# Confluent rewriting system

Template:Rewriting system property

## Contents

## Definition

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

### Stronger properties

### Weaker properties

## Metaproperties

### Dependence only on reduction graph

*This property of a rewriting system depends only on the reduction graph associated with the rewriting system, viz it can be reduces to testing the reduction graph for a directed graph property*

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.

### Free product-closedness

*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.