Critical Pair
Let 
and 
be two rules of a term rewriting system, and suppose these rules have no variables in common. If they do, rename the variables. If 
is a subterm of 
(or the term 
itself) such that it is not a variable, and the pair 
is unifiable with the most general unifier 
, then 
and the result of replacing 
in 
by 
are called a critical pair.
The fact that all critical pairs of a term rewriting system are joinable, i.e., can be reduced to the same expression, implies that the system is locally confluent.
For instance, if 
and 
, then 
and 
would form a critical pair because they can both be derived from 
.
Note that it is possible for a critical pair to be produced by one rule, used in two different ways. For instance, in the string rewrite "AA" -> "B", the critical pair ("BA", "AB") results from applying the one rule to "AAA" in two different ways.
REFERENCES
Baader, F. and Nipkow, T. Term Rewriting and All That. Cambridge, England: Cambridge University Press, 1999.
Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1037, 2002.
