We present a concurrent Completion procedure based 
on the use of SOUR graphs as data structures. The 
procedure has the following characteristics. It is 
asynchronous, there is no need of a global memory 
or global control, equations are stored in a SOUR
graph with maximal structure sharing, and each vertex 
is a process, representing a term.  Therefore, the 
parallelism is at the term level.  Each edge is a 
communication link, representing a (subterm, ordering,
unification or rewrite) relation between terms.  
Completion is performed on the graph as local graph 
transformations by cooperation between processes. 
We  show that this concurrent Completion procedure is 
correct and complete with respect to the sequential one, 
provided that the information is locally time stamped 
in order to detect out of date information.