A related, but weaker notion than the existence of normal forms is that of two objects being joinable: x and y are said to be joinable if there exists some z with the property that x \stackrel{*}{\rightarrow} z \stackrel{*}{\leftarrow} y.
One of the important problems that may be formulated in an ARS is the word problem: given x and y, are they equivalent under \stackrel{*}{\leftrightarrow}?