, computer science
, and logic
, rewriting covers a wide range of (potentially non-deterministic
) methods of replacing subterms of a formula
with other terms. The objects of focus for this article include rewriting systems (also known as rewrite systems, rewrite engines
or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects.
Rewriting can be 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 theorem prover
s and declarative programming language
s are based on term rewriting.
, the procedure for obtaining the conjunctive normal form
(CNF) of a formula can be implemented as a rewriting system.
The rules of an example of such a system would be:
(double negation elimination
(De Morgan's laws
[This variant of the previous rule is needed since the commutative law ''A''∨''B'' = ''B''∨''A'' cannot be turned into a rewrite rule. A rule like ''A''∨''B'' → ''B''∨''A'' would cause the rewrite system to be nonterminating.]
where the symbol (
) indicates that an expression matching the left hand side of the rule can be rewritten to one formed by the right hand side, and the symbols each denote a subexpression. In such a system, each rule is chosen so that the left side is equivalent
to the right side, and consequently when the left side matches a subexpression, performing a rewrite of that subexpression from left to right maintains logical consistency and value of the entire expression.
Term rewriting systems can be employed to compute arithmetic operations on natural number
To this end, each such number has to be encoded as a term
The simplest encoding is the one used in the Peano axioms
, based on the constant 0 (zero) and the successor function
for example, the numbers 0, 1, 2, and 3 are represented by the terms 0, S(0), S(S(0)), and S(S(S(0))), respectively.
The following term rewriting system can then be used to compute sum and product of given natural numbers.
For example, the computation of 2+2 to result in 4 can be duplicated by term rewriting as follows:
where the rule numbers are given above the ''rewrites-to'' arrow.
As another example, the computation of 2⋅2 looks like:
where the last step comprises the previous example computation.
, rewrite rules, also called phrase structure rule
s, are used in some systems of generative grammar
as a means of generating the grammatically correct sentences of a language. Such a rule typically takes the form A → X, where A is a syntactic category
label, such as noun phrase
, and X is a sequence of such labels or morpheme
s, expressing the fact that A can be replaced by X in generating the constituent structure of a sentence. For example, the rule S → NP VP means that a sentence can consist of a noun phrase followed by a verb phrase
; further rules will specify what sub-constituents a noun phrase and a verb phrase can consist of, and so on.
Abstract rewriting systems
From the above examples, it is 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. (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).
[Book and Otto, p. 10]
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 or just reduction.
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.
Example 1. Suppose the set of objects is ''T'' = 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. In a sense, ''c'' is 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.
is the transitive closure
, where = is the identity relation
is the smallest preorder
. It is also called the reflexive transitive closure
, that is the union of the relation → with its converse relation
, also known as the symmetric closure
is the transitive closure
, that is
is the smallest equivalence relation
. It is also known as the reflexive transitive symmetric closure
Normal forms, joinability and the word problem
An object ''x'' in ''A'' is called reducible if there exists some other ''y'' in ''A'' such that
; otherwise it is called irreducible or a normal form. An object ''y'' is called a normal form of ''x'' if
, and ''y'' is irreducible. If ''x'' has a ''unique'' normal form, then this is usually denoted with
. In example 1 above, ''c'' is a normal form, and
. If every object has at least one normal form, the ARS is called normalizing.
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
. From this definition, it is apparent that one may define the joinability relation as
is the composition of relations
. Joinability is usually denoted, somewhat confusingly, also with
, but in this notation the down arrow is a binary relation, i.e. we write
if ''x'' and ''y'' are joinable.
One of the important problems that may be formulated in an ARS is the word problem: given ''x'' and ''y'', are they equivalent under
? This is a very general setting for formulating the 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
. The word problem for an ARS is undecidable
The Church–Rosser property and confluence
An ARS is said to possess the Church–Rosser property if
. 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; hence the name of the property. (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.
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. An ARS
* confluent if for all ''w'', ''x'', and ''y'' in ''A'',
. 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.
* locally confluent if for all ''w'', ''x'', and ''y'' in ''A'',
. This property is sometimes called weak confluence.
Theorem. For an ARS the following conditions are equivalent: (i) it has the Church–Rosser property, (ii) it is confluent.
Corollary. In a confluent ARS if
* If both ''x'' and ''y'' are normal forms, then .
* If ''y'' is a normal form, then
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. 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
. This definition, found in Book and Otto, is equivalent to common one given here in a confluent system, but it is more inclusive
[i.e. it considers more objects as a normal form of ''x'' than our definition]
more in a non-confluent ARS.
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.
is locally confluent, but not confluent, as
are equivalent, but not joinable.
Termination and convergence
An abstract rewriting system is said to be terminating or noetherian if there is no infinite chain
. 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
, 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.
Theorem (Newman's Lemma
): A terminating ARS is confluent if and only if it is locally confluent.
String rewriting systems
A string rewriting system (SRS), also known as semi-Thue system, exploits the free monoid
structure of the strings
(words) over an alphabet
to extend a rewriting relation,
, to ''all'' strings in the alphabet that contain left- and respectively right-hand sides of some rules as substring
s. Formally a semi-Thue system is a tuple
is a (usually finite) alphabet, and
is a binary relation between some (fixed) strings in the alphabet, called rewrite rules. The one-step rewriting relation relation