Inverting Implication

All generalisations within logic involve inverting implication. Yet, ever since Plotkin's work in the early 1970's methods of generalising rst-order clauses have involved inverting the clausal subsumption relationship. However , even Plotkin realised that this approach was incomplete. Since inversion of subsumption is central to many Inductive Logic Programming approaches, this form of incompleteness has been propagated to techniques such as Inverse Resolution and Relative Least General Generalisation. A more complete approach to inverting implication has been attempted with some success recently by Lapointe and Matwin. In the present paper the author derives general solutions to this problem from rst principles. It is shown that clausal subsumption is only incomplete for self-recursive clauses. Avoiding this incompleteness involves algorithms which nd \nth roots" of clauses. Completeness and correctness results are proved for a non-deterministic algorithms which constructs nth roots of clauses. It is shown how this algorithm can be used to invert implication in the presence of background knowledge. In conclusion the relationship between these results and Hoare's logical deenition of programming from speciications is discussed.