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In
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
and
theoretical computer science Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Associati ...
, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviated ARS) is a formalism that captures the quintessential notion and properties of
rewriting In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduc ...
systems. In its simplest form, an ARS is simply a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
(of "objects") together with a
binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
, traditionally denoted with \rightarrow; this definition can be further refined if we index (label) subsets of the binary relation. Despite its simplicity, an ARS is sufficient to describe important properties of rewriting systems like normal forms, termination, and various notions of
confluence In geography, a confluence (also ''conflux'') occurs where two or more watercourses join to form a single channel (geography), channel. A confluence can occur in several configurations: at the point where a tributary joins a larger river (main ...
. Historically, there have been several formalizations of rewriting in an abstract setting, each with its idiosyncrasies. This is due in part to the fact that some notions are equivalent, see below in this article. The formalization that is most commonly encountered in monographs and textbooks, and which is generally followed here, is due to
Gérard Huet Gérard Pierre Huet (; born 7 July 1947) is a French computer scientist, linguist and mathematician. He is senior research director at INRIA and mostly known for his major and seminal contributions to type theory, programming language theory and ...
(1980).


Definition

An ''abstract reduction system'' (''ARS'') is the most general (unidimensional) notion about specifying a set of objects and rules that can be applied to transform them. More recently, authors use the term ''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.) An ARS is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''A'', whose elements are usually called objects, together with a
binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
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. In some contexts it may be beneficial to distinguish between some subsets of the rules, i.e. some subsets of the reduction relation →, e.g. the entire reduction relation may consist of
associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
and
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a p ...
rules. Consequently, some authors define the reduction relation → as the indexed union of some relations; for instance if = , the notation used is (A, →1, →2). As a mathematical object, an ARS is exactly the same as an unlabeled
state transition system In theoretical computer science, a transition system is a concept used in the study of computation. It is used to describe the potential behavior of discrete systems. It consists of states and transitions between states, which may be labeled wi ...
, and if the relation is considered as an indexed union, then an ARS is the same as a labeled state transition system with the indices being the labels. The focus of the study, and the terminology are different however. In a
state transition system In theoretical computer science, a transition system is a concept used in the study of computation. It is used to describe the potential behavior of discrete systems. It consists of states and transitions between states, which may be labeled wi ...
one is interested in interpreting the labels as actions, whereas in an ARS the focus is on how objects may be transformed (rewritten) into others.


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'' to get ''c''. Furthermore, nothing can be applied to ''c'' to transform it any further. Such a property is clearly an important one.


Basic notions

First define some basic notions and notations. * \stackrel is the
transitive closure In mathematics, the transitive closure of a homogeneous binary relation on a set (mathematics), set is the smallest Relation (mathematics), relation on that contains and is Transitive relation, transitive. For finite sets, "smallest" can be ...
of \rightarrow. * \stackrel is the
reflexive transitive closure In mathematics, a subset of a given set is closed under an operation on the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but ...
of \rightarrow, i.e. the
transitive closure In mathematics, the transitive closure of a homogeneous binary relation on a set (mathematics), set is the smallest Relation (mathematics), relation on that contains and is Transitive relation, transitive. For finite sets, "smallest" can be ...
of (\rightarrow) \cup (=), where = is the
identity relation In mathematics, a homogeneous relation (also called endorelation) on a set ''X'' is a binary relation between ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...
. Equivalently, \stackrel is the smallest
preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive relation, reflexive and Transitive relation, transitive. The name is meant to suggest that preorders are ''almost'' partial orders, ...
containing \rightarrow. * Similarly, \stackrel, and \stackrel are closures of , the converse relation of . * \leftrightarrow is the
symmetric closure In mathematics, the symmetric closure of a binary relation R on a set X is the smallest symmetric relation A symmetric relation is a type of binary relation. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: : \forall a, b \in ...
of \rightarrow, that is, the union of \rightarrow with . * \stackrel is the reflexive transitive symmetric closure of \rightarrow, i.e. the
transitive closure In mathematics, the transitive closure of a homogeneous binary relation on a set (mathematics), set is the smallest Relation (mathematics), relation on that contains and is Transitive relation, transitive. For finite sets, "smallest" can be ...
of (\leftrightarrow) \cup (=). Equivalently, \stackrel is the smallest
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
containing \rightarrow.


Normal forms

An object ''x'' in ''A'' is called ''reducible'' if there exist some other ''y'' in ''A'' and x \rightarrow y; otherwise it is called ''irreducible'' or a ''normal form''. An object ''y'' is called a normal form of ''x'' if x \stackrel y and ''y'' is irreducible. If ''x'' has a ''unique'' normal form, then this is usually denoted with x\downarrow. In example 1 above, ''c'' is a normal form, and c = a\downarrow = b\downarrow. If every object has at least one normal form, the ARS is called ''normalizing''.


Joinability

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 x \stackrel z \stackrel y. From this definition, it's apparent one may define the joinability relation as \stackrel \circ \stackrel, where \circ is the
composition of relations In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
. Joinability is usually denoted, somewhat confusingly, also with \downarrow, but in this notation the down arrow is a binary relation, i.e. we write x\mathrely if ''x'' and ''y'' are joinable.


The Church–Rosser property and notions of confluence

An ARS is said to possess the ''Church–Rosser property'' if and only if x \stackrel y implies x\mathrely for all objects ''x'', ''y''. Equivalently, the Church–Rosser property means that the reflexive transitive symmetric closure is contained in the joinability relation.
Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is bes ...
and
J. Barkley Rosser John Barkley Rosser Sr. (December 6, 1907 – September 5, 1989) was an American logician, a student of Alonzo Church, and known for his part in the Church–Rosser theorem in lambda calculus. He also developed what is now called the " Rosser ...
proved in 1936 that
lambda calculus In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using var ...
has this property; hence the name of the property. 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. Various properties, simpler than Church–Rosser, are equivalent to it. The existence of these equivalent properties allows one to prove that a system is Church–Rosser with less work. Furthermore, the notions of confluence can be defined as properties of a particular object, something that's not possible for Church–Rosser. An ARS (A,\rightarrow) is said to be, * ''confluent'' if and only if for all ''w'', ''x'', and ''y'' in ''A'', x \stackrel w \stackrel y implies x\mathrely. 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. * ''semi-confluent'' if and only if for all ''w'', ''x'', and ''y'' in ''A'', x \leftarrow w \stackrel y implies x\mathrely. This differs from confluence by the single step reduction from ''w'' to ''x''. * ''locally confluent'' if and only if for all ''w'', ''x'', and ''y'' in ''A'', x \leftarrow w \rightarrow y implies x\mathrely. This property is sometimes called ''weak confluence''. Theorem. For an ARS the following three conditions are equivalent: (i) it has the Church–Rosser property, (ii) it is confluent, (iii) it is semi-confluent. Corollary. In a confluent ARS if x \stackrel y then * If both ''x'' and ''y'' are normal forms, then ''x'' = ''y''. * If ''y'' is a normal form, then x \stackrel y. Because of these equivalences, a fair bit of variation in definitions is encountered in the literature. For instance, in Terese 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, one may define a normal form ''y'' of ''x'' as an irreducible ''y'' with the property that x \stackrel y. This definition, found in Book and Otto, is equivalent to the common one given here in a confluent system, but it is more inclusive 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. The typical counterexample is \, which is locally confluent but not confluent (cf. picture).


Termination and convergence

An abstract rewriting system is said to be terminating or ''noetherian'' if there is no infinite chain x_0 \rightarrow x_1 \rightarrow x_2 \rightarrow \cdots. (This is just saying that the rewriting relation is a Noetherian relation.) 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 a \rightarrow b \rightarrow a \rightarrow b \rightarrow \cdots, even though the system is normalizing. A confluent and terminating ARS is called canonical, or convergent. In a convergent ARS, every object has a unique normal form. 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. Theorem (
Newman's lemma In theoretical computer science, specifically in term rewriting, Newman's lemma, also commonly called the diamond lemma, is a criterion to prove that an abstract rewriting system is Confluence (abstract rewriting), confluent. It states that Confluen ...
): A terminating ARS is confluent if and only if it is locally confluent. The original 1942 proof of this result by Newman was rather complicated. It wasn't until 1980 that Huet published a much simpler proof exploiting the fact that when \rightarrow is terminating we can apply well-founded induction.


See also

*
Word problem (mathematics) In computational mathematics, a word problem is the decision problem, problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word problem for groups, but there ...
— particularly the section on abstract rewriting systems


Notes


References

* A textbook suitable for undergraduates. *
Nachum Dershowitz Nachum Dershowitz () is an Israeli computer scientist, known e.g. for the Dershowitz–Manna ordering and the Path_ordering_(term_rewriting), multiset path ordering used to prove Rewriting#Termination, termination of term rewrite systems. Educat ...
and Jean-Pierre Jouannaudbr>''Rewrite Systems''
Chapter 6 in
Jan van Leeuwen Jan van Leeuwen (born 17 December 1946 in Waddinxveen) is a Dutch computer scientist and emeritus professor of computer science at the Department of Information and Computing Sciences at Utrecht University.
(Ed.), ''Handbook of Theoretical Computer Science, Volume B: Formal Models and Semantics'', Elsevier and MIT Press, 1990, , pp. 243–320. The
preprint In academic publishing, a preprint is a version of a scholarly or scientific paper that precedes formal peer review and publication in a peer-reviewed scholarly or scientific journal. The preprint may be available, often as a non-typeset versi ...
of this chapter is freely available from the authors, but it misses the figures. * * This is a comprehensive monograph. It uses, however, a fair deal of notations and definitions not commonly encountered elsewhere. For instance the Church–Rosser property is defined to be identical with confluence. * Abstract rewriting from the practical perspective of solving problems in equational logic. *
Gérard Huet Gérard Pierre Huet (; born 7 July 1947) is a French computer scientist, linguist and mathematician. He is senior research director at INRIA and mostly known for his major and seminal contributions to type theory, programming language theory and ...
, ''Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems'', Journal of the ACM ( JACM), October 1980, Volume 27, Issue 4, pp. 797–821. Huet's paper established many of the modern concepts, results and notations. * Sinyor, J.
"The 3x+1 Problem as a String Rewriting System"
''International Journal of Mathematics and Mathematical Sciences'', Volume 2010 (2010), Article ID 458563, 6 pages. * * {{cite journal , last1=Church , first1=Alonzo , last2=Rosser , first2=J. B. , title=Some Properties of Conversion , journal=Transactions of the American Mathematical Society , date=1936 , volume=39 , issue=3 , pages=472–482 , doi=10.2307/1989762 , jstor=1989762 , issn=0002-9947, doi-access=free Formal languages Logic in computer science Rewriting systems