindawi.com/journals/ijmms/2010/458563.pdf "The 3x+1 Problem as a String Rewriting System"], ''International Journal of Mathematics and Mathematical Sciences'', Volume 2010 (2010), Article ID 458563, 6 pages.
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| == External links ==
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| {{Wiktionary}}
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| * The [http://rewriting.loria.fr/ Rewriting Home Page]
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| * [http://verify.rwth-aachen.de/IFIP-WG1.6/ IFIP Working Group 1.6]
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| * [http://cl-informatik.uibk.ac.at/~ami/research/rr/index.php Researchers in rewriting] by [[Aart Middeldorp]], [[University of Innsbruck]]
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| * [http://www.termination-portal.org/ Termination Portal]
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| [[Category:Formal languages]]
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| [[Category:Logic in computer science]]
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| [[Category:Mathematical logic]]
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| [[Category:Rewriting systems]]
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