On the representation and querying of sets of possible worlds

Reviewer: Felipe Carino

Representations of the set of possible worlds for querying information databases that have both possible and certain facts are covered in this interesting paper. The paper is a terse exegesis of the data complexity for querying databases with null values relative to the authors representations. The representations range from the simple Codd table to more complex views, for which the authors derived upper and lower bounds on the data complexity for testing containment, membership, and uniqueness for sets of possible worlds. The paper defines representations consisting of: Codd table (no duplicate values) i-table (having conjunction of inequalities) e-table (having conjunction of equalities) g-table (global table that is an e-table with inequalities) c-table (g-table with an extra column of local conditions of equalities or inequalities attached to a tuple) instance of the set of possible worlds represented by a c-table (results from instantiating the variables with constants and satisfying the conditions) view of a set of possible worlds (obtained by applying a given query program to every possible world described by a c-table) Containment, membership, and uniqueness answer the following questions, respectively: “Is a set of possible worlds a subset of another set of possible worlds ” “Is a complete information database one of a set of possible worlds ” and “Is a set of possible worlds the same as a complete information database ” The authors contributions and results concerning containment are described using a summary table that appears in the paper. The vertical dimension is the set of worlds tested for containment in the set of worlds of the horizontal dimension. The standard complexity classes polynomial time (P), NP, co- NP, and P p 2 are used. This excellent and well-refereed paper clearly explains previous work, its results, and limitations. The examples support the authors points, and the algorithms are easy to follow.