Modular stratification and magic sets for Datalog programs with negation

A class of “modularly stratified” logic programs is defined. Modular stratification generalizes stratification and local stratification, while allowing programs that are not expressible as stratified programs. For modularly stratified programs, the well-founded semantics coincides with the stable model semantics and makes every ground literal true or false. Modularly stratified programs are weakly stratified, but the converse is false. Unlike some weakly stratified programs, modularly stratified programs can be evaluated in a subgoal-at-a time fashion. An extension of top-down methods with memoing that handles this broader class of programs is presented. A technique for rewriting a modularly stratified program for bottom-up evaluation is demonstrated and extended to include magic-set techniques. The rewritten program, when evaluated bottom-up, gives correct answers according to the well-founded semantics, but much more efficiently than computing the complete well-founded model. A one-to-one correspondence between steps of the extended top-down method and steps during the bottom-up evaluation of the magic-rewritten program is exhibited, demonstrating that the complexity of the two methods is the same. Extensions of modular stratification to other operators such as set-grouping and aggregation, which have traditionally been stratified to prevent semantic difficulties, are discussed.