Complexity of matching problems

The associative-commutative matching problem is shown to be NP-complete; more precisely, the matching problem for terms in which some function symbols are uninterpreted and others are both associative and commutative, is NP-complete. It turns out that the similar problems of associative-matching and commutative-matching are also NP-complete. However, if every variable appears at most once in a term being matched, then the associative-commutative matching problem is shown to have an upper-bound of O ( | s | * | t |3), where | s | and | t | are respectively the sizes of the pattern 8 and the subject t