Improving the efficiency of AC matching and unification

This report consists of three independent parts that each study important steps in the matching and unification process for AC theories. In the first part we consider the problem of AC matching where there is just a single (variadic) AC symbol and no free function symbols in the pattern and subject. We show that even this restricted problem is NP-complete. We give some search methods and empirical results. In the second part we consider the full AC matching problem where there is no restriction on AC and free functions symbols allowed in the pattern and subject. Our approach is to build a hierarchy of bipartite graph matching problems which encodes all the possible solutions of subproblems. Certain sets of solutions to the graph problems are then used to construct simplified AC systems which are solved by a constrained search. In the final part we focus on one of the computationally intensive steps in current AC unification algorithms : the extraction of potential unifiers from a diophantine basis. We show that certain sub-problems are NP-complete and derive a new search algorithm which is shown to be at worst equivalent to the best published algorithm and which is potentially much better