Completing open logic programs by constructive induction

We consider part of the problem of schema-biased inductive synthesis of recursive logic programs from incomplete speciications, such as clausal evidence (for instance, but not necessarily , ground positive and negative examples). After synthesizing the base clause and introducing recursive call(s) to the recursive clause, it remains to combine the overall result from the partial results obtained through recursion, so as to complete the recursive clause. Evidence for this combination relation can be abduced from the initially given evidence for the top-level relation. A program for this combination relation can be anything, from a single clause performing a uniication (such as for lastElem) to multiple guarded clauses performing uniications (such as for ltering programs) to recursive programs (such as for naive reverse). Existing methods cannot induce guarded clause programs for this combination relation from the abduced evidence. Some existing methods cannot even detect that the combination program itself may have to be recursive and thus they then do not recursively invoke themselves the overall recursive program synthesizer. We introduce our Program Completion Method as a suitable extension and generalization of the existing methods.