Double - Exponential Complexity

A new algorithm for computing a complete set of uniiers for two terms involving associative-commutative function symbols is presented. The algorithm is based on a non-deterministic algorithm given by the authors in 1986 to show the NP-completeness of associative-commutative uniiability. The algorithm is easy to understand, its termination can be easily established. More importantly, its complexity can be easily analyzed and is shown to be doubly exponential in the size of the input terms. The analysis also shows that there is a double-exponential upper bound on the size of a complete set of uniiers of two input terms. Since there is a family of simple associative-commutative uniication problems which have complete sets of uniiers whose size is doubly exponential, the algorithm is optimal in its order of complexity in this sense. This is the rst associative-commutative unii-cation algorithm whose complexity has been completely analyzed. The approach can also be used to show a single exponential complexity for computing a complete set of uniiers for terms involving associative-commutative function symbols which also have the identity. Furthermore, for uniication in the presence of associative-commutative-idempotent operators we get a doubly exponential bound.