Confluence (abstract rewriting)
In computer science, confluence is a property of rewriting systems, describing that terms in this system can be rewritten in more than one way, to yield the same result. This article describes the properties in the most abstract setting of an abstract rewriting system.
Motivating example 
Consider the rules of regular arithmetic. We can think of these rules as forming a rewriting system. Suppose we are given the expression
We can rewrite this expression in two ways—either simplifying the first bracket, or the second. Simplifying the first bracket, we have
Simplifying the second, gives
We get the same result from rewriting the term in two different ways. This suggests that the rewriting system formed from regular arithmetic is a confluent rewriting system.
General case and theory 
||This article may be confusing or unclear to readers. (August 2009)|
A rewriting system can be expressed as a directed graph in which nodes represent expressions and edges represent rewrites. So, for example, if the expression can be rewritten into , then we say that is a reduct of (alternatively, reduces to , or is an expansion of ). This is represented using arrow notation; indicates that reduces to . Intuitively, this means that the corresponding graph has a directed edge from to .
If there is a path between two graph nodes (let's call them and ), then the intermediate nodes form a reduction sequence. So, for instance, if , then we can write , indicating the existence of a reduction sequence from to .
With this established, confluence can be defined as follows. Let a, b, c ∈ S, with a →* b and a →* c. a is deemed confluent if there exists a d ∈ S with b →* d and c →* d. If every a ∈ S is confluent, we say that → is confluent, or has the Church-Rosser property. This property is also sometimes called the diamond property, after the shape of the diagram shown on the right. Caution: other presentations reserve the term diamond property for a variant of the diagram with single reductions everywhere; that is, whenever a → b and a → c, there must exist a d such that b → d and c → d. The single-reduction variant is strictly stronger than the multi-reduction one.
Local confluence 
An element a ∈ S is said to be locally (or weakly) confluent if for all b, c ∈ S with a → b and a → c there exists d ∈ S with b →* d and c →* d. If every a ∈ S is locally confluent, we say that → is locally (or weakly) confluent, or has the weak Church-Rosser property. This is different from confluence in that b and c must be reduced from a in one step. In analogy with this, confluence is sometimes referred to as global confluence.
We may view →*, which we introduced as a notation for reduction sequences, as a rewriting system in its own right, whose relation is the transitive closure of R. Since a sequence of reduction sequences is again a reduction sequence (or, equivalently, since forming the transitive closure is idempotent), →** = →*. It follows that → is confluent if and only if →* is locally confluent.
A rewriting system may be locally confluent without being globally confluent. However, Newman's lemma states that if a locally confluent rewriting system has no infinite reduction sequences (in which case it is said to be terminating or strongly normalizing), then it is globally confluent.
The definition of local confluence differs from that of global confluence in that only elements reached from a given element in a single rewriting step are considered. By considering one element reached in a single step and another element reached by an arbitrary sequence, we arrive at the intermediate concept of semi-confluence: a ∈ S is said to be semi-confluent if for all b, c ∈ S with a → b and a →* c there exists d ∈ S with b →* d and c →* d; if every a ∈ S is semi-confluent, we say that → is semi-confluent.
A semi-confluent element need not be confluent, but a semi-confluent rewriting system is necessarily confluent.
Strong confluence 
Strong confluence is another variation on local confluence that allows us to conclude that a rewriting system is globally confluent. An element a ∈ S is said to be strongly confluent if for all b, c ∈ S with a → b and a → c there exists d ∈ S with b →* d and either c → d or c = d; if every a ∈ S is strongly confluent, we say that → is strongly confluent.
A strongly confluent element need not be confluent, but a strongly confluent rewriting system is necessarily confluent.
Examples of confluent systems 
- Reduction of polynomials modulo an ideal is a confluent rewrite system provided one works with a Gröbner basis.
- Matsumoto's theorem follows from confluence of the braid relations.
See also 
- Term Rewriting Systems, Terese, Cambridge Tracts in Theoretical Computer Science, 2003.
- Term Rewriting and All That, Franz Baader and Tobias Nipkow, Cambridge University Press, 1998