Chapter 29. Theory of Quantified Boolean Formulas

Quantified Boolean formulas (QBF) are a generalization of propositional formulas by allowing universal and existential quantifiers over variables. This enhancement makes QBF a concise and natural modeling language in which problems from many areas, such as planning, scheduling or verification, can often be encoded in a more compact way than with propositional formulas. We introduce in this chapter the syntax and semantics of QBF and present fundamental concepts. This includes normal form transformations and Q-resolution, an extension of the propositional resolution calculus. In addition, Boolean function models are introduced to describe the valuation of formulas and the behavior of the quantifiers. We also discuss the expressive power of QBF and provide an overview of important complexity results. These illustrate that the greater capabilities of QBF lead to more complex problems, which makes it interesting to consider suitable subclasses of QBF. In particular, we give a detailed look at quantified Horn formulas (QHORN) and quantified 2-CNF (Q2-CNF).

Knowledge Refinement via Rule Selection

In several different applications, including data transformation and entity resolution, rules are used to capture aspects of knowledge about the application at hand. Often, a large set of such rules is generated automatically or semi-automatically, and the challenge is to refine the encapsulated knowledge by selecting a subset of rules based on the expected operational behavior of the rules on available data. In this paper, we carry out a systematic complexity-theoretic investigation of the following rule selection problem: given a set of rules specified by Horn formulas, and a pair of an input database and an output database, find a subset of the rules that minimizes the total error, that is, the number of false positive and false negative errors arising from the selected rules. We first establish computational hardness results for the decision problems underlying this minimization problem, as well as upper and lower bounds for its approximability. We then investigate a bi-objective optimization version of the rule selection problem in which both the total error and the size of the selected rules are taken into account. We show that testing for membership in the Pareto front of this bi-objective optimization problem is DP-complete. Finally, we show that a similar DP-completeness result holds for a bi-level optimization version of the rule selection problem, where one minimizes first the total error and then the size.