Rewriting

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 or abstract rewriting system (abbreviated ARS). An ARS is simply a set A of objects, together with a binary relation → on A called the reduction relation, rewrite relation or just reduction.

Many notions and notations can be defined in the general setting of an ARS. ${\displaystyle \stackrel{\ast }{\to }}$ is the reflexive transitive closure of ${\displaystyle \to }$. ${\displaystyle \leftrightarrow }$ is the symmetric closure of ${\displaystyle \to }$. ${\displaystyle \stackrel{\ast }{\leftrightarrow }}$ is the reflexive transitive symmetric closure of ${\displaystyle \to }$. The word problem for an ARS is determining, given x and y, whether ${\displaystyle x\stackrel{\ast }{\leftrightarrow }y}$. An object x in A is called reducible if there exists some other y in A such that ${\displaystyle x\to y}$; otherwise it is called irreducible or a normal form. An object y is called a "normal form of x" if ${\displaystyle x\stackrel{\ast }{\to }y}$, and y is irreducible. If the normal form of x is unique, then this is usually denoted with ${\displaystyle x\downarrow }$. If every object has at least one normal form, the ARS is called normalizing. ${\displaystyle x\downarrow y}$ or x and y are said to be joinable if there exists some z with the property that ${\displaystyle x\stackrel{\ast }{\to }z\stackrel{\ast }{\leftarrow }y}$. An ARS is said to possess the Church–Rosser property if ${\displaystyle x\stackrel{\ast }{\leftrightarrow }y}$ implies ${\displaystyle x\downarrow y}$. An ARS is confluent if for all w, x, and y in A, ${\displaystyle x\stackrel{\ast }{\leftarrow }w\stackrel{\ast }{\to }y}$ implies ${\displaystyle x\downarrow y}$. An ARS is locally confluent if and only if for all w, x, and y in A, ${\displaystyle x\leftarrow w\to y}$ implies ${\displaystyle x\downarrow y}$. An ARS is said to be terminating or noetherian if there is no infinite chain ${\displaystyle x_0\to x_1\to x_2\to \cdots }$. A confluent and terminating ARS is called convergent or canonical.

Important theorems for abstract rewriting systems are that an ARS is confluent iff it has the Church–Rosser property, Newman's lemma (a terminating ARS is confluent if and only if it is locally confluent), and that the word problem for an ARS is undecidable in general.

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, ${\displaystyle R}$, to all strings in the alphabet that contain left- and respectively right-hand sides of some rules as substrings. Formally a semi-Thue system is a tuple ${\displaystyle (\mathrm{\Sigma },R)}$ where ${\displaystyle \mathrm{\Sigma }}$ is a (usually finite) alphabet, and ${\displaystyle R}$ is a binary relation between some (fixed) strings in the alphabet, called the set of rewrite rules. The one-step rewriting relation ${\displaystyle \underset{R}{\to }}$ induced by ${\displaystyle R}$ on ${\displaystyle {\mathrm{\Sigma }}^{\ast }}$ is defined as: if ${\displaystyle s,t\in {\mathrm{\Sigma }}^{\ast }}$ are any strings, then ${\displaystyle s\underset{R}{\to }t}$ if there exist ${\displaystyle x,y,u,v\in {\mathrm{\Sigma }}^{\ast }}$ such that ${\displaystyle s=xuy}$, ${\displaystyle t=xvy}$, and ${\displaystyle uRv}$. Since ${\displaystyle \underset{R}{\to }}$ is a relation on ${\displaystyle {\mathrm{\Sigma }}^{\ast }}$, the pair ${\displaystyle ({\mathrm{\Sigma }}^{\ast },\underset{R}{\to })}$ fits the definition of an abstract rewriting system. Since the empty string is in ${\displaystyle {\mathrm{\Sigma }}^{\ast }}$, ${\displaystyle R}$ is a subset of ${\displaystyle \underset{R}{\to }}$. If the relation ${\displaystyle R}$ is symmetric, then the system is called a Thue system.

In a SRS, the reduction relation ${\displaystyle \stackrel{\ast }{\underset{R}{\to }}}$ is compatible with the monoid operation, meaning that ${\displaystyle x\stackrel{\ast }{\underset{R}{\to }}y}$ implies ${\displaystyle uxv\stackrel{\ast }{\underset{R}{\to }}uyv}$ for all strings ${\displaystyle x,y,u,v\in {\mathrm{\Sigma }}^{\ast }}$. Similarly, the reflexive transitive symmetric closure of ${\displaystyle \underset{R}{\to }}$, denoted ${\displaystyle \stackrel{\ast }{\underset{R}{\leftrightarrow }}}$, is a congruence, meaning it is an equivalence relation (by definition) and it is also compatible with string concatenation. The relation ${\displaystyle \stackrel{\ast }{\underset{R}{\leftrightarrow }}}$ is called the Thue congruence generated by ${\displaystyle R}$. In a Thue system, i.e. if ${\displaystyle R}$ is symmetric, the rewrite relation ${\displaystyle \stackrel{\ast }{\underset{R}{\to }}}$ coincides with the Thue congruence ${\displaystyle \stackrel{\ast }{\underset{R}{\leftrightarrow }}}$.

The notion of a semi-Thue system essentially coincides with the presentation of a monoid. Since ${\displaystyle \stackrel{\ast }{\underset{R}{\leftrightarrow }}}$ is a congruence, we can define the factor monoid ${\displaystyle {\mathcal{M}}_R={\mathrm{\Sigma }}^{\ast }/\stackrel{\ast }{\underset{R}{\leftrightarrow }}}$ of the free monoid ${\displaystyle {\mathrm{\Sigma }}^{\ast }}$ by the Thue congruence. If a monoid ${\displaystyle \mathcal{M}}$ is isomorphic with ${\displaystyle {\mathcal{M}}_R}$, then the semi-Thue system ${\displaystyle (\mathrm{\Sigma },R)}$ is called a monoid presentation of ${\displaystyle \mathcal{M}}$.

We immediately get some very useful connections with other areas of algebra. For example, the alphabet ${\displaystyle \{a,b\}}$ with the rules ${\displaystyle \{ab\to \epsilon ,ba\to \epsilon \}}$, where ${\displaystyle \epsilon }$ is the empty string, is a presentation of the free group on one generator. If instead the rules are just ${\displaystyle \{ab\to \epsilon \}}$, then we obtain a presentation of the bicyclic monoid. Thus semi-Thue systems constitute a natural framework for solving the word problem for monoids and groups. In fact, every monoid has a presentation of the form ${\displaystyle (\mathrm{\Sigma },R)}$, i.e. it may always be presented by a semi-Thue system, possibly over an infinite alphabet.

The word problem for a semi-Thue system is undecidable in general; this result is sometimes known as the Post–Markov theorem.

Term rewriting systems

A term rewriting system (TRS) is a rewriting system whose objects are terms, which are 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 ${\displaystyle (\vee )}$ and ${\displaystyle (\wedge )}$ and the unary operator ${\displaystyle (\mathrm{\neg })}$. Also present in the rules are variables, which represent any possible term (though a single variable always represents the same term throughout a single rule).

In contrast to string rewriting systems, whose objects are sequences of symbols, the objects of a term rewriting system 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. As a formalism, term rewriting systems have the full power of Turing machines, that is, every computable function can be defined by a term rewriting system.

Some programming languages are based on term rewriting. One such example is Pure, a functional programming language for mathematical applications.

Formal definition

A rewrite rule is a pair of terms, commonly written as ${\displaystyle l\to r}$, to indicate that the left-hand side l can be replaced by the right-hand side r. A term rewriting system is a set R of such rules. A rule ${\displaystyle l\to r}$ can be applied to a term s if the left term l matches some subterm of s, that is, if there is some substitution ${\displaystyle \sigma }$ such that the subterm of ${\displaystyle s}$ rooted at some position p is the result of applying the substitution ${\displaystyle \sigma }$ to the term l. The subterm matching the left hand side of the rule is called a redex or reducible expression. The result term t of this rule application is then the result of replacing the subterm at position p in s by the term ${\displaystyle r}$ with the substitution ${\displaystyle \sigma }$ applied, see picture 1. In this case, ${\displaystyle s}$ is said to be rewritten in one step, or rewritten directly, to ${\displaystyle t}$ by the system ${\displaystyle R}$, formally denoted as ${\displaystyle s{\to }_{R}t}$, ${\displaystyle s\underset{R}{\to }t}$, or as ${\displaystyle s\stackrel{R}{\to }t}$ by some authors.

If a term ${\displaystyle t_1}$ can be rewritten in several steps into a term ${\displaystyle t_n}$, that is, if ${\displaystyle t_1\underset{R}{\to }{t}_2\underset{R}{\to }\cdots \underset{R}{\to }{t}_n}$, the term ${\displaystyle t_1}$ is said to be rewritten to ${\displaystyle t_n}$, formally denoted as ${\displaystyle t_1\stackrel{+}{\underset{R}{\to }}{t}_n}$. In other words, the relation ${\displaystyle \stackrel{+}{\underset{R}{\to }}}$ is the transitive closure of the relation ${\displaystyle \underset{R}{\to }}$; often, also the notation ${\displaystyle \stackrel{\ast }{\underset{R}{\to }}}$ is used to denote the reflexive-transitive closure of ${\displaystyle \underset{R}{\to }}$, that is, ${\displaystyle s\stackrel{\ast }{\underset{R}{\to }}t}$ if ${\displaystyle s=t}$ or ${\displaystyle s\stackrel{+}{\underset{R}{\to }}t}$. A term rewriting given by a set ${\displaystyle R}$ of rules can be viewed as an abstract rewriting system as defined above, with terms as its objects and ${\displaystyle \underset{R}{\to }}$ as its rewrite relation.

For example, ${\displaystyle x\ast (y\ast z)\to (x\ast y)\ast z}$ is a rewrite rule, commonly used to establish a normal form with respect to the associativity of ${\displaystyle \ast }$. That rule can be applied at the numerator in the term ${\displaystyle \frac{a\ast ((a+1)\ast (a+2))}{1\ast (2\ast 3)}}$ with the matching substitution ${\displaystyle \{x\mapsto a,y\mapsto a+1,z\mapsto a+2\}}$, see picture 2. Applying that substitution to the rule's right-hand side yields the term ${\displaystyle (a\ast (a+1))\ast (a+2)}$, and replacing the numerator by that term yields ${\displaystyle \frac{(a\ast (a+1))\ast (a+2)}{1\ast (2\ast 3)}}$, which is the result term of applying the rewrite rule. Altogether, applying the rewrite rule has achieved what is called "applying the associativity law for ${\displaystyle \ast }$ to ${\displaystyle \frac{a\ast ((a+1)\ast (a+2))}{1\ast (2\ast 3)}}$" in elementary algebra. Alternately, the rule could have been applied to the denominator of the original term, yielding ${\displaystyle \frac{a\ast ((a+1)\ast (a+2))}{(1\ast 2)\ast 3}}$.

Termination

Termination issues of rewrite systems in general are handled in Abstract rewriting system#Termination and convergence. For term rewriting systems in particular, the following additional subtleties are to be considered.

Termination even of a system consisting of one rule with a linear left-hand side is undecidable. Termination is also undecidable for systems using only unary function symbols; however, it is decidable for finite ground systems.

The following term rewrite system is normalizing, but not terminating, and not confluent: $${\displaystyle \begin{array}{cc}f(x,x)& \to g(x),\\ f(x,g(x))& \to b,\\ h(c,x)& \to f(h(x,c),h(x,x)).\end{array}}$$

The following two examples of terminating term rewrite systems are due to Toyama:

${\displaystyle f(0,1,x)\to f(x,x,x)}$

and

${\displaystyle g(x,y)\to x,}$

${\displaystyle g(x,y)\to y.}$

Their union is a non-terminating system, since

$${\displaystyle \begin{array}{cc}& f(g(0,1),g(0,1),g(0,1))\\ \to & f(0,g(0,1),g(0,1))\\ \to & f(0,1,g(0,1))\\ \to & f(g(0,1),g(0,1),g(0,1))\\ \to & \cdots \end{array}}$$ This result disproves a conjecture of Dershowitz, who claimed that the union of two terminating term rewrite systems ${\displaystyle R_1}$ and ${\displaystyle R_2}$ is again terminating if all left-hand sides of ${\displaystyle R_1}$ and right-hand sides of ${\displaystyle R_2}$ are linear, and there are no "overlaps" between left-hand sides of ${\displaystyle R_1}$ and right-hand sides of ${\displaystyle R_2}$. All these properties are satisfied by Toyama's examples.

See Rewrite order and Path ordering (term rewriting) for ordering relations used in termination proofs for term rewriting systems.

Higher-order rewriting systems

Higher-order rewriting systems are a generalization of first-order term rewriting systems to lambda terms, allowing higher order functions and bound variables. Various results about first-order TRSs can be reformulated for HRSs as well.

Graph rewriting systems

Graph rewrite systems are another generalization of term rewrite systems, operating on graphs instead of (ground-) terms / their corresponding tree representation.

Trace rewriting systems

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.