Proof-tree transformation theorems and their applications

For certain sets of logical rules, one can demonstrate that for every proof tree there is another tree proving the same fact and having a special form. One technique for detecting such opportunities is to reduce the question to one of conjunctive-query containment. A more powerful technique is to test whether one conunctive query is contained in the infinite union of conjunctive queries formed by expanding a set of recursive rules. We discuss two applications of these techniques. First, we give tests for commutativity of linear rules. When linear rules commute, we can reduce the complexity of 'counting' methods for query evaluation rom exponential to polynomial; commutativity also implies separability in the sense of Naughton. A second application is the discovery of linear rules that are equivalent to given nonlinear rules.