Confluent rewriting system
A rewriting system is said to be confluent if it satisfies the following condition: whenever and are multi-step reductions in the rewriting system, then there exists a word such that there exist multi-step reductions and .
In other words, any two things from the same source finally get together again.
The term confluent rewriting system can also be used for a rewriting system for a group. Note that the free group rewriting system is confluent. A group that possesses a confluent rewriting system is termed a confluent group.
Relation with other properties
Dependence only on reduction graph
Whether a rewriting system is confluent or not, can be reduced to checking a property of the associated reduction graph )assuming we remove the identity rewrite.
This property of a rewriting system is free product-closed. In other words, if we have two rewriting systems satisfying the property, the natural free product of these rewriting systems also satisfies the property
A free product of confluent rewriting systems is confluent. This is essentially because reductions in the various free factors do not interfere with one another, and hence commute.