Approximations and Refinements of Certain Answers via Many-Valued Logics

Computing certain answers is the preferred way of answering queries in scenarios involving incomplete data. This, however, is computationally expensive, so practical systems use efficient techniques based on a particular three-valued logic, even though this often leads to incorrect results. Our goal is to provide a general many-valued framework for correctly approximating certain answers. We do so by defining the semantics of many-valued answers and queries, following the principle that additional knowledge about the input must translate into additional knowledge about the output. This framework lets us compare query outputs and evaluation procedures in terms of their informativeness. For each many-valued logic with a knowledge ordering on its truth values, one can build a syntactic evaluation procedure for all first-order queries, that correctly approximates certain answers; additional truth values are used to refine information about certain answers. For concrete examples, we show that a recently proposed approach fixing some of the inconsistencies of SQL query evaluation is an immediate consequence of our framework, and we further refine it by adding a fourth truth value. We show that no evaluation procedure based on Boolean logic delivers correctness guarantees. Finally, we study the relative power of evaluation procedures based on the informativeness of the answers they produce.