An Efficient ATMS for Equivalence Relations

We introduce a specialized ATMS for efficiently computing equivalence relations in multiple contexts. This specialized ATMS overcomes the problems with existing solutions to reasoning with equivalence relations. The most direct implementation of an equivalence relation in the ATMS-encoding the reflexive, transitive and symmetric rules in the consumer architectureproduces redundant equality derivations and requires O(n3) label update attempts (where n is the number of terms in an equivalence class). An alternative implementation is one that employs simple equivalence classes. However, this solution is unacceptable, since the number of classes grows exponentially with the number of distinct assumptions. The specialized ATMS presented here produces no redundant equality derivations, requires only O(n2) label update attempts, and is most efficient when there are many distinct assumptions. This is achieved by exploiting a special relationship that holds among the labels of the equality assertions because of transitivity. The standard dependency structure construction and traversal is replaced by a single pass over each label in a weaker kind of equivalence class. The specialized ATMS has been implemented as part of the logic programming language FORLOG.