Caching and non-Horn inference in model elimination theorem provers

Caching in an inference procedure holds the promise of replacing exponential search with constant-time lookup, at a cost of slightly-increased overhead for each node expansion. Caching will be useful if subgoals are repeated often enough during proofs. In experiments on solving queries using a backward chainer on Horn theories, caching appears to be very helpful on average. When trying to extend this success to first-order theories, however, intuition suggests that subgoal caches are no longer useful. The cause is that complete first-order backward chaining requires goal-goal resolutions in addition to resolutions with the database, and this introduces a context-sensitivity into the proofs for a subgoal. A cache is only feasible if the solutions are independent of context, so that they may be copied from one part of the space to another. It is shown here that a full exploration of a subgoal in one context actually provides complete information about the solutions to the same subgoal in all other contexts of the proof. In a straightforward way, individual solutions from one context may be copied over directly. More importantly, non-Horn failure caching is also feasible, so no additional solutions in the new context (that might affect the query) are possible and therefore there is no need to re-explore the space in the new context. Thus most Horn clause caching schemes may be used with minimal changes in a non-Horn setting. In addition, a new Horn clause caching scheme is proposed: postponement caching. This new scheme involves exploring the inference space as a graph instead of as a tree, so that a given literal will only occur once in the proof space. Despite the previous extension of failure caching to non-Horn theories, postponement caching is incomplete in the non-Horn case. A counterexample is presented, and possible enhancements to reclaim completeness are investigated.