SOUR Graphs for E cient Completion

We introduce a data structure called SOUR Graphs and present an eecient Knuth-Bendix completion procedure based on it. SOUR Graphs allow for a maximal structure sharing of terms in rewriting systems. The term representation is a dag representation, except that edges are labelled with equational constraints and variable renamings. The rewrite rules correspond to rewrite edges, the uniication problems to uniication edges. The Critical Pair and Simpliication inferences are recognized as patterns in the graph and are performed as local graph transformations. Our algorithm avoids duplicating term structure while performing inferences, which causes exponential behavior in the standard procedure. This approach gives a basis to design other completion algorithms, such as goal-oriented completion, concurrent completion and group completion procedures.